FUCHSIAN GROUPS, QUASICONFORMAL GROUPS, CONICAL LIMIT SETS

Peter W. Jones

Lesley Ward

Abstract We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.

1. Introduction

A classical result of Hopf says that the normalized one-dimensional Lebesgue measure of the conical limit set of any finitely generated Fuchsian group is either zero or one. This has been extended to infinitely generated Fuchsian groups, to conformal and to uniformly quasiconformal groups acting discontinuously on the unit ball in (real) dimension three or higher, and to discrete groups of complex hyperbolic isometries of the unit ball in several complex variables. In this article it is shown that this dichotomy does not hold for uniformly quasiconformal groups acting on the unit disc in the complex plane. For each λ with 0 ≤ λ ≤ 1 we explicitly construct a quasiconformal group, acting on the disc, whose conical limit set has normalized one-dimensional Lebesgue measure equal to λ. The main step is to construct a doubling measure supported on the conical limit set of a certain Fuchsian group, where this conical limit set has Lebesgue measure zero.

A group G of homeomorphisms of the unit disc, in dimension two, or of the unit ball, in dimension greater than two, is said to act discontinuously if each point in the disc or ball has a neighbourhood U such that only finitely many of the images g(U), g ∈ G, intersect U. The limit set L(G) of such a group is the set of accumulation points of the orbit of the origin under the action of G. The same limit set is obtained using any point in the disc in place of the origin. The discontinuity of G implies that L(G) is a subset of the unit circle or sphere. A point x ∈ L(G) is a conical limit point of G if there is a sequence of orbit points which converges to x inside a Euclidean cone with vertex at x, axis perpendicular to the circle or sphere, and opening angle less than π/2, so that the sides of the cone are not tangent to the circle or sphere. The conical limit set L_{c}(G) of G is the set of conical limit points of G.

Jones was partially supported by NSF grant #DMS-92-13595. Ward’s research at MSRI was supported in part by NSF grant #DMS-90-22140.

A Fuchsian group is a group of Möbius transformations which acts discontinuously on the unit disc ID in the complex plane. For simplicity we assume that the group has no elliptic elements. Hopf [H] showed that the conical limit set of a finitely generated Fuchsian group has normalized Lebesgue measure zero or one. This depends on whether the Poincaré series for the group converges or diverges at the exponent 1:

|L_{c}(G)| = \{ \begin{matrix}0, & if ∑_{g∈G}(1 − |g(0)|) < ∞; \\ 1, & if ∑_{g∈G}(1 − |g(0)|) = ∞; \end{matrix}  (1.1)

the result also holds for infinitely generated Fuchsian groups.

For Fuchsian groups, and for groups of Möbius transformations acting discontinuously on the n-dimensional ball when n ≥ 3, there are several dichotomies which are equivalent to the zero-one dichotomy for the normalized (n−1)-dimensional measure of the conical limit set. The equivalence of the following conditions was proved by Sullivan [S1, S2]:

1) |L_{c}(G)| = 0;

2) The Poincaré series converges at the exponent n − 1: ∑_{g∈G}(1 − |g(0)|)^{n−1} < ∞;

3) Green’s function for the Laplace-Beltrami operator exists for the hyperbolic manifold IB^{n}/G; and

4) the geodesic flow on IB^{n}/G is transient. Otherwise, the conical limit set has full Lebesgue measure in the (n − 1)-sphere; the Poincaré series diverges at the exponent n−1; there is no Green’s function for IB^{n}/G (that is, the integral of the heat kernel for IB^{n}/G does not converge); and the geodesic flow on IB^{n}/G is recurrent and completely ergodic. See Nicholls’ book [N2] for further information. Analogous results hold in several complex variables [Kam1, Kam2, MW].

A homeomorphism φ : Ω → Ω′ of complex domains is called K-quasiconformal if it is in the Sobolev class W_{loc}^{1,2} and its directional derivatives satisfy

max_{α} |∂_{α}φ(z)| ≤ K min_{α} |∂_{α}φ(z)|  (1.2)

for almost every z ∈ Ω. Geometrically, this means that for almost every z ∈ Ω, infinitesimal circles centred at z are mapped by φ to infinitesimal ellipses centred at φ(z), whose eccentricities are uniformly bounded below by 1/K. A quasiconformal group is a group of K-quasiconformal maps acting discontinuously on the disc, for some fixed K ≥ 1.

One can define quasiconformal mappings in dimension n ≥ 3 by generalizing the geometric definition above, using spheres and ellipsoids instead of circles and ellipses. Garnett, Gehring, and Jones [GGJ], and independently Tukia [T3], showed that results analogous to Hopf’s on the measure of the conical limit set and the convergence of the Poincaré series at the exponent n−1 hold for uniformly quasiconformal groups when the dimension is at least three: If G is a group of K-quasiconformal mappings acting discontinuously on the n-dimensional unit ball, where n ≥ 3, then

|L_{c}(G)| = \{ \begin{matrix}0, & if ∑_{g∈G}(1 − |g(0)|)^{n−1} < ∞; \\ 1, & if ∑_{g∈G}(1 − |g(0)|)^{n−1} = ∞. \end{matrix}  (1.3)

In this article we show that, in sharp contrast to the higher dimensional case, there is no zero-one dichotomy for the normalized Lebesgue measure of the conical limit set of a quasiconformal group acting on the two-dimensional unit disc.

Theorem 1.1. For each number λ ∈ [0, 1], there is a quasiconformal group Γ acting on the unit disc such that the normalized one-dimensional Lebesgue measure of the conical limit set of Γ is λ.

A doubling measure μ on the circle is a positive measure satisfying μ(Ĩ) ≤ c μ(I), for some constant c > 0, whenever I is an arc of the circle and Ĩ is the arc with the same centre and twice the length. Equivalently, μ(I) ≤ c′μ(J) for some uniform constant c′ > 0 whenever I and J are adjacent arcs of equal length.

Theorem 1.1 is a consequence of the following result:

Theorem 1.2. There exist a Fuchsian group G and a doubling measure μ such that the conical limit set of G has one-dimensional Lebesgue measure zero, and μ is supported on the conical limit set of G.

Our result is perhaps rather surprising given that Sullivan [S2] and independently Tukia [T1] have proven that every K-quasiconformal group acting on the disc is conjugate by a quasiconformal mapping to a Möbius group. The corresponding statement is false in higher dimensions [T2]. On the other hand, boundary values of quasiconformal homeomorphisms of the disc (in other words, quasisymmetric maps) have much less regularity than is true in higher dimensions. If φ : IB^{n} → IB^{n} is K-quasiconformal and n ≥ 3, the restriction of φ to ∂IB^{n} is also K-quasiconformal and hence lies in the Sobolev space W^{1,n−1}(∂IB^{n}). In particular φ|_{∂IB^{n}} takes sets of (n − 1)-dimensional Lebesgue measure zero to sets of (n − 1)-dimensional Lebesgue measure zero. Quasisymmetric mappings do not have this property. It is this lack of regularity for quasisymmetric mappings that we exploit to construct our examples.

Theorem 1.1 is an easy consequence of Theorem 1.2. This argument, carried out in Section 3, uses little more than the Beurling-Ahlfors theorem: A homeomorphism f of the circle is the restriction of a quasiconformal mapping of the disc to itself if and only if the distributional derivative \frac{∂f}{∂θ} on the circle is a doubling measure.

Theorem 1.2 gives some information about the possible geometry of limit sets and of sets which support doubling measures. Roughly speaking, the support of a doubling measure must be rather evenly distributed. We give a Fuchsian group whose conical limit set is evenly enough distributed to support a doubling measure, even though it has zero Lebesgue measure. (In fact, the Hausdorff dimension of this conical limit set is strictly less than one.) We construct a doubling measure that is tailored so its support is in this small set. Such constructions were first carried out by Kahane [K], and our method follows his philosophy, though the details are necessarily more complicated.

As José Fernández has pointed out to us, the Patterson-Sullivan measure associated to the group G is a natural candidate for the doubling measure of Theorem 1.2. Unfortunately, certain technical difficulties are encountered in this approach, and we have not yet understood how to overcome them.

It is interesting to note that the result of Theorem 1.2 cannot be achieved if the quasisymmetric map associated to the doubling measure μ is in the Teichmüller space of the group G, or if G is finitely generated. The first statement follows from the fact that the existence of Green’s function is a quasiconformal invariant. If G is finitely generated and of the first kind, then L_{c}(G) has full measure in the unit circle. If G is finitely generated and of the second kind, then ∂ID \ L_{c}(G) contains an arc, and so L_{c}(G) cannot support a doubling measure.

Section 4 contains a rather full outline and summary of our proof of Theorem 1.2. The rest of the paper is organized as follows.

In Section 2 we give some definitions and notation. In Section 3, we prove Theorem 1.1 as a corollary of Theorem 1.2. In Section 5 we establish some basic properties of the Fuchsian group G used in our proof of Theorem 1.2. In Section 6, we prove that a certain simple construction yields doubling measures which are supported on small sets. Sections 7 to 11 contain the construction of our doubling measure, up to the specification of some parameters. In Section 7 we fix a set of fundamental domains in the universal cover ID of the Riemann surface ID/G. In Section 8 we define a collection of Whitney intervals in the Riemann surface, and establish some properties of its lift to the universal cover. In Section 9 we construct a grid of intervals in the circle, on which our doubling measure will be built. Section 10 contains estimates which control the geometric distortion caused by the covering map. In Section 11 we define a sequence of density functions which, by the lemma in Section 6, yields a doubling measure. In Section 12 we define some auxiliary functions and outline the rest of the proof. Sections 13 and 14 contain estimates of the expectation and second moment, with respect to the doubling measure, of the auxiliary functions. In Section 15, an argument involving a constrained random walk on the Riemann surface, together with the estimates of the previous two sections, proves that the measure is supported on the conical limit set of G.

It is a pleasure for the second author to acknowledge the support of MSRI.

2. Definitions and Notation The conical limit set is also known as the radial or non-tangential limit set. Hed-

lund [He] introduced conical limit points in connection with his study of horocyclic transitive points. The conical limit set may be characterized in terms of spherical caps. A spherical cap on a point x in the unit disc ID is an arc of the unit circle ∂ID of the form

Cap (x, a) = {y ∈ ∂ID | |y − x| ≤ a (1 − |x|)},  (2.1)

where 1 < a < ∞. A non-tangential cone on a point y ∈ ∂ID is a subset of ID of the form

Cone (y, b) = {x ∈ ID | |y − x| ≤ b (1 − |x|)},  (2.2)

where 1 < b < ∞. Clearly y ∈ Cap (x, a) if and only if x ∈ Cone (y, a). Non-tangential cones are comparable to the Euclidean cones discussed in the introduction, in the sense that each such Euclidean cone contains a non-tangential cone and is contained in a nontangential cone. It follows that y is the non-tangential limit of {g_{j}(0)}_{j=1}^{∞} if and only if there is an a such that y lies in infinitely many spherical caps Cap (g_{j}(0), a). Taking a countable union over wider and wider opening angles of the cones, the conical limit set is given by

L_{c}(G) = ⋃_{l=2}^{∞} ⋂_{k=1}^{∞} ⋃_{j=k}^{∞} Cap (g_{j}(0), l).  (2.3)

A set K in the complex plane is uniformly perfect if there is a constant c > 0 such that for each z_{0} ∈ K and for all r such that 0 < r < diam K,

K ∩ {z | cr ≤ |z − z_{0}| ≤ r} ≠ ∅.  (2.4)

(Here diam K is the diameter of K.) In other words, there is an upper bound on the moduli of annuli lying in the complement of K.

The Poincaré series for a discrete group G is the series

∑_{g∈G}(1 − |g(0)|)^{s},  (2.5)

where s is positive. There is a critical exponent δ = δ(G), called the exponent of convergence, such that the series converges for all s > δ and diverges for all s < δ.

Let Ω be a domain in \overline{C}, let E be a Borel subset of ∂Ω, and let z be a point in Ω. The harmonic measure at z of E in Ω is the Perron solution w(z) = w(z, E, Ω) of the Dirichlet problem in Ω for the boundary values 1_{E}. As a function of z, w(z, E, Ω) is harmonic in Ω. For fixed z, w(z, E, Ω) is a probability measure on ∂Ω; w(z, E, Ω) is the probability that a Brownian traveller from z first hits ∂Ω in the set E. Harmonic measure is monotonic in the domain Ω: if z ∈ Ω_{1} ⊂ Ω_{2} and E ⊂ ∂Ω_{1} ∩ ∂Ω_{2}, then w(z, E, Ω_{1}) ≤ w(z, E, Ω_{2}). It is also monotonic in E: if E_{1} ⊂ E_{2} ⊂ ∂Ω, then w(z, E_{1}, Ω) ≤ w(z, E_{2}, Ω).

The hyperbolic metric on the disc ID is given by the element of arclength ds = \frac{2}{1−|z|^{2}} |dz|. The Möbius transformations g : ID → ID are the isometries for this metric. They are of the form

g(z) = e^{iθ} \frac{z − z_{0}}{1 − \overline{z}_{0}z}  (2.6)

where θ ∈ [0, 2π] and z_{0} ∈ ID. The hyperbolic geodesics are the orthocircular arcs: that is, arcs of circles which meet the unit circle ∂ID at right angles. The hyperbolic distance from 0 to a ∈ ID is

d_{hyp}(0, a) = ∫_{0}^{|a|} \frac{2}{1 − |z|^{2}} |dz| = log [\frac{1 + |a|}{1 − |a|}] .  (2.7)

Let Ω be a domain in \overline{C} whose universal covering space is the disc, and let π : ID → Ω be the covering map. The hyperbolic metric on ID can be projected via π to a metric on Ω given by λ_{Ω}(w)|dw| = λ_{Ω}(π(z))|π′(z)||dz| = \frac{2}{1−|z|^{2}} |dz|, where w = π(z). The boundary ∂Ω of Ω is uniformly perfect if and only if there is a constant c_{Ω} > 0 such that the function λ_{Ω} satisfies

\frac{c_{Ω}}{dist (w, ∂Ω)} ≤ λ_{Ω}(w) ≤ \frac{2}{dist (w, ∂Ω)}.  (2.8)

See [BP, Po2]. Here dist denotes Euclidean distance.

In dimension two, there is a connection between quasiconformal maps and doubling measures. Let φ : C → C be a K-quasiconformal map which preserves the upper half plane. Then φ maps IR to itself, and the restriction f of φ to IR is an increasing homeomorphism which satisfies

\frac{1}{M} ≤ \frac{f(x + t) − f(x)}{f(x) − f(x − t)} ≤ M  (2.9)

with a constant M depending on K, for all x ∈ IR and all t > 0. Functions with this property are called M-quasisymmetric. Quasisymmetry is a necessary and sufficient condition for an increasing homeomorphism f of IR to be the boundary values of a quasiconformal mapping which preserves the upper half plane [BA]. Note that different quasiconformal mappings may have the same boundary values.

Let f : IR → IR be an increasing homeomorphism. Define a measure μ on IR by setting μ([a, b]) = f(b) − f(a) for intervals [a, b]. Then f is M-quasisymmetric if and only if μ satisfies 1/M ≤ μ(I)/μ(J) ≤ M whenever I and J are adjacent intervals of equal length. In other words, f is M-quasisymmetric if and only if μ is doubling.

To summarize, there is a many-to-one correspondence between quasiconformal selfmappings of the upper half plane and doubling measures on the real line. There is a similar correspondence between quasiconformal self-mappings of the disc and doubling measures on the circle.

Some references for the material above are [N1] and [L].

Two quantities A and B are comparable, denoted A ∼ B, if there is a constant c > 0 such that \frac{1}{c}B ≤ A ≤ c B. We also write A ∼^{c} B if A and B are comparable with constant c. Throughout the paper we normalize Lebesgue measure so that the unit n-sphere has measure one.

In the figures we denote the unit circle ∂ID by S^{1}.

3. Proof of Theorem 1.1 assuming Theorem 1.2

Fix a number λ with 0 ≤ λ ≤ 1. Let μ be the measure given by Theorem 1.2, and take c > 0 such that μ(2I) ≤ c μ(I) for all arcs I in the unit circle ∂ID. Set ν(·) = λμ(·) + (1 − λ)| · |, where | · | denotes normalized Lebesgue measure on ∂ID. Then ν is a doubling measure on the circle, since for each arc I in ∂ID,

\begin{aligned}ν(2I) & = λ μ(2I) + (1 − λ) |2I| \\ & ≤ c λ μ(I) + 2 (1 − λ) |I| \\ & ≤ max (c, 2) ν(I). \end{aligned}  (3.1)

The measure ν assigns the fraction λ of its mass to the conical limit set of G.

Define f : ∂ID → ∂ID so that ν([a, b]) = f(b) − f(a) for all arcs [a, b] in ∂ID. f is a quasisymmetric mapping of the circle to itself, and |f(E)| = ν(E) for any measurable subset E of the circle. The map f may be extended to a quasiconformal mapping φ of the unit disc onto itself [BA, DE] and further to a quasiconformal self-mapping of \overline{C}. Conjugating the Fuchsian group G of Theorem 1.2 by φ yields a quasiconformal group Γ = φ ∘ G ∘ φ^{−1} which acts on the disc.

It remains to show that the conical limit set of Γ is the image under φ of the conical limit set of the Fuchsian group G. This is a special case of a much more general fact; see [M] for example. We give a direct proof. It is sufficient to show that a quasiconformal map φ : ID → ID maps any non-tangential cone into another non-tangential cone, perhaps with larger opening angle. Since φ^{−1} is also quasiconformal, this implies that L_{c}(Γ) = φ(L_{c}(G)).

Define the cross ratio of any four distinct points α, β, γ, and δ in \overline{C} by

τ = |α, β, γ, δ| = \frac{|α − γ|}{|α − δ|} · \frac{|β − δ|}{|β − γ|},  (3.2)

with the convention that if δ = ∞,

|α, β, γ, ∞| = \frac{|α − γ|}{|β − γ|}.  (3.3)

Let φ : \overline{C} → \overline{C} be K-quasiconformal. Let φ(τ) = |φ(α), φ(β), φ(γ), φ(δ)|. Väisälä has shown [V] that there is an increasing function Φ_{K} : (0, ∞) → (0, ∞), depending only on K, such that φ(τ) ≤ Φ_{K}(τ) for all τ.

Let z be a point in Cone (y, a) = {z ∈ ID | |z − y| ≤ a (1 − |z|)}, where y is a point in ∂ID and 1 < a < ∞. Let w be the point in ∂ID such that 1 − |φ(z)| = |φ(w) − φ(z)|.

Suppose φ^{−1}(∞) = ∞. Then

\begin{aligned}\frac{|φ(y) − φ(z)|}{1 − |φ(z)|} & = \frac{|φ(y) − φ(z)|}{|φ(w) − φ(z)|} \\ & = |φ(y), φ(w), φ(z), ∞| \\ & ≤ Φ_{K}(|y, w, z, ∞|) \\ & = Φ_{K} (\frac{|y − z|}{|w − z|}) \\ & ≤ Φ_{K} (\frac{|y − z|}{1 − |z|}) \\ & ≤ Φ_{K}(a). \end{aligned}  (3.4)

Here the second last inequality holds because |w − z| ≥ 1 − |z| and Φ_{K} is increasing. Therefore φ(z) lies in Cone (φ(y), Φ_{K}(a)).

Now suppose φ^{−1}(∞) ≠ ∞. Let w and y be any two points in the unit circle. Since φ^{−1}(∞) is not in the closed unit disc, we have

|w − φ^{−1}(∞)| ≤ \frac{2 + d}{d} |y − φ^{−1}(∞)|  (3.5)

where d = dist (φ^{−1}(∞), \overline{ID}) > 0. Then

\begin{aligned}\frac{|φ(y) − φ(z)|}{1 − |φ(z)|} & = |φ(y), φ(w), φ(z), ∞| \\ & ≤ Φ_{K}(|y, w, z, φ^{−1}(∞)|) \\ & = Φ_{K} (\frac{|y − z|}{|w − z|} · \frac{|w − φ^{−1}(∞)|}{|y − φ^{−1}(∞)|}) \\ & ≤ Φ_{K} (\frac{|y − z|}{1 − |z|} · \frac{2 + d}{d}) \\ & ≤ Φ_{K}(a · \frac{2 + d}{d}). \end{aligned}  (3.6)

Hence φ(z) lies in Cone (φ(y), Φ_{K}(a · \frac{2+d}{d})).

It follows that L_{c}(Γ) = φ(L_{c}(G)), and |L_{c}(Γ)| = |φ(L_{c}(G))| = ν(L_{c}(G)) = λ. In other words Γ is a quasiconformal group, acting on the unit disc, whose conical limit set supports the fraction λ of the mass of Lebesgue measure on the unit circle. This completes the proof of Theorem 1.1.

4. Outline of the proof of Theorem 1.2

Theorem 1.2. There exist a Fuchsian group G and a doubling measure such that the conical limit set L_{c}(G) of G has one-dimensional Lebesgue measure zero, and the doubling measure is supported on L_{c}(G).

In this section we present an outline of the proof of Theorem 1.2. Let K ⊂ [0, 1] be the classical ternary Cantor set, and let Ω = \overline{C} \ K be the complement of K in the extended complex plane. Let G be the Fuchsian group, acting on the disc ID, which uniformizes Ω; in other words G is the covering group of Ω and ID/G is conformally equivalent to Ω. Let π : ID → Ω be the covering map, normalized so that π(0) = ∞. The conical limit set L_{c}(G) is the set of non-tangential accumulation points of the orbit {g(0)}_{g∈G}. For this group G, the conical limit set has measure zero. (See Section 5.) Our aim is to construct a doubling measure which is supported on L_{c}(G).

We begin with a model example of a doubling measure μ, obtained as the limit of a sequence of measures μ_{n} of the form dμ_{n} = F_{n} ⋯ F_{1} dx, such that μ is supported on a set of Lebesgue measure zero. This type of example is due to Kahane [K]. Divide the unit interval I_{0} into 5 equal subintervals I_{1,1}, …, I_{1,5}, 1 ≤ k ≤ 5, numbered from left to right, and let

F_{1}(x) = \{ \begin{matrix}1, & for x ∈ I_{1,1} ∪ I_{1,5}; \\ 1/2, & for x ∈ I_{1,2} ∪ I_{1,4}; and \\ 2, & for x ∈ I_{1,3}. \end{matrix}  (4.1)

Then . The large fraction 2/5 of the area under the graph of F_{1} lies above the middle fifth I_{1,3} of I_{0}. . F_{1} has mean value one on I_{0}; . F_{1} is a positive function whose values lie between 1/2 and 2; and . F_{1} ≡ 1 on subintervals of length |I_{0}|/5 at each end of I_{0}.

Let F_{n}(x) = F_{n−1}(5x mod 1), for n ≥ 2. On each of the 5^{n−1} subintervals I_{n−1,j} of I_{0} of length 5^{−n+1}|I_{0}|, F_{n} is a dilation of F_{1}, with the same properties on I_{n−1,j} as F_{1} has on I_{0}. We refer to the intervals I_{n,j} as 5-ary intervals. The effect of F_{n} is to concentrate the large fraction 2/5 of the mass assigned to each I_{n−1,j} onto the middle fifth of I_{n−1,j}.

The measure μ_{n} given by dμ_{n} = F_{n} ⋯ F_{1} dx satisfies 1/4 ≤ μ_{n}(I)/μ_{n}(J) ≤ 4 for all adjacent 5-ary intervals I and J of equal length. Therefore μ_{n} is doubling. The measures μ_{n} converge weakly to a unique limit μ, which is also doubling. The support of this measure is a set E of Hausdorff dimension less than one, concentrated near the centres of the 5-ary intervals which appear in the construction. (More precisely, this model set E consists of those points x in I_{0} such that the proportions of 0’s, 1’s, 2’s, 3’s, and 4’s in the base five expansion of x are asymptotically equal to 1/5, 1/10, 2/5, 1/10, and 1/5 respectively. Its Hausdorff dimension is 1 − \frac{1}{5} \frac{log 2}{log 5} ≈ 0.914.)

It is convenient to recast this construction in terms of a random walk on a tree. We use a 5-ary tree as a discrete model of the disc. Put a vertex at the origin, with five equally spaced edges emanating from it. Put a vertex at the end of each edge; call these the children of the original vertex. Add five more edges emanating from each new vertex.

Repeat, with the new edges always pointing outwards towards the circle. Each branch of the tree corresponds to a point of the circle; one can think of the tip of the branch landing somewhere on the circle.

We distinguish a proper subset V of vertices of the tree. Each vertex v in the tree has five children, v_{1}, …, v_{5}; let V consist of all middle children v_{3}.

Consider a particle performing a random walk on the vertices of the tree, with the following constraints:

. the particle may jump from one vertex to another if and only if the second is a child of the first: in other words they are connected by an edge and the second vertex is below the first (i.e. closer to the circle); and . when the particle is at the vertex v, it jumps to the children v_{k}, 1 ≤ k ≤ 5, of v with jump probabilities p_{k}, 1 ≤ k ≤ 5, respectively, where p_{1} = p_{5} = 1/5, p_{2} = p_{4} = 1/10, and p_{3} = 2/5.

The particle always moves away from the origin, towards the tips of the branches.

We recover the 5-ary intervals by projecting the tree radially outwards onto the circle, so that each vertex corresponds to an interval in the circle. (The construction of the measure μ given above can be done on the unit circle instead of the unit interval.) Let the origin correspond to the whole circle. Project the five vertices which are the children of the origin to five intervals of equal length, which together cover the circle. And so on: for each vertex v, project the children of v to five equal subintervals of the interval corresponding to v. The collection of all these intervals forms a 5-adic grid of nested intervals in the circle; we refer to these intervals as grid intervals. We say a grid interval is in the n^{th} layer of the grid if it is the projection of a vertex which is separated from the origin by n edges in the tree.

Each path which the particle may take, given by a sequence of vertices v_{1} → v_{2} → ⋯ in the tree, corresponds to a point x in the circle, and also to a nested, decreasing sequence of grid intervals I_{1} ⊃ I_{2} ⊃ ⋯ ∋ x in the circle.

Clearly the jump probabilities {p_{k}} and the functions {F_{n}} defined on the grid intervals encode the same information. We can recover the functions {F_{n}} as follows. Let v be a vertex which is n − 1 edges away from the origin. Let v_{k}, 1 ≤ k ≤ 5, be the children of v, let p_{k} be the probability of jumping from v to v_{k}, and let I and I_{k} be the grid intervals which are the projections of v and v_{k} respectively. Then, on I, F_{n} is simply the function which is constant on each I_{k} and satisfies

p_{k} = Prob (v → v_{k}) = \frac{∫_{I_{k}} F_{n} dx}{∫_{I} F_{n} dx}  (4.2)

for 1 ≤ k ≤ 5.

In this setting, the measure μ which is the limit of the μ_{n}’s defined by dμ_{n} = F_{n} ⋯ F_{1} dx is the hitting measure for the random walk on the tree: the measure of a subset A of the circle is exactly the probability that a particle which starts from the origin will hit the circle somewhere in A. Note that the support of μ contains those points in the circle ∂ID which correspond to paths in the tree which pass through infinitely many vertices from V.

As noted above, the support of μ is a small set E which is distributed in a very regular way. If we had a Fuchsian group whose conical limit set were exactly this set E, then we would have established Theorem 1.2. However, it is not apparent how to build a Fuchsian group with pre-determined conical limit set. Instead, we start with the particular Fuchsian group described above, and modify the Kahane-type construction to produce a doubling measure whose support E is contained in L_{c}(G).

In Section 6 we generalize the Kahane-type construction, showing that one obtains a doubling measure under rather mild conditions on the functions {F_{n}}. In particular, the 5-adic grid of intervals in the example above may be replaced by one where each interval I_{n−1,j} at the (n − 1)^{th} level is divided into finitely or infinitely many intervals I_{n,k} at the n^{th} level, such that F_{n} is constant on each I_{n,k}. It suffices (Lemma 6.3) to assume that

. F_{n} has mean value one on each I_{n−1,j} at the (n − 1)^{th} level;

. the functions F_{n} are positive and uniformly bounded away from zero and infinity; and . F_{n} ≡ 1 on subintervals at each end of each interval I_{n−1,j}, whose lengths are at least some definite fraction (independent of n and j) of the length of I_{n−1,j}.

Then the measures μ_{n} given by dμ_{n} = F_{n} ⋯ F_{1} dx converge to a doubling measure μ.

The idea is now to distinguish a collection V of grid intervals in a new, less regular grid, such that points in ∂ID which lie in infinitely many of the intervals in V are actually conical limit points, and to define the functions F_{n} so that they satisfy the three hypotheses above, are large on the intervals in V, and are small elsewhere. The result is a doubling measure which assigns full measure to L_{c}(G).

In the setting of random walks on trees, our goal is to define a tree in the disc, in which the orbit points {g(0)}_{g∈G} appear as a proper subset V of the vertices, and to define the jump probabilities for the edges in this tree so that the subset of paths which hit V infinitely often carries full measure. Here the measure on the space of paths is that which gives, to each family of paths with a common initial segment, measure equal to the product of the jump probabilities for the edges in that segment.

The branches of the tree which contain infinitely many vertices from V correspond to points in the circle which lie in the limit set of G. This is almost by definition: for each such point x in the circle there is a sequence of infinitely many orbit points which approaches x. When we implement this abstract procedure by specifying the vertices and edges in the tree, we will get something more: these points x will actually be conical limit points of G. This is shown in Section 15.

We make a minor simplification of the goal stated above. We impose on the particle the additional constraint that:

. whenever the particle reaches a vertex in V, it stops there. Suppose the particle is started at some vertex v, with this stopping condition. Let V (v) be the subset of vertices in V which can be reached from v. In other words, V (v) consists of those w ∈ V below v such that there are no other vertices from V between v and w. Then it suffices to define the jump probabilities so that, from each starting vertex v, the particle reaches V (v) with probability one. We show below how to do this; then by removing the stopping condition we obtain the measure we require.

In the setting of the grid of intervals, this simplification means that it is sufficient to define the functions F_{n} so that, beginning with Lebesgue measure on any grid interval I, we obtain a doubling measure supported on the maximal grid intervals in I which correspond to orbit points g(0) in the tree.

In the rest of this section we describe:

. a tree in the disc, in which the orbit points {g(0)}_{g∈G} appear as a proper subset V of the vertices; . the adjacencies in the tree (which pairs of vertices are connected by edges); . a projection P from vertices to grid intervals in the circle; . a definition of F_{n} (equivalently, of the jump probabilities); and . a model calculation showing that this definition of F_{n} implies that the resulting measure μ is supported on L_{c}(G).

We now implement the scheme described above, making extensive use of the wellunderstood geometry of the complement of the Cantor set K.

Let D be the closure of a simply connected domain in the disc such that π(D) is either the upper or lower half plane. We call such a region D a half fundamental domain for Ω. The boundary of D consists of a closed subset ∂D ∩ ∂ID of the unit circle, and a union of disjoint arcs of circles which meet the unit circle at right angles. We call these orthocircular arcs. The covering map π maps ∂D ∩ ∂ID onto the Cantor set K, and it maps the orthocircular arcs in ∂D onto the components of \overline{IR} \ K. The normal fundamental domain F for Ω is the union of the two half fundamental domains whose boundaries contain the origin. Its images {g(F)}_{g∈G} tile the disc. We use instead the tiling in which each fundamental domain g(F), except F itself, is divided into the two half fundamental domains which it contains.

Divide each orthocircular arc in the boundaries of the half fundamental domains in this tiling into infinitely many intervals, all of comparable hyperbolic length. These intervals will be the vertices of the tree. A convenient way to divide the arcs is to make a Whitney decomposition of the components of \overline{IR} \ K, in the Riemann surface Ω, and then to pull it back to the disc via the branches of the inverse of the covering map.

To this end, let L be a component of [0, 1] \ K. L is an open interval of length 3^{−k}, for some k ≥ 1. Partition L into subintervals J so that there are 2^{j} J’s of length 3^{−j}|L|, for each j ≥ 1, and so that each J satisfies the following Whitney condition:

|J| = 2 dist (J, K).  (4.3)

This can be done by putting the two J’s of length |J| = 3^{−1}|L| in the middle of L, and then adjoining the smaller J’s at each end in decreasing order of size. The Euclidean length of each J is a negative power of 3.

For the component L = \overline{IR} \ [0, 1] of \overline{IR} \ K, we make a similar decomposition, except that there is one interval J_{∞} which contains the point at infinity. Fix a small positive number σ, and let J_{∞} = \overline{IR} \ (−σ, 1 + σ). Partition the intervals (−σ, 0] and [1, 1 + σ) into J’s satisfying (4.3) so that the length of each J is a negative power of 3 and the lengths decrease as the J’s approach [0, 1]. Make the convention that |J_{∞}| = 1/3.

Let the vertices of the tree be the images I of these intervals J under all branches of the inverse of the covering map. Also, let V consist of those intervals I in the disc such that π(I) = J_{∞}. The orbit {g(0)}_{g∈G} is mapped by the covering map to the point at infinity. So each interval I such that π(I) = J_{∞} contains an orbit point, and each orbit point appears in the tree as a vertex in the set V.

The random walk on the tree in the disc can be pushed forward by the covering map to a random walk in Ω = \overline{C} \ K, with the intervals J as vertices. Our goal becomes: to define the jump probabilities for this random walk so that if the particle starts at any initial vertex, it reaches J_{∞} with probability one.

The idea is that J_{∞} is the largest interval (in the Euclidean sense), and we define the {p_{k}} so that from any interval, the most likely outcome is that the particle jumps to a much larger interval (in the Euclidean sense).

For each interval J, we specify the intervals J′ to which the particle may jump from J. The particle may jump only to the intervals J′ in a particular segment E_{J} of \overline{IR} near to J. See Figure 4 in Section 9 below. This segment E_{J} is much larger than J. We define E_{J} as follows for most grid intervals J, and we call these J standard. The Euclidean length of E_{J} is |E_{J}| = 2 · 3^{N}|J|, where N is a fixed large integer, independent of J. E_{J} comprises one of the scaled copies of the Cantor set K, of length 3^{N}|J|, which arise in the construction of K, together with an adjacent gap in K, also of length 3^{N}|J|. By a gap in K, we mean an interval which is a component of \overline{IR} \ K. The segment E_{J} contains infinitely many intervals J′. Counting, we see that the two largest have Euclidean length |J′| = 3^{N−1}|J|, and there are 2^{k} intervals J′ in E_{J} of length |J′| = 3^{N−k}|J| = 2^{−1} ·3^{−k}|E_{J}|, for each k ≥ 1.

For non-standard grid intervals J, the segments E_{J} take different forms. In particular, from some intervals J it is possible to jump directly to J_{∞}. (Full definitions of the segments E_{J} for standard and non-standard intervals are made in Section 9.)

There are now infinitely many intervals J′ accessible from J. Pulling back to the disc, we obtain a tree in which each vertex has infinite valence. We define the adjacencies in the tree, in other words which pairs of vertices are joined by edges, as follows. Let I be a vertex in the tree, and let J = π(I). Pull E_{J} back to the disc by the branch of π^{−1} which takes J to I. One can show (Section 10) that this pre-image π^{−1}(E_{J}) lies roughly below I, that is, between I and the nearest part of the unit circle. The Whitney intervals J′ in E_{J} are pulled back to vertices I′ = π^{−1}(J′) in π^{−1}(E_{J}). Let these I′ be the children of I; put an edge between I and each such I′.

We also use the segments π^{−1}(E_{J}) to define the grid intervals corresponding to the vertices of the tree. We define the projection P from the vertices to the grid intervals as follows. Since the endpoints of E_{J} lie in K, the endpoints of π^{−1}(E_{J}) lie in the unit circle. Let P (I) be the arc of the unit circle which has the same endpoints as π^{−1}(E_{J}) and which lies below π^{−1}(E_{J}); in other words P (I) is the shorter arc of the unit circle between the endpoints of π^{−1}(E_{J}). This interval P (I) is the grid interval corresponding to the vertex I. The projection P is almost radial, but not quite.

We define some auxiliary functions X_{i} on the circle which keep track of the size of the Whitney intervals J along a path in Ω travelled by the particle. (See Section 12.) Let I_{0} be a grid interval in the circle, corresponding to a vertex v_{0} in the tree. Let x be a point in I_{0}. It corresponds to a path v_{0} → v_{1} → v_{2} → ⋯ in the tree; to a nested, decreasing sequence of grid intervals I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ ⋯ ∋ x in the circle; and to a path J_{0} → J_{1} → J_{2} → ⋯ in Ω. Define

X_{1}(x) = log_{3} [\frac{|J_{1}|}{|J_{0}|}] .  (4.4)

The logarithm is simply in order to make X_{1} integer-valued; recall that the Euclidean lengths of the intervals J are all powers of 3. X_{1} is positive if and only if J_{1} is larger than J_{0}. Define

X_{i}(x) = \{ \begin{matrix}log_{3} [\frac{|J_{i}|}{|J_{i−1}|}] , & if J_{1}, …, J_{i−1} ≠ J_{∞}; \\ 1, & otherwise, \end{matrix}  (4.5)

for i ≥ 1, with the convention that |J_{∞}| = 1/3. This function measures the difference in size of the intervals where the particle is before and after the i^{th} jump, unless the particle has already reached J_{∞}. If the particle reaches J_{∞}, then it stays there, and all subsequent X_{i}’s simply record a 1.

Let

S_{k}(x) = ∑_{i=1}^{k} X_{i}(x).  (4.6)

Then S_{k}(x) → +∞ if and only if the path in Ω which corresponds to x reaches J_{∞} (and therefore stops there).

We wish to ensure that a particle starting from any vertex in the tree reaches V with probability one. Let v be a vertex, and let I_{0} be the corresponding grid interval. Let V (I_{0}) denote the union of the maximal grid intervals in I_{0} which correspond to J_{∞}. These maximal grid intervals are the projections to the circle of those vertices v′ in V which can be reached from v by a particle which stops when it first hits V. Then

V (I_{0}) = {x ∈ I_{0} | S_{k}(x) → +∞},  (4.7)

and our problem is reduced to defining the functions {F_{n}} in such a way that S_{k}(x) → +∞ a.e. dμ on I_{0}, where μ is the limit of the measures μ_{n}, and dμ_{n} = F_{n} ⋯ F_{1} dx.

To do this, it is sufficient to define the {F_{n}} so that the following estimates hold:

EX_{i} = \frac{1}{μ(I)} ∫_{I} X_{i}(x) dμ ≥ c_{1},  (4.8)

and

EX_{i}^{2} = \frac{1}{μ(I)} ∫_{I} X_{i}^{2}(x) dμ ≤ c_{2},  (4.9)

with uniform positive constants c_{1} and c_{2}, for all grid intervals I in the (n − 1)^{th} layer of the grid, for all n ≥ 1. Here X_{i} is the auxiliary function, defined in (4.5) above, for a particle which has not yet reached J_{∞} and which makes its (i − 1)^{th} jump from the vertex corresponding to I. In other words, the expectations of the X_{i}’s with respect to μ are uniformly large, and the second moments are uniformly bounded. Then an appeal to the strong law of large numbers for martingales (see Section 15) yields the result that S_{k}(x) → +∞ a.e. dμ on I_{0}, as required.

In the remainder of this section, we show how to define F_{n} for those grid intervals I which correspond to standard intervals J. (Full definitions of F_{n} are given in Section 11, for standard intervals, and in Section 14, for non-standard intervals.) We also give a model calculation for the estimate (4.8) on the expectation of X_{i}.

Let I be a grid interval, from the (n − 1)^{th} layer of the grid, whose corresponding interval J in \overline{IR} \ K is standard. The grid intervals I′ in I from the next (n^{th}) layer of the grid correspond to those vertices in the tree which are accessible from the vertex coresponding to I. Therefore these I′ are exactly those grid intervals which correspond to the intervals J′ in E_{J} in the Riemann surface Ω. In other words, we pull back the segment E_{J} from \overline{IR} to the disc via the appropriate branch of the inverse of the covering map, and then project it to the interval I in the circle using the projection P defined above.

The key idea is that the geometry of the segment E_{J} is not distorted very much by this operation. In fact, we now assume that it is not distorted at all, so that the n^{th} layer of grid intervals in I has exactly the same structure as E_{J}. Namely, one half of I contains an exact scaled copy K̃ of the ternary Cantor set K, of length |I|/2, and the other half of I corresponds to a gap in K. Again, we are assuming that the intervals I′ in I \ K̃ are exactly Whitney intervals with |I′| = 2 dist (I′, K̃), just like the Whitney intervals J′ in E_{J}. In particular, I \ K̃ contains 2^{k} intervals I′ of length |I′| = 2^{−1} · 3^{−k}|I|, for each k ≥ 1.

Much of the paper (especially in Sections 8, 10, 11, 13, and 14) is devoted to proving estimates which control the distortion of the segments E_{J} by P ∘π^{−1}. These estimates rely on the fact that on the boundary of a chord-arc domain, one can use harmonic measure to estimate arclength (via A_{∞}-equivalence). For the remainder of this section, however, we simply assume that there is no distortion.

Let B_{1} denote the union of the two largest grid intervals I′ in I from the n^{th} layer of the grid; they satisfy |I′| = |I|/6. Fix a small positive number ε, independent of I. Define the function F_{n} on I so that:

. F_{n} ≡ (1 − ε) |I|/|B_{1}| on B_{1};

. F_{n} ≡ 1 on small segments I_{l} and I_{r} at each end of I, where |I_{l}|/|I| and |I_{r}|/|I| are fixed quantities independent of I, and the endpoints of I_{l} and I_{r} are endpoints of intervals from the n^{th} layer of the grid; and . F_{n} ≡ δ on the rest of I, where δ is a small number chosen to ensure that F_{n} has mean value 1 on I.

The force of the first condition is that μ(B_{1})/μ(I) = 1−ε; the effect of F_{n} is to concentrate almost all the mass of I onto B_{1}. This is because of the way that μ is constructed: the μ-measure of an interval from the n^{th} layer of the grid is simply its μ_{n}-measure, since all F_{l} with l > n have mean value 1 on the grid interval. So

\begin{aligned}\frac{μ(B_{1})}{μ(I)} & = \frac{μ_{n}(B_{1})}{μ_{n−1}(I)} \\ & = \frac{∫_{B_{1}} F_{n}(x) ⋯ F_{1}(x) dx}{∫_{I} F_{n−1}(x) ⋯ F_{1}(x) dx} \\ & = (1 − ε) \frac{|I|}{|B_{1}|} \frac{∫_{B_{1}} F_{n−1}(x) ⋯ F_{1}(x) dx}{∫_{I} F_{n−1}(x) ⋯ F_{1}(x) dx} \\ & = 1 − ε, \end{aligned}  (4.10)

since the product F_{n−1} ⋯ F_{1} is constant on I.

In Sections 11 and 14 we show that these functions F_{n} are uniformly bounded away from zero and infinity, which is a hypothesis of Lemma 6.3 on doubling measures.

The second condition above is another hypothesis of Lemma 6.3. However, in order to simplify the model calculation below, we temporarily forget that F_{n} = 1 near the ends of I, and assume that F_{n} = δ on all of I \ B_{1}.

Let

\begin{aligned}B_{k} & = \{x ∈ I | X_{i}(x) = N − k\} \\ & = ⋃ \{I′ ⊂ I | |I′| = 2^{−1} · 3^{−k}|I|\}, \end{aligned}  (4.11)

for k ≥ 1. We know that B_{k} consists of exactly 2^{k} intervals I′, each of length 2^{−1} · 3^{−k}|I|, and so

|B_{k}| = \frac{1}{2} (\frac{2}{3})^{k} |I|.  (4.12)

The function X_{i} takes the constant value N − k on B_{k}, for k ≥ 1. The function F_{n} is also constant on each B_{k}; we are assuming that F_{n} ≡ δ on I \ B_{1}, and forgetting that actually F_{n} = 1 on the parts of some B_{k}’s which are very near the endpoints of I. We know that μ(B_{k}) = μ_{n}(B_{k}), and μ(I) = μ_{n−1}(I). Therefore, in computing the ratio μ(B_{k})/μ(I), we need only take account of the value of F_{n} on B_{k}; the F_{l}’s with l ≠ n are irrelevant. We see that μ(B_{k})/μ(I) is just δ |B_{k}|/|I|, if k ≥ 2. Therefore

\begin{aligned}EX_{i} & = \frac{1}{μ(I)} ∫_{I} X_{i}(x) dμ \\ & = \frac{1}{μ(I)} ∑_{k=1}^{∞} ∫_{B_{k}} X_{i}(x) dμ \\ & = ∑_{k=1}^{∞} (N − k) \frac{μ(B_{k})}{μ(I)} \\ & = (N − 1) \frac{μ(B_{1})}{μ(I)} + δ ∑_{k=2}^{∞} (N − k) \frac{|B_{k}|}{|I|} \\ & ≥ (N − 1) (1 − ε) − δ ∑_{k=2}^{∞} k \frac{|B_{k}|}{|I|} \\ & = (N − 1) (1 − ε) − δ ∑_{k=2}^{∞} k \frac{1}{2} (\frac{2}{3})^{k} . \end{aligned}  (4.13)

The last series converges, so we can ensure that EX_{i} ≥ c_{1} > 0 by choosing N sufficiently large with respect to δ.

When we correct for the fact that F_{n} = 1, not δ, near the endpoints of I, we get some other terms involving similar convergent series, and we can still make the whole argument work by choosing N large. More crucially, we prove estimates (see Sections 10, 13, and 14, especially Lemmas 13.1 and 14.1) showing that, even allowing for the distortion of lengths produced by the inverse of the covering map and by the projection from the disc to the circle, the ratio |B_{k}|/|I| is controlled by β^{k}, where β is smaller than 1 and independent of I, so that the relevant series converge and a calculation like the one above can be done.

The estimate EX_{i}^{2} ≤ c_{2} is similar. The estimates (4.8) and (4.9) are established in Section 13 for standard intervals, and in Section 14 for non-standard intervals.

5. Choice of the Fuchsian group

Let K ⊂ [0, 1] be the classical ternary Cantor set. Let Ω = \overline{C} \ K be the complement of K in the extended complex plane. Ω is an infinitely connected planar Riemann surface. Since K = ∂Ω contains more than two points, the universal covering space of Ω is the unit disc ID, and Ω is conformally equivalent to ID/G, where G is a Fuchsian group. We choose this infinitely generated Fuchsian group G as our example.

In this section we show that the conical limit set L_{c}(G) of G has one-dimensional Lebesgue measure zero, and moreover the Hausdorff dimension of L_{c}(G) satisfies 1/2 ≤ dim (L_{c}(G)) < 1. The remainder of the paper is devoted to the construction of a doubling measure which is supported on L_{c}(G).

Since K has positive logarithmic capacity, Green’s function exists for Ω, and the result cited in the introduction on the measure of the conical limit set of a Fuchsian group implies that |L_{c}(G)| = 0. Fernández [F] has shown that if the boundary ∂Ω of a planar domain Ω is uniformly perfect, then the exponent of convergence of the corresponding Fuchsian group is strictly less than one. The Cantor set K is uniformly perfect. For Fuchsian groups the exponent of convergence is equal to the Hausdorff dimension of the conical limit set [Pa1, S1]. (See also [BJ], where this is proved in much greater generality.) We may therefore conclude that in fact dim (L_{c}(G)) < 1.

A Fuchsian group G is of fully accessible type if there is a measurable fundamental set B ⊂ ∂ID for the action of G on ∂ID. In other words, B contains no two G-equivalent points, and ∂ID is a.e. equal to the union of the G-images of B: ∑_{g∈G} |g(B)| = 1. G is of accessible type if there is a measurable set B ⊂ ∂ID, containing no two G-equivalent points, such that ∑_{g∈G} |g(B)| > 0.

The normal fundamental domain F_{0} (see Section 7) for Ω = \overline{C} \ K has the property that |∂F_{0} ∩ ∂ID| = 0. We outline the well-known proof. The covering map π takes ∂F_{0} to [0, 1]. Since ∂F_{0} is a rectifiable curve, π preserves sets of zero length. (This is the F. and M. Riesz theorem; see [Po3].) Since K has Lebesgue measure zero, ∂F_{0} ∩π^{−1}(K) = ∂F_{0} ∩ ∂ID has measure zero.

This implies that the group G is not of accessible type [Po1]. On the other hand, Patterson [Pa2] showed that if δ(G) < 1/2 then G is of fully accessible type. It follows that our group G must have δ(G) = dim (L_{c}(G)) ≥ 1/2.

6. A Lemma on Doubling Measures

In this section we follow the philosophy first laid out by Kahane [K] to construct doubling measures on an interval I_{0}. Our measure is the limit of a sequence of measures whose densities are step functions. In Lemma 6.3 we show that mild conditions on these step functions ensure that the limit measure is doubling.

Simple constructions of this type can yield doubling measures which are supported on very small sets. In particular there are examples of such measures which are supported on sets of arbitrarily small, but positive, Hausdorff dimension. Such a measure is equivalent to the derivative of a quasisymmetric mapping that takes a set of small Hausdorff dimension to a set of full one-dimensional Lebesgue measure.

The two interrelated ingredients of this construction are a grid of nested subintervals of I_{0} and a sequence of suitable density functions.

Definition 6.1. A grid of subintervals of an interval I_{0} is a collection H = ⋃_{n=1}^{∞} H_{n} of subintervals of I_{0} satisfying

i) H_{0} = I_{0};

ii) for each n ≥ 0, the intervals in H_{n} have disjoint interiors, and |I_{0} \ ⋃_{I∈H_{n}} | = 0; and

iii) for each n ≥ 1, for each interval J ∈ H_{n} there is an interval I ∈ H_{n−1} such that J ⊂ I.

The collections H_{n} are called the layers of intervals in the grid H.

Definition 6.2. A function F defined on a interval I is a step function if it is constant on each interval in some finite or infinite partition of I. We say F is (δ, η)-suitable for I if F is a step function satisfying

i) \frac{1}{|I|} ∫_{I} F (x) dx = 1;

ii) 0 < δ ≤ F (x) ≤ 1/δ for all x ∈ I; and

iii) F (x) ≡ 1 on subintervals I_{l} and I_{r} of I such that I_{l} has the same left endpoint as I, I_{r} has the same right endpoint as I, |I_{l}| ≥ η|I|, and |I_{r}| ≥ η|I|.

See Figure 1.

Fix an interval I_{0}, and numbers δ and η such that 0 < δ ≤ 1 and 0 < η ≤ 1/2. We construct a sequence {μ_{n}} of measures on I_{0}, whose densities are of the form dμ_{n} = F_{n}(x) ⋯ F_{1}(x) dx. We define the functions F_{n} simultaneously with a related grid H of subintervals of I_{0}. Each F_{n} is (δ, η)-suitable for each interval from the (n−1)^{th} layer H_{n−1} of the grid.

Figure 1. F (x) is (δ, η)-suitable for I = [0, 1].

Let μ_{0} be Lebesgue measure on I_{0}. Let F_{1} be any function which is (δ, η)-suitable for I_{0}. Define μ_{1} by dμ_{1} = F_{1}(x) dx. Since F_{1} has mean value one on I_{0}, μ_{1} has the same total mass as μ_{0}. Take any partition of I_{0} into subintervals J such that F_{1} is constant on each J. Note that we allow F_{1} to take the same value on adjacent J’s; the J’s need not be maximal. Also, there may be finitely or infinitely many J’s. Let H_{1}, the first layer of intervals in the grid, be the collection of subintervals in this partition.

We define the measures μ_{n} inductively for n ≥ 2. Suppose we have already defined functions F_{1}, …, F_{n−1}, measures μ_{1}, …, μ_{n−1}, and the layers H_{1}, …, H_{n−1} of the grid. In particular, F_{n−1} is constant on each interval I ∈ H_{n−1}. Let F_{n} be any function which is (δ, η)-suitable for each interval I ∈ H_{n−1}. Define μ_{n} by

dμ_{n} = F_{n}(x) dμ_{n−1} = F_{n}(x) ⋯ F_{1}(x) dx.  (6.1)

Again, the fact that F_{n} has mean value one implies that μ_{n} has the same total mass as μ_{0}. To define the next layer H_{n} of the grid, we partition each I ∈ H_{n−1} into subintervals J such that F_{n} is constant on each J. Let H_{n} be the collection of all these subintervals J.

Note that the collection H = ⋃_{n=1}^{∞} H_{n} of subintervals of I_{0} does form a grid according to the definition above: each interval J ∈ H_{n} is contained in some I ∈ H_{n−1}, and each layer H_{n} is a partition of I_{0}, so |I_{0} \ ⋃_{I∈H_{n}} I | = 0 and the intervals in H_{n} have disjoint interiors. Also, on each J ∈ H_{n}, μ_{n} is a constant multiple of Lebesgue measure: dμ_{n} = (μ_{n}(J)/|J|) dx on J. Furthermore, the total mass assigned to an interval J ∈ H_{n} does not change after the n^{th} step: μ_{n}(J) = μ_{n+1}(J) = · · ·.

Lemma 6.3. Fix an interval I_{0} and numbers δ and η with 0 < δ ≤ 1 and 0 < η ≤ 1/2. Let {μ_{n}} be a sequence of measures and H a grid of subintervals of I_{0} as described above, so that for each n ≥ 1, dμ_{n} = F_{n}(x) ⋯ F_{1}(x) dx and F_{n}(x) is (δ, η)-suitable for each interval I ∈ H_{n−1}. Then the sequence {μ_{n}} converges weakly to a measure μ on I_{0} which has the same total mass as μ_{0} and which is doubling with a constant depending only on δ and η.

Proof. The μ_{n} are a sequence of measures, all with the same finite total mass, on a compact interval I_{0}. Therefore there is a subsequence converging weakly to a measure μ on I_{0} with the same mass. It is an exercise to see that this limit measure is unique; in other words the full sequence μ_{n} converges to μ. The μ-measure of any interval I ∈ H_{n} is given by μ(I) = μ_{n}(I).

Let J and K be adjacent intervals, of equal length, contained in I_{0}. If, for each n, the function F_{n} is constant on J ∪ K, then μ(J) = μ(K). Otherwise, let m be the first integer such that F_{m} is not constant on J ∪ K. In particular,

μ_{m}(J) ∼^{1/δ} μ_{m−1}(J) = μ_{m−1}(K) ∼^{1/δ} μ_{m}(K).  (6.2)

We show that μ(J) ∼ μ_{m}(J) and μ(K) ∼ μ_{m}(K).

We may assume that F_{m} is not constant on J. Let x ∈ J be a point where the value of F_{m} changes. Then for each n ≥ m + 1 there is a neighbourhood of x on which F_{n} ≡ 1.

Let J_{r} be the part of J to the right of x. If J_{r} does not contain any whole interval from H, then F_{n} ≡ 1 on J_{r} for all n ≥ m + 1, and so μ(J_{r}) = μ_{m}(J_{r}). Otherwise, let

L_{m} = ⋃_{j} {I_{m,j} ∈ H_{m} | I_{m,j} ⊂ J_{r}}.  (6.3)

Then μ(L_{m}) = μ_{m}(L_{m}). (Therefore if J_{r} consists only of whole intervals from H_{m}, in other words if J_{r} = L_{m}, then μ(J_{r}) = μ_{m}(J_{r}) and we are done.)

The function F_{m+1} is constant on each interval in H_{m+1}. Suppose there is an interval I_{m+1,j} ∈ H_{m+1} which meets J_{r} \ L_{m} and on which F_{m+1} ≠ 1. The interval I_{m,k} ∈ H_{m} which contains I_{m+1,j} is adjacent to L_{m}, or has x as its left endpoint in case L_{m} is empty, and it contains J_{r} \ L_{m}. Let I be the largest subinterval of I_{m,k}, with the same left endpoint as I_{m,k}, on which F_{m+1} ≡ 1. I is a union of intervals in H_{m+1}, |I| ≥ η |I_{m,k}|, and I ⊂ J_{r} \ L_{m} ⊂ I_{m,k}. Then

\begin{aligned}μ(J_{r} \ L_{m}) & ≥ μ_{m+1}(I) \\ & = μ_{m}(I) \\ & = \frac{μ_{m}(I_{m,k})}{|I_{m,k}|} |I| \\ & ≥ η μ_{m}(I_{m,k}) \\ & ≥ η μ_{m}(J_{r} \ L_{m}), \end{aligned}  (6.4)

and

\begin{aligned}μ(J_{r} \ L_{m}) & ≤ μ_{m}(I_{m,k}) \\ & ≤ \frac{μ_{m}(I_{m,k})}{|I_{m,k}|} η^{−1} |I| \\ & = η^{−1} μ_{m}(I) \\ & ≤ η^{−1} μ_{m}(J_{r} \ L_{m}). \end{aligned}  (6.5)

Therefore μ(J_{r} \ L_{m}) ∼^{1/η} μ_{m}(J_{r} \ L_{m}), and so μ(J_{r}) ∼^{1/η} μ_{m}(J_{r}).

On the other hand, suppose there is no interval in H_{m+1} which meets J_{r} \ L_{m} and on which F_{m+1} ≠ 1. Then F_{m+1} ≡ 1 on J_{r} \ L_{m}. If J_{r} \ L_{m} contains an interval I_{m+1,j} from H_{m+1} on which F_{m+1} ≡ 1, then the μ-measure of I_{m+1,j} is not only equal to its μ_{m+1}-measure but also to its μ_{m}-measure. Let

L_{m+1} = ⋃_{j} {I_{m+1,j} ∈ H_{m+1} | I_{m+1,j} ⊂ J_{r} \ L_{m}; F_{m+1} ≡ 1 on I_{m+1,j}} .  (6.6)

be the union of all such intervals. L_{m+1} is an interval to the right of the interval L_{m} and adjacent to it, if both are non-empty. Now μ(L_{m+1}) = μ_{m}(L_{m+1}). If J_{r} = L_{m}∪L_{m+1}, then μ(J_{r}) = μ_{m}(J_{r}) and we are done. Otherwise, we have reduced the problem to estimating the μ-measure of J_{r} \ (L_{m} ∪ L_{m+1}).

Apply this argument to J_{r} \ ⋃_{n=m}^{m+i} L_{n} for i ≥ 1, where L_{m+i} is defined inductively by

L_{m+i} = ⋃_{j} {I_{m+i,j} ∈ H_{m+i} | I_{m+i,j} ⊂ J_{r} \ ( ⋃_{n=m}^{m+i−1} L_{n}); F_{m+i} ≡ 1 on I_{m+i,j}} . (6.7)

Suppose that for some integer i ≥ 1 there is an interval I_{m+i+1,j} ∈ H_{m+i+1} which meets J_{r} \ ⋃_{n=m}^{m+i} L_{n} and on which F_{m+i+1} ≠ 1. Let l be the first such integer. As in the case l = 1 above, let I_{m+l,k} be the interval from H_{m+l} which contains I_{m+l+1,j} and let I be the largest subinterval of I_{m+l,k} on which F_{m+l+1} ≡ 1. Then I ⊂ J_{r} \ ⋃_{n=m}^{m+l} L_{n} ⊂ I_{m+l,k}

and so, by the analogues of (6.4) and (6.5), μ(J_{r}) ∼^{1/η} μ_{m}(J_{r}), and we are done.

On the other hand, suppose there is no such integer l ≥ 1. If the intervals L_{m+i}, i ≥ 1, exhaust J_{r}, then μ(J_{r}) = μ_{m}(J_{r}), and we are done. Otherwise, J_{r} \ ⋃_{n=m}^{∞} L_{n} is a non-empty interval which has the same right endpoint as J_{r} and which contains no whole interval from H. So F_{m+i} ≡ 1 on J_{r} \ ⋃_{n=m}^{∞} L_{n} for all i ≥ 1. Therefore μ(J_{r} \ ⋃_{n=m}^{∞} L_{n}) = μ_{m}(J_{r} \ ⋃_{n=m}^{∞} L_{n}), and so μ(J_{r}) = μ_{m}(J_{r}). This is the last possible

case; we have shown that μ(J_{r}) ∼^{1/η} μ_{m}(J_{r}) in all cases.

The same argument shows that μ(J_{l}) ∼^{1/η} μ_{m}(J_{l}), where J_{l} is the part of J to the left

of x. Therefore μ(J) ∼^{1/η} μ_{m}(J).

If the function F_{m} is not constant on the adjacent interval K then μ(K) ∼^{1/η} μ_{m}(K)

by the argument above. Also, if F_{n} ≡ 1 on K for all n ≥ m + 1, then μ(K) ∼^{1/η} μ_{m}(K). Otherwise, either F_{n} is constant (although not necessarily ≡ 1) on K for each n ≥ m + 1, or we let l ≥ m + 1 be the first integer such that F_{l} is not constant on K. In these cases it is enough to estimate the number of integers n ≥ m + 1 for which F_{n} ≡ constant ≠ 1 on K. We show that this number is bounded by a constant c = c(η). Then in the first

case μ(K) ∼^{c′} μ_{m}(K), and in the second μ(K) ∼^{1/η} μ_{l}(K) ∼^{c′} μ_{m}(K), where the constants c′ depend on c and on δ.

Suppose there are at least k integers n_{1} < ⋯ < n_{k} greater than m for which F_{n_{j}} ≡ constant ≠ 1 on the interval K. We may assume that K is to the right of J. For each j with 1 ≤ j ≤ k, there is an interval I_{j} ∈ H_{n_{j}} which contains K and whose left endpoint lies in J. These intervals are nested: I_{1} ⊃ ⋯ ⊃ I_{k}. Let L_{j} (respectively R_{j}) be the largest subinterval of I_{j}, with the same left (respectively right) endpoint as I_{j}, on which F_{n_{j}} ≡ 1, and let M_{j} = I_{j} \ (L_{j} ∪ R_{j}). Then |M_{j}| ≤ (1 − 2η)|I_{j}|, and I_{j} ⊂ M_{j−1} for 2 ≤ j ≤ k. See Figure 2.

Figure 2. Graphs of F_{n_{1}+1}(x) and F_{n_{2}+1}(x).
Also, L_{1} ⊂ J and η |I_{1}| ≤ |L_{1}| ≤ |J|. So
|K| ≤ |M_{k}| ≤ (1 − 2η) |I_{k}| ≤ (1 − 2η) |M_{k−1}|.  (6.8)

After k steps we see that

|K| ≤ (1 − 2η)^{k} |I_{1}| ≤ (1 − 2η)^{k} η^{−1} |J| = (1 − 2η)^{k} η^{−1} |K|.  (6.9)

Therefore k ≤ log η/ log (1 − 2η).

We have shown that μ(J) ∼^{c} μ(K), where c depends only on δ and η, for every pair J, K of adjacent intervals of equal length. In other words μ is a doubling measure, and the doubling constant depends only on δ and η.

7. Fundamental domains for Ω We begin the construction of the doubling measure by describing the fundamental

domains for Ω = \overline{C} \ K, where K ⊂ [0, 1] is the classical ternary Cantor set. G is the Fuchsian group such that ID/G is conformally equivalent to Ω. G acts as the covering group of Ω on the universal covering space ID of Ω. Let π : ID → Ω be the covering map; it is a many-to-one conformal mapping which takes each G-orbit in ID to a single point in Ω. Normalize π, by composition with a Möbius transformation of the disc, so that π(0) = ∞.

A fundamental domain for Ω is a domain F in ID such that \overline{F} contains at least one point from every G-orbit, and F contains no two G-equivalent points. The domain Ω = \overline{C} \ K is a Denjoy domain; that is, it is the complement in \overline{C} of a closed linear set. Denjoy domains have fundamental domains which have a particularly simple form; namely, they are orthocircular. This means that the boundary is of the form ∂F = E ∪ ⋃_{n=1}^{∞} A_{n}, where E is a closed subset of the unit circle ∂ID, and the A_{n}’s (called orthocircular arcs) are disjoint arcs of circles which meet ∂ID at right angles. The covering map takes the set ∂F ∩∂ID to ∂Ω, and it takes the orthocircular arcs comprising ∂F \∂ID to the components of \overline{IR}\∂Ω. The images \{g(F)\}_{g∈G} are orthocircular domains which tile the disc. (See [RR]; they consider the case where the boundary ∂Ω has positive length, but a Denjoy domain Ω with |∂Ω| = 0 also has an orthocircular fundamental domain F, with |∂F ∩ ∂ID| = 0.)

We refer to the images in the disc of the upper and lower half planes under the branches of π^{−1} as half fundamental domains for Ω = \overline{C} \ K. Each half fundamental domain D is orthocircular. To fix ideas, suppose that π(D) is the upper half plane. The boundary \overline{IR} of the upper half plane lifts via π^{−1} to the boundary of D. The Cantor set K lifts to the closed subset ∂D ∩ ∂ID of the circle, and the open intervals which are the components of \overline{IR} \ K lift to the orthocircular arcs in the boundary of D.

We now fix a tiling of the disc by half fundamental domains for Ω, together with one fundamental domain for Ω. Since π(0) = ∞, one of the preimages under π of the open interval \overline{IR} \ [0, 1] is a diameter γ of ID. Let D be the preimage of the upper half plane whose boundary contains γ. Let D′ be the reflection of D in γ. D′ is a preimage of the lower half plane. Then D ∪ γ ∪ D′ is a fundamental domain for Ω; it contains the origin; and it is symmetric with respect to the diameter γ. We denote this fundamental domain by F_{0}; it is, by symmetry, the normal fundamental domain for Ω. Its image under the covering map is π(F_{0}) = \overline{C}\[0, 1]. Each open interval in [0, 1]\K lifts to two orthocircular arcs in ∂F_{0}.

Tile the remainder of the disc, ID \ F_{0}, by the half fundamental domains which are the G-images of D and D′. These half fundamental domains are indexed in a natural way by the number of orthocircular boundary arcs separating them from the origin. Write Ω_{1,j} for any half fundamental domain which is adjacent to F_{0}; that is, which is separated from the origin 0 by a single orthocircular arc. Similarly, write Ω_{n,j} for a half fundamental domain which is separated from 0 by n orthocircular arcs in the boundaries of the half fundamental domains in the fixed tiling of ID.

A note on orientation: we think of the direction from a point in the disc towards the unit circle as down, and of the direction towards the origin as up. Terms like below and above are to be understood in the same way; below means closer to the unit circle. Given a half fundamental domain Ω_{n,j}, we refer to the large orthocircular arc, denoted A_{n,j}, which separates Ω_{n,j} from 0 as the upper part of the boundary of Ω_{n,j}, and to ∂Ω_{n} \ A_{n,j} as the lower part of the boundary of Ω_{n,j}.

Let A_{1} = ∂F_{0} \ ∂ID be the union of the orthocircular arcs in the boundary of F_{0}. Each arc A_{1,j} ⊂ A_{1} is the upper part of some ∂Ω_{1,j}. For each n ≥ 2, let A_{n} be the union, over all j, of the orthocircular arcs in the lower parts of the boundaries of the Ω_{n−1,j}:

A_{n} = [⋃_{j} ∂Ω_{n−1,j} \ ∂ID] \ A_{n−1}.  (7.1)

The union A = ⋃_{n=1}^{∞} A_{n} is the collection of all orthocircular boundary arcs which appear in the tiling of ID \ F_{0} by half fundamental domains.

We show in Section 10 that the orthocircular boundary arcs are uniformly hyperbolically separated: the hyperbolic distance between any two of these arcs is greater than a fixed positive constant. It follows from this observation that the boundaries of the half fundamental domains are chord-arc curves, with a uniform chord-arc constant. We will not need this fact, but we will need to show (Sections 13 and 14) that certain subsets of the half fundamental domains are chord-arc with uniform chord-arc constants.

8. Whitney decompositions

In this section we make a Whitney decomposition of the components of \overline{IR} \ K, and show that it lifts via π^{−1} to a Whitney decomposition of the orthocircular arcs in the boundaries of the half fundamental domains in ID. The intervals in this lifted decomposition are the vertices in the tree described in Section 4.

The Cantor set K is formed by removing the open middle third (\frac{1}{3}, \frac{2}{3}) from [0, 1], then removing the open middle thirds (\frac{1}{9}, \frac{2}{9}) and (\frac{7}{9}, \frac{8}{9}) from the closed intervals [0, \frac{1}{3}] and [\frac{2}{3}, 1] which remain, and so on. We refer to the closed intervals [0, 1], [0, \frac{1}{3}], [\frac{2}{3}, 1], [0, \frac{1}{9}], [\frac{2}{9}, \frac{1}{3}], … which appear in this procedure as closed construction intervals of K.

A Whitney decomposition of an open interval L in IR is a partition of L into closed intervals J, with disjoint interiors, such that the Euclidean length of each J is comparable to the Euclidean distance from J to the nearest endpoint of L. In other words, there is a constant c > 0 such that |J| ∼^{c} dist (J, IR \ L) for each J. This also implies that for each point x in such an interval J, the distance from x to IR \ L is comparable to the distance from J to IR \ L. The intervals J in a Whitney decomposition are called Whitney intervals. We will also deal with Whitney decompositions of orthocircular arcs; these are defined analogously, so that the length of each Whitney interval in the arc is comparable to its distance from the nearest end of the arc.

Write \overline{IR} \ K as a countable union of disjoint open intervals; let L denote any one of these intervals. We now fix a Whitney decomposition of each component L of \overline{IR} \ K.

First consider a component L = (a, b) of [0, 1] \ K. L is an open interval of length 3^{−k}, for some k ≥ 1. Partition L into subintervals of length 3^{−n−k}, n ≥ 1, as follows. For n ≥ 1, let

\begin{aligned}J_{n} & = [b − 3^{−n+1} \frac{|L|}{2}, b − 3^{−n} \frac{|L|}{2}] ,  and \\ J_{−n} & = [a + 3^{−n} \frac{|L|}{2}, a + 3^{−n+1} \frac{|L|}{2}] . \end{aligned}  (8.1)

Then J_{1} is the interval whose left endpoint is the midpoint of L and which extends twothirds of the way towards the right endpoint of L; for n ≥ 2, J_{n} is the interval whose left endpoint is the right endpoint of J_{n−1} and which extends two-thirds of the way towards the right endpoint of L; and for n ≥ 1, J_{−n} is the reflection of J_{n} in the midpoint of L. The intervals {J_{±n}}_{n≥1} form a Whitney decomposition of L, |J_{±n}| = 2 dist (J_{±n}, K), and |J_{±n}| = 3^{−n} |L| = 3^{−n−k}.

Let N be a large integer; its value will be fixed below. It is convenient to amalgamate the 2N intervals J_{−N}, …, J_{N} which are closest to the midpoint of L into a single interval: Let

J_{c} = J_{−N} ∪ ⋯ ∪ J_{N}.  (8.2)

This interval has length |J_{c}| = (1 − 3^{−N})|L|, and its distance from K is 2^{−1} · 3^{−N}|L|, so it satisfies |J_{c}| = 2 (3^{N} − 1) dist (J_{c}, K).

The intervals {J_{±n}}_{n≥N+1} and J_{c} form a Whitney decomposition of L, with constant 2 (3^{N} − 1).

Now consider the remaining component L = \overline{IR} \ [0, 1]. Fix the small number σ = (2 · 3^{N+2})^{−1}; the reasons for this particular choice will become apparent later. Let

J_{∞} = \overline{IR} \ (−σ, 1 + σ).  (8.3)

J_{∞} is an interval of large but finite hyperbolic length, which contains the point at infinity. For n ≥ N + 3, let

\begin{aligned}J_{n} & = [−\frac{1}{2} \frac{1}{3^{n−1}} , −\frac{1}{2} \frac{1}{3^{n}} ] ,  and \\ J_{−n} & = [1 + \frac{1}{2} \frac{1}{3^{n}} , 1 + \frac{1}{2} \frac{1}{3^{n−1}} ] . \end{aligned}  (8.4)

Then J_{N+3} = [−σ, −σ/3], and for n ≥ N + 4, J_{n} is the interval whose left endpoint is the right endpoint of J_{n−1} and which extends two-thirds of the way towards 0. For n ≥ N + 3, J_{−n} is the reflection of J_{n} through the midpoint of [0, 1]. The intervals {J_{±n}}_{n≥N+3} form a Whitney decomposition of [−σ, 0) ∪ (1, 1 + σ], in which |J_{±n}| = 3^{−n} = 2 dist (J_{±n}, K).

Make the convention that

|J_{∞}| = 1/3.  (8.5)

With this convention, the intervals {J_{±n}}_{n≥N+3} and J_{∞} form a Whitney decomposition of L = \overline{IR} \ [0, 1], with constant 2 · 3^{N+3}.

We use the phrase the Whitney decomposition of \overline{IR} \ K to denote the collection of Whitney intervals {J_{±n}}_{n≥N+1} and J_{c} in the components of [0, 1] \ K, together with the Whitney intervals \{J_{±n}\}_{n≥N+3} and J_{∞} in \overline{IR} \ [0, 1]. This decomposition will remain fixed for the rest of the construction. We call the intervals {J_{±n}} standard and the intervals J_{c} and J_{∞} non-standard.

The inverse π^{−1} of the covering map lifts the components of \overline{IR}\K to the orthocircular arcs ⋃_{n} A_{n} which appear in the boundaries of the half fundamental domains in our tiling of ID. π^{−1} lifts the Whitney decomposition of \overline{IR} \ K to a decomposition of ⋃_{n} A_{n} into closed intervals with disjoint interiors.

The main result of this section is that this is a Euclidean Whitney decomposition of the orthocircular arcs ⋃_{n} A_{n}.

Lemma 8.1. The Whitney decomposition of \overline{IR} \ K lifts via π^{−1} to a Whitney decomposition of the orthocircular arcs ⋃_{n} A_{n}.

Proof. Since ∂Ω = K is uniformly perfect, there is a constant c_{Ω} > 0 such that the function λ_{Ω} satisfies

\frac{c_{Ω}}{dist (w, K)} ≤ λ_{Ω}(w) ≤ \frac{2}{dist (w, K)}  (8.6)

for all w in Ω, where the element of hyperbolic arclength in Ω is ds = λ_{Ω}(w) |dw|. Combined with the relation |J| = 2 dist (J, K) for standard Whitney intervals J, this implies that ℓ_{hyp}(J) ∼ 1 with a uniform constant for all standard J in \overline{IR} \ K. Similarly, since the ratio |J_{c}|/dist (J_{c}, K) is the same for all non-standard intervals J_{c}, ℓ_{hyp}(J_{c}) ∼ 1 for all such J_{c}. Also, the hyperbolic length of J_{∞} depends only on the constant σ.

These observations imply that the intervals I in the preimage of the Whitney decomposition of \overline{IR} \ K are all of comparable hyperbolic length, because the covering map π preserves hyperbolic length. So 1/r ≤ ℓ_{hyp}(I) ≤ r for all I, where the constant r depends on the uniformly perfect constant of K, on the constant of the Whitney decomposition of \overline{IR} \ K, and on the choice of σ in the definition of J_{∞}.

It follows that for each I,

\frac{1}{2r} (1 − |z_{1}|) ≤ |I| ≤ 2re^{r} (1 − |z_{1}|),  (8.7)

where z_{1} is the endpoint of I closest to ∂ID. Therefore the intervals I form a Whitney decomposition of the orthocircular arcs ⋃_{n} A_{n}.

9. Construction of palm leaves and the grid of intervals

In this section we construct a grid H of subintervals of the unit circle, of the kind described in Section 6. We already have a collection of Whitney intervals which form a Whitney decomposition of the orthocircular arcs ⋃_{n} A_{n} in the boundaries of the half fundamental domains in the tiling of the unit disc. The idea is to project these Whitney intervals onto the unit circle, using a projection map P defined below. The projections of the intervals in A_{n} form the n^{th} layer H_{n} of the grid. Heuristically, the projection P is almost radial projection. Let denote the inverse of the projection P: given a grid interval I, Î denotes the unique Whitney interval in ⋃_{n} A_{n} such that P (Î) = I.

Recall that A_{1} = ∂F_{0} \ ∂ID is the collection of orthocircular arcs in the boundary of the fundamental domain F_{0} which contains the origin; and that for n ≥ 2, A_{n} consists of the collection of orthocircular boundary arcs, from the tiling of ID\F_{0} by half fundamental domains, which are separated from 0 by exactly n − 1 other orthocircular boundary arcs.

In order to define the projection P, we make a construction in Ω = \overline{C} \ K and lift it to the disc via π^{−1}. An outline follows. Given any Whitney interval Î ⊂ A_{n}, we must specify which of the intervals Î_{n+1,k} ⊂ A_{n+1} should project via P to subintervals of P (Î). (In terms of the tree described in Section 4, whose vertices are the Whitney intervals in ⋃_{n} A_{n}, we are now specifying the adjacencies between vertices; in other words which pairs of vertices are connected by edges.) Let A_{n,j} be the orthocircular arc containing Î, and let Ω_{n,j} be the half fundamental domain below A_{n,j}. The covering map π maps Î to a Whitney interval J in \overline{IR} \ K; it maps A_{n,j} to the component L of \overline{IR} \ K which contains J = π(Î); it maps Ω_{n,j} to either the upper or lower half plane; and it maps ∂Ω_{n,j} to \overline{IR}. See Figure 3.

To each interval J in the Whitney decomposition of L we associate an interval E_{J} ⊂ \overline{IR} \ L, in such a way that the E_{J} for all J in L have disjoint interiors and their union is \overline{IR} \ L. (The rest of this section contains precise definitions of these intervals E_{J}.) These intervals E_{J} are of a certain form; in particular, for most J the Euclidean length |E_{J}| is a large fixed multiple of |J|, and the endpoints of E_{J} always lie in the Cantor set K.

Let Ĩ be the segment of ∂Ω_{n,j} such that π(Ĩ) = E_{J}. Ĩ lies in the lower part of the boundary of Ω_{n,j}, and its endpoints lie in the unit circle. Let P (Î) be the arc of the unit circle which has the same endpoints as Ĩ and which lies below Ĩ. With this definition the intervals Î_{n+1,k} ⊂ A_{n+1} whose projections P (Î_{n+1,k}) are contained in P (Î) are precisely those intervals Î_{n+1,k} which are contained in Ĩ.

The projections {P (Î_{n,j})}_{j} of the Whitney intervals Î_{n,j} in A_{n} constitute the n^{th} layer H_{n} of the grid of subintervals of the circle. With the definition of P outlined above, each interval P (Î_{n+1,k}) in H_{n+1} is contained in some P (Î_{n,j}) from the previous layer H_{n}. Also, since the intervals E_{J} have pairwise disjoint interiors for all J in each L, the P (Î_{n,j}) at the n^{th} level have pairwise disjoint interiors. Finally, by the remarks in Section 5 above, the part of ∂ID below the orthocircular arcs in A_{n} has full Lebesgue measure in ∂ID for each n ≥ 1, so ⋃_{_{I}⊂A_{n}} P (Î) has full Lebesgue measure in ∂ID for each n ≥ 1. In other

words, the union of the intervals P (Î) in H_{n} covers ∂ID, up to a set of measure zero. Thus the layers H_{n} of intervals form a grid H of subintervals of ∂ID, according to the definition in Section 6.

If R̂ is a segment of the boundary of some half fundamental domain Ω_{n,j}, consisting of a collection of whole Whitney intervals {Î_{n+1,k}}_{k}, possibly together with a subset E of ∂Ω_{n,j} ∩ ∂ID, then we define P (R̂) to be E ∪ ⋃_{k} P (Î_{n+1,k}).

Figure 3. Fan of palm leaves Λ_{J} based at the Whitney intervals J = π(Î) in L; and a preimage in ID of one leaf Λ_{J}.

We now give the details of the construction of the grid H of intervals, in particular defining the intervals E_{J} associated to the Whitney intervals J in \overline{IR} \ K. Let N >> 1 be the same large integer as in the previous section. Until the last part of the paper we regard N as fixed, and we make all our geometric constructions using this fixed value.

Let L be a component of \overline{IR} \ K. We define a decomposition of the upper half plane into infinitely many regions Λ_{J}, indexed by the Whitney intervals J in L. See Figure 3. We refer to these regions as palm leaves based at the Whitney intervals J in L, and to the collection of these regions as a fan of palm leaves, based at L. The boundary of the leaf Λ_{J} based at the Whitney interval J consists of two intervals in \overline{IR}, J itself and E_{J}, together with two non-intersecting semicircles which join the endpoints of J to the endpoints of E_{J} and which meet IR at right angles. We call J the base and E_{J} the tip of the leaf Λ_{J}. The union over all J in L of the intervals E_{J} covers \overline{IR} \ L.

Once the entire construction of the fans of palm leaves is complete, we reflect them in the real axis to obtain analogous fans in the lower half plane.

We distinguish two cases: when the interval L is a component of [0, 1] \ K, and when L = \overline{IR} \ [0, 1]. Case 1: Let L be a component of \overline{IR} \ K which lies in [0, 1] and has length |L| = 3^{−l} for some integer l ≥ 1. Write L = (a, a+3^{−l}). Let K_{l} = [a−3^{−l}, a] and K_{r} = [a+3^{−l}, a+2·3^{−l}] be the closed construction intervals of K of length |K_{l}| = |K_{r}| = |L| = 3^{−l} such that K_{l} is immediately to the left of L and K_{r} is immediately to the right of L. The Whitney intervals in L are enumerated from left to right as …, J_{−N−2}, J_{−N−1}, J_{c}, J_{N+1}, J_{N+2}, … Recall that |J_{±n}| = 3^{−n}|L|, for n ≥ N + 1, and |J_{c}| = (1 − 3^{−N})|L|.

We refer to the Whitney intervals J_{±n} in L such that n ≥ N + 1 as standard intervals. For each standard interval J we define a leaf Λ_{J} based at J such that the tip E_{J} of the leaf has length |E_{J}| = 2 · 3^{N}|J|. We begin with the standard Whitney interval J_{N+1}, which has length |J_{N+1}| = 3^{−N−1}|L| and is in the right half of L.

Let E_{J_{N+1}} be the closed interval which has the same right endpoint as K_{r} and whose length is two-thirds the length of K_{r}. E_{J_{N+1}} consists of a closed construction interval of K and the closure of an adjacent open interval in [0, 1] \ K of the same length. Now |E_{J_{N+1}}| = \frac{2}{3}|K_{r}| = \frac{2}{3}|L| = \frac{2}{3} · 3^{N+1}|J_{N+1}| = 2 · 3^{N}|J_{N+1}|. Join the left endpoint of J_{N+1} to the right endpoint of E_{J_{N+1}} by a semicircle in the upper half plane. Join the right endpoint of J_{N+1} to the left endpoint of E_{J_{N+1}} in the same way. Let the leaf Λ_{J_{N+1}} be the region in the upper half plane bounded by J_{N+1}, E_{J_{N+1}}, and the two semicircles. See Figure 4.

Figure 4. Standard palm leaf Λ_{J} for J = J_{N+1}. Not to scale: |E_{J}| = 2 · 3^{N}|J| and N >> 1.

Consider the next standard Whitney interval J_{N+2}; it has length 3^{−N−2}|L| and is immediately to the right of J_{N+1}. Construct the leaf Λ_{J_{N+2}} and its tip E_{J_{N+2}} exactly as for J_{N+1}: Let E_{J_{N+2}} be the right two-thirds of K_{r} \ E_{J_{N+1}}, and join the endpoints of J_{N+2} to those of E_{J_{N+2}} by semicircles. The leaf Λ_{J_{N+2}} bounded by J_{N+2}, E_{J_{N+2}}, and these semicircles is a copy of Λ_{J_{N+1}}, shrunk by a factor of one-third. The smaller semicircle in the boundary of Λ_{J_{N+1}} is the larger semicircle in the boundary of Λ_{J_{N+2}}. Clearly |E_{J_{N+2}}| = 2 · 3^{N}|J_{N+2}|.

Repeat this construction for the intervals J_{N+k}, k ≥ 1, obtaining a sequence of leaves Λ_{J_{N+k}} based at J_{N+k} which satisfy |E_{J_{N+k}}| = 2 · 3^{N}|J_{N+k}|. The leaves Λ_{J_{N+k}} are all congruent to each other via dilations by powers of three. The intervals E_{J_{N+k}}, k ≥ 1, have disjoint interiors and their union is K_{r}, the closed construction interval of length |L| to the right of L.

Define leaves Λ_{J_{−N−k}}, k ≥ 1, for the standard intervals at the left end of L, by reflecting the leaves Λ_{J_{N+k}} through the perpendicular bisector of L. The leaf Λ_{J_{−N−k}} based at J_{−N−k} is the mirror image of the leaf Λ_{J_{N+k}} based at J_{N+k}. The intervals E_{J_{−N−k}}, k ≥ 1, cover K_{l}, the closed construction interval of length |L| to the left of L.

We refer to the leaves we have just constructed for the standard intervals J_{±(N+k)}, k ≥ 1, as standard leaves, and to their tips as standard E_{J}’s. We also refer to any grid interval I such that J = π(Î) is standard as a standard grid interval. For any standard leaf Λ_{J}, |E_{J}| = 2 · 3^{N}|J|; the endpoints of E_{J} lie in the Cantor set K; and E_{J} consists of a closed construction interval of length 3^{N}|J| together with the closure of an adjacent open interval in \overline{IR} \ K, also of length 3^{N}|J|.

It remains to define the leaf for the central interval J_{c} = J_{−N} ∪ ⋯ ∪ J_{N}. Let E_{J_{c}} = \overline{IR} \ (K_{l} ∪ L ∪ K_{r}). Define the leaf Λ_{J_{c}} to be the region in the upper half plane bounded by J_{c}, E_{J_{c}}, and the two non-intersecting semicircles joining the endpoints of J_{c} to those of E_{J_{c}}. The endpoints of E_{J_{c}} lie in the Cantor set K. Notice that when L = (\frac{1}{3}, \frac{2}{3}), E_{J_{c}} is exactly \overline{IR} \ [0, 1], since the tips of the leaves for the standard intervals in (\frac{1}{3}, \frac{2}{3}) take up all of [0, \frac{1}{3}] ∪ [\frac{2}{3}, 1]. For smaller intervals L, E_{J_{c}} contains \overline{IR} \ [0, 1] and part of [0, 1]. We use the term non-standard to refer to J_{c}, Λ_{J_{c}}, E_{J_{c}}, and any grid interval I such that π(Î) = J_{c}.

To summarize: given an open interval L which is a component of [0, 1] \ K, we have defined a fan of palm leaves Λ_{J} based at the Whitney intervals J in L, such that:

. the Λ_{J}’s tile the upper half plane;

. the tips E_{J} of the leaves are intervals whose union is \overline{IR} \ L;

. the E_{J} have disjoint interiors;

. the endpoints of each E_{J} lie in the Cantor set K;

. |E_{J}| = 2 · 3^{N}|J| for all standard J, i.e. for all but the central interval J_{c} in L; and . each standard E_{J} consists of a closed construction interval of K and the closure of an adjacent open interval in \overline{IR} \ K of the same length.

Case 2: L is the component \overline{IR} \ [0, 1] of \overline{IR} \ K. Then L consists of the large interval J_{∞} = \overline{IR} \ (−σ, 1 + σ) and Whitney intervals \{J_{±n}\}_{n≥N+3} in [−σ, 0) ∪ (1, 1 + σ], where σ = (2 · 3^{N+2})^{−1}.

For the interval J_{∞} = \overline{IR} \ (−σ, 1 + σ), define E_{J_{∞}} = [\frac{1}{9}, \frac{8}{9}]. Join the endpoints of J_{∞} to the endpoints of [\frac{1}{9}, \frac{8}{9}] by non-intersecting semicircles in the upper half plane, and let the leaf Λ_{J_{∞}} be the region in the upper half plane bounded by J_{∞}, E_{J_{∞}}, and the two semicircles. See Figure 5(a).

Figure 5(a). Non-standard palm leaf Λ_{J_{∞}} for J = J_{∞}.

Figure 5(b). Standard palm leaves for J_{1} = J_{N+3}, J_{2} = J_{N+4} in [−σ, 0).

The Whitney intervals in [−σ, 0) are enumerated from left to right as J_{N+3}, J_{N+4}, …, in order of decreasing size. See Figure 5(b). For J_{N+3} = [−σ, −σ/3], let E_{J_{N+3}} = [3^{−3}, 3^{−2}]. Form the leaf Λ_{J_{N+3}} by joining the endpoints of J_{N+3} to those of E_{J_{N+3}} by two non-intersecting semicircles which meet IR at right angles. Then |J_{N+3}| = \frac{2}{3}σ = 3^{−(N+3)} = (2 · 3^{N})^{−1}|E_{J_{N+3}}|. So Λ_{J_{N+3}} and E_{J_{N+3}} are of the standard form.

Similarly, for J_{N+k} = [−σ/3^{k−3}, −σ/3^{k−2}], k ≥ 3, let E_{J_{N+k}} = [3^{−k}, 3^{−k+1}] and define Λ_{J_{N+k}} as usual as the region in the upper half plane bounded by J_{N+k}, E_{J_{N+k}}, and the two non-intersecting semicircles joining the endpoints of J_{N+k} to those of E_{J_{N+k}}. Then |E_{J_{N+k}}| = 2·3^{N}|J_{N+k}| for k ≥ 3; the J_{N+k}’s are standard intervals with standard E_{J_{N+k}}’s and Λ_{J_{N+k}}’s; the E_{J_{N+k}}’s cover [0, \frac{1}{9}], and the Λ_{J_{N+k}}’s tile the half-disc bounded by [−σ, \frac{1}{9}] and the semicircle in the upper half plane which joins −σ to \frac{1}{9}. See Figure 5(b).

The Whitney intervals in (1, 1 + σ] are enumerated from right to left in order of decreasing size, as J_{−(N+3)}, J_{−(N+4)}, … Define leaves Λ_{J_{−(N+k)}} for J_{−(N+k)}, k ≥ 3, by reflecting the leaves Λ_{J_{N+k}} through the line x = \frac{1}{2}. The leaf Λ_{J_{−(N+k)}} based at J_{N+k} is the mirror image of the leaf Λ_{J_{N+k}} based at J_{N+k}. All these leaves and their E_{J_{N+k}}’s are standard. The E_{J_{N+k}}, k ≥ 3, cover [\frac{8}{9}, 1].

In the case L = \overline{IR} \ [0, 1] we have defined standard leaves Λ_{J} for all the Whitney intervals J in L, except for J_{∞} = \overline{IR} \ (−σ, 1 + σ), which has a non-standard leaf. For all intervals J ⊂ L, the endpoints of E_{J} lie in the Cantor set K. The palm leaves have the same properties as those summarized at the end of Case 1.

For each component L of \overline{IR} \ K, reflect the fan of palm leaves based at L through the real axis, obtaining a fan of palm leaves, also based at L, which tiles the lower half plane.

10. Distortion estimates for Whitney intervals

Let Î be a Whitney interval in the upper orthocircular boundary arc A_{n,j} of a half fundamental domain Ω_{n,j}. Let I = P (Î) be the corresponding grid interval, and let Ĩ be the segment of ∂Ω_{n,j}, below A_{n,j}, with the same endpoints as I. The purpose of this section is to show that the Euclidean lengths of Î, Ĩ, and I are comparable to each other, with constants which are uniform for all Î.

We prove a preliminary lemma.

Lemma 10.1. The orthocircular arcs in ⋃_{n} A_{n} are uniformly hyperbolically separated.

Proof. It is sufficient to prove that the components of \overline{IR} \ K are uniformly hyperbolically separated, since these components lift via π^{−1} to ⋃_{n} A_{n}, and the conformal map π^{−1} is a hyperbolic isometry.

Recall that the hyperbolic metric on Ω = \overline{C} \ K, given by λ_{Ω}(w)|dw|, satisfies

\frac{c_{Ω}}{dist (w, K)} ≤ λ_{Ω}(w) ≤ \frac{2}{dist (w, K)}  (10.1)

where c_{Ω} > 0 depends only on the uniformly perfect constant of the Cantor set K.

Let L and L′ be components of \overline{IR} \ K, with |L| ≤ |L′|. Let γ be an arc joining L to L′. Then the Euclidean length of γ is at least |L|, since between L and L′ there is a closed construction interval of K of length at least |L|.

If each point in γ is within distance 2|L| of K, then

ℓ_{hyp}(γ) = ∫_{γ} λ_{Ω}(w)|dw|

≥ c_{Ω} ∫_{γ} \frac{|dw|}{dist (w, K)}

≥ c_{Ω} ∫_{γ} \frac{|dw|}{2|L|}

≥ c_{Ω}/2.

(10.2)

If some point in γ is not within distance 2|L| of K, then there is a segment γ′ of γ of length |L|, which has one endpoint at distance 2|L| from K, and which stays within distance 2|L| of K. Then

ℓ_{hyp}(γ) ≥ ℓ_{hyp}(γ′)

= ∫_{γ′} λ_{Ω}(w) |dw|

≥ c_{Ω} ∫_{γ′} \frac{|dw|}{dist (w, K)}

≥ \frac{c_{Ω}}{2|L|} |L|

= \frac{c_{Ω}}{2} .

(10.3)

Therefore the hyperbolic distance between any two orthocircular arcs in ⋃_{n} A_{n} is at least c_{Ω}/2.

Lemma 10.2. Let I ∈ H be any grid interval. Then |Î| ∼^{c} |Ĩ| ∼^{\frac{π}{2}} |I|, and the constant c > 0 is independent of I.

We split the proof into several sublemmas, showing that |Î| ≤ c |I| for all standard and non-standard grid intervals, and then that |Î| ≥ c |I| for all standard and non-standard grid intervals. As usual, c denotes constants which may change from line to line.

Note that if A_{n,j} is any orthocircular arc, then its length is comparable with constant π/2 to the length of the arc of ∂ID below A_{n,j} which has the same endpoints as A_{n,j}. For each I ∈ H, the endpoints of Ĩ are the endpoints of I, and they lie in ∂ID. Therefore Ĩ consists of a subset of ∂ID together with a union of whole orthocircular arcs, and so |Ĩ| ∼ |I| with constant π/2.

Lemma 10.3. |Î| ≤ c |I| for all standard intervals I ∈ H.

Proof. Let A_{n,j} be the orthocircular arc containing Î, and let Ω_{n,j} be the half fundamental domain below A_{n,j}. Without loss of generality, assume that π(Ω_{n,j}) is the upper half plane. Let F be the fundamental domain which consists of Ω_{n,j}, the arc A_{n,j}, and the half fundamental domain immediately above A_{n,j}. Let L = π(A_{n,j}). Then π(F) is the union of the open upper and lower half planes together with the component L of \overline{IR} \ K; its boundary is ∂π(F) = \overline{IR} \ L.

Let J = π(Î); J is a standard Whitney interval in \overline{IR} \ K. As usual, let E_{J} be the tip of the standard leaf based at J; E_{J} = π(Ĩ). See Figure 6.

Figure 6. Standard grid interval I; definition of z_{J}.

Let w_{J} be the midpoint of J and let w_{I} be the point in Î such that π(w_{I}) = w_{J}. Let α ≤ |J|/2 be a small number, which will be fixed later. Let z_{J} be the point in the upper half plane, directly above the midpoint w_{J} of J, such that |z_{J} − w_{J}| = α |J|. Let z_{I} be the point in Ω_{n,j} such that π(z_{I}) = z_{J}.

The hyperbolic distance from z_{I} to w_{I} is bounded away from zero and infinity, uniformly for all Î. The Euclidean ball of radius |J|/2 centred at w_{J}, which contains z_{J} and J, lies in the set {z | dist (J, K) ≤ dist (z, K) ≤ 3 dist (J, K)}, since |J| = 2 dist (J, K). It follows that

d_{hyp}(z_{I}, w_{I}) = d_{hyp}(z_{J}, w_{J})

≤ |z_{J} − w_{J}| \frac{2}{dist (J, K)}

= α |J| \frac{2}{|J|/2}

= 4α.

(10.4)

Also,

d_{hyp}(z_{J}, w_{J}) ≥ |z_{J} − w_{J}| \frac{c_{Ω}}{3 dist (J, K)}

= \frac{2}{3} c_{Ω}α.

(10.5)

So \frac{2}{3} c_{Ω}α ≤ d_{hyp}(z_{I}, w_{I}) ≤ 4α for all I. This implies that the Euclidean distance from z_{I} to w_{I} is less than c(α) (1 − |w_{I}|), where c(α) depends only on α and decreases to zero with α.

Since z_{I} and w_{I} are close in the hyperbolic metric, harmonic measure on ∂F with basepoint z_{I} is close to harmonic measure on ∂F with basepoint w_{I}. Let u(z) = ω(z, E, F), where E is any Borel subset of ∂F. The function u(z) is harmonic in F. The hyperbolic distance from w_{I} to ∂F is at least c_{Ω}/2, by Lemma 10.1. The hyperbolic ball of radius c_{Ω}/2 centred at w_{I} contains a Euclidean ball of radius c′_{Ω} (1 − |w_{I}|), centred at w_{I}, where c′_{Ω} depends only on c_{Ω}. The Euclidean distance from z_{I} to w_{I} decreases to zero with the hyperbolic distance from z_{I} to w_{I}. Choose the number α, where |z_{I} − w_{I}| = α |J| and |z_{I} − w_{I}| ≤ c(α) (1 − |w_{I}|), small enough that the Euclidean distance from z_{I} to w_{I} is less than (c′_{Ω}/3) (1 − |w_{I}|). By Harnack’s inequality,

\frac{1}{2}u(w_{I}) ≤ u(z_{I}) ≤ 2u(w_{I}).  (10.6)

In particular, setting E = Ĩ, we obtain ω(z_{I}, Ĩ, F) ∼^{2} ω(w_{I}, Ĩ, F) for all I.

After these preliminaries, we now use estimates on harmonic measure based at z_{I} and at w_{I} to show that |Ĩ| ≤ c (1 − |w_{I}|). This is sufficient to establish Lemma 10.3, since |Î| ∼ 1 − |w_{I}| with a constant independent of I (by Lemma 8.1).

For standard intervals J, the shapes of the leaves Λ_{J} and the position of z_{J} within Λ_{J} are all identical, up to dilations and reflections. Specifically, |E_{J}| = 2 · 3^{N}|J|; the distance from J to E_{J} is also a fixed multiple of |J|; and z_{J} is always at height α|J| above the midpoint of J. Therefore the harmonic measure of E_{J} in the upper half plane U, taken with respect to the basepoint z_{J}, is constant for all standard J: ω(z_{J}, E_{J}, U) = c > 0.

Let F_{I} be the unit disc without the fundamental domains below Ĩ. Let γ be the orthocircular arc joining the endpoints of Ĩ, and let F_{γ} be the disc without the “bite” below γ. So Ω_{n,j} ⊂ F ⊂ F_{γ} ⊂ F_{I}. Then

\begin{aligned}c & = ω(z_{J}, E_{J}, U) \\ & = ω(z_{I}, Ĩ, Ω_{n,j}) \\ & ≤ ω(z_{I}, Ĩ, F) \\ & ≤ 2 ω(w_{I}, Ĩ, F) \\ & ≤ 2 ω(w_{I}, Ĩ, F_{I}) \\ & = 2 [1 − ω(w_{I}, ∂F_{I}\ Ĩ, F_{I})] \\ & ≤ 2 [1 − ω(w_{I}, ∂F_{I}\ Ĩ, F_{γ})] \\ & = 2 [1 − ω(w_{I}, ∂F_{γ} \ γ, F_{γ})] \\ & = 2 ω(w_{I}, γ, F_{γ}). \end{aligned}  (10.7)

Figure 7. Möbius transformations τ, σ : ID → ID.

We have used the conformal invariance of harmonic measure; the observations made above; and the fact that harmonic measure ω(z, E, Ω) is monotonic in the domain Ω: if z ∈ Ω ⊂ Ω′ and E ⊂ ∂Ω ∩ ∂Ω′, then ω(z, E, Ω) ≤ ω(z, E, Ω′).

We assumed here that w_{I} was not on or below the orthocircular arc γ which joins the endpoints of I. Otherwise, |I| ≥ |γ|/2 ≥ 1 − |w_{I}| and we are done.

Let τ : ID → ID be the Möbius transformation τ(z) = (z − w_{I})/(1 − \overline{w}_{I}z) which takes w_{I} to 0. See Figure 7. Then

\begin{aligned}ω(w_{I}, γ, F_{γ}) & = ω(0, τ(γ), τ(F_{γ})) \\ & = 2 ω(0, τ(I), ID) \\ & = π^{−1}|τ(I)|. \end{aligned}  (10.8)

The second step can be justified by mapping τ(γ) to a diameter of ID via a Möbius transformation σ : ID → ID (Figure 7). Consider the probability that a Brownian traveller from σ(0) in ID first hits ∂ID somewhere on σ ∘ τ(I). By symmetry, this is exactly half the probability that the traveller first hits the boundary of the half disc σ ∘ τ(F_{γ}) somewhere on the diameter σ ∘ τ(γ). That is,

\begin{aligned}ω(0, τ(I), ID) & = ω(σ(0), σ ∘ τ(I), ID) \\ & = 2^{−1} ω(σ(0), σ ∘ τ(γ), σ ∘ τ(F_{γ})) \\ & = 2^{−1} ω(0, τ(γ), τ(F_{γ})), \end{aligned}  (10.9)

as required.

We have shown that c ≤ 2 ω(w_{I}, γ, F_{γ})| ≤ 2π^{−1}|τ(γ)|. Now τ′(z) = (1 − |w_{I}|^{2})/(1 − w_{I}z)^{2}, and so

\begin{aligned}|τ(I)| & ≤ |I| max_{z∈I} |τ′(z)| \\ & ≤ |I| \frac{2}{1 − |w_{I}|}. \end{aligned}  (10.10)

It follows that |I| ≥ c (1 − |w_{I}|), where the constant c is independent of I, and therefore |I| ≥ c |Î|, with c independent of I.

Lemma 10.4. |Î| ≤ c |I| for all non-standard intervals I ∈ H.

Proof. We use the same proof as for Lemma 10.3 above. The key point is to choose a suitable point z_{J} in the upper half plane U for each J.

Let J be a non-standard Whitney interval. J is of one of two types: either J is the central Whitney interval J_{c} in an open component L of [0, 1] \ K, or J = J_{∞}. In the first case, define z_{J} as in the proof of Lemma 10.3, with the same α. See Figure 8(a). The leaves Λ_{J} for these intervals are identical up to dilations, so the harmonic measure of E_{J} in U, taken from z_{J}, is uniformly bounded below for all these J. (The bound is not necessarily the same as that in Lemma 10.3.) Also, z_{J} is close enough to w_{J} in the hyperbolic metric that the other estimates on harmonic measure hold with the same constants as in Lemma 10.3.

Figure 8(a). Definition of z_{J} for non-standard J = J_{c}.

For J = J_{∞}, fix a point w_{J} ∈ J_{∞} which is far from [0, 1]. See Figure 8(b).

Figure 8(b). Definition of z_{J} for non-standard J = J_{∞}.

Let z_{J} be a point in U, directly above w_{J}, which is close to w_{J} in the hyperbolic metric. Then the harmonic measure of E_{J} in U, taken from z_{J}, is a positive constant, not necessarily the same as the lower bound in the earlier cases. Choose z_{J} sufficiently close to w_{J} that the other harmonic measure estimates hold with the same constants as in the proof of Lemma 10.3.

Lemma 10.5. |Î| ≥ c |I| for all standard intervals I ∈ H.

Proof. We use the same notation as in the proof of Lemma 10.3. In particular, w_{I} is the point in Î such that w_{J} = π(w_{I}) is the midpoint of the Whitney interval J = π(Î). Without loss of generality assume that w_{I} is in the left half of the orthocircular arc A_{n,j} containing Î. Let a and b be the left and right endpoints, respectively, of I. See Figure 9.

It is enough to show that |b−w_{I}| ≤ c (1−|w_{I}|) with a constant c independent of I. For I is trapped between b and the left endpoint a′ of A_{n,j}, which satisfies |a′−w_{I}| ≤ 2 (1−|w_{I}|). So then

\begin{aligned}|I| & ≤ |a′ − b| \\ & ≤ |a′ − w_{I}| + |w_{I} − b| \\ & ≤ c (1 − |w_{I}|) \\ & ≤ c |Î| \end{aligned}  (10.11)

(using Lemma 8.1), with a constant c independent of I.

Let B be the shorter segment of ∂F between the right endpoint b of I and the right endpoint b′ of A_{n,j}. See Figure 9. The following calculation is justified below:

|w_{I} − b| ≤ c dist (w_{I}, B)

≤ c \frac{dist (w_{I}, ∂F)}{ω(w_{I}, B, F)^{2}}

≤ c dist (w_{I}, ∂F) ≤ c (1 − |w_{I}|),

(10.12)

where dist denotes Euclidean distance, and c denotes constants independent of I.

Figure 9. Ĩ is near Î; definition of B.

The Euclidean distance from w_{I} to B is |w_{I} − b|, unless there is a point in an orthocircular arc in B which lies closer to w_{I} than b does. If |b − w_{I}| ≤ 10 (1 − |w_{I}|) the lemma is proved. If not, B is far to the right of w_{I} and the distance from w_{I} to any point in B is at least \frac{1}{2}|w_{I} − b|, justifying the first line of (10.12).

By Beurling’s Lemma [A], there is a constant C > 0 such that for each point z_{0} in F, and for all M > 0,

ω(z_{0}, \{z | |z − z_{0}| ≥ M dist (z_{0}, ∂F)\}, F) ≤ CM^{− \frac{1}{2}} .  (10.13)

Setting z_{0} = w_{I} and M = dist (w_{I}, B)/dist (w_{I}, ∂F), we find

ω(w_{I}, B, F) ≤ ω(w_{I}, {z | |z − w_{I}| ≥ dist (z_{0}, B)}, F)

≤ C [\frac{dist (w_{I}, ∂F)}{dist(w_{I}, B)}]^{\frac{1}{2}} ,

(10.14)

and so

dist (w_{I}, B) ≤ C \frac{dist (w_{I}, ∂F)}{ω(w_{I}, B, F)^{2}},  (10.15)

which is the second line of (10.12).

The harmonic measure of B in F, measured from w_{I}, is bounded below by a positive constant independent of I. With z_{I} as in the proof of Lemma 10.3,

\begin{aligned}ω(w_{I}, B, F) & ≥ 2^{−1} ω(z_{I}, B, F) \\ & = 2^{−1} ω(z_{J}, π(B), π(F)) \\ & ≥ 2^{−1} ω(z_{J}, π(B), U), \end{aligned}  (10.16)

by (10.6) and since π(F) contains U. π(B) is one of the two components of \overline{IR} \ (L ∪ E_{J}), where L is the open component of \overline{IR} \ K which contains J. Again, since the picture of U with J, z_{J}, and E_{J} is invariant up to dilations and reflections for all standard J, the harmonic measure of π(B) in U, taken from z_{J}, is bounded below by a constant c > 0 independent of J, for all standard J. This justifies the third line of (10.12). The last line of (10.12) is true because Î is a Whitney interval in A_{n,j}.

Lemma 10.6. |Î| ≥ c |I| for all non-standard intervals I ∈ H.

Proof. We use the same method as for Lemma 10.5. Define z_{J} for each non-standard J as in the proof of Lemma 10.4. The interval π(B) is one of the two components of \overline{IR}\(L∪E_{J}). See Figures 8(a) and 8(b). By the usual arguments we may conclude that the harmonic measure of B in F, taken from w_{I}, is bounded below by a positive constant independent of I. The rest of the proof applies without change.

This completes the proof of Lemma 10.2: |Î| ∼ |Ĩ| ∼ |I| for all grid intervals I ∈ H, with constants which are independent of I.

11. Definition of the density functions F_{n} ⋯ F_{1}

In this section we define the functions F_{n} whose products F_{n} ⋯ F_{1} give the densities of the measures μ_{n}. Here we define the F_{n} on all standard grid intervals I ∈ H_{n−1}, n ≥ 1, up to the values of certain parameters, Q and ε, which will be fixed later. We show that these functions F_{n} are (δ, η)-suitable for each standard I ∈ H_{n−1}, with constants δ and η independent of I and n. For non-standard intervals I ∈ H_{n−1}, we define the functions F_{n} and prove that they are (δ, η)-suitable in Section 14.

We first prove a lemma giving a rough estimate of the distortion caused by the composition P ∘ π^{−1} of a branch of the inverse π^{−1} of the covering map with the almost radial projection P of the disc onto the circle.

Lemma 11.1. For each standard Whitney interval J in \overline{IR} \ K, let S_{J} be a segment of E_{J}, and let A_{1} and A_{2} be the components (possibly empty) of E_{J} \ S_{J}. Suppose that |S_{J}|/|E_{J}| = c_{1}, |A_{1}|/|E_{J}| = c_{2}, and |A_{2}|/|E_{J}| = 1 − c_{1} − c_{2}, where c_{1} and c_{2} are constants independent of J. Suppose also that S_{J} is a union of whole Whitney intervals, possibly together with a subset of K. Let I be any standard grid interval such that π(Î) = J, and let R̂_{I} be the segment of Î such that π(R̂_{I}) = S_{J}. Let R_{I} = P (R̂_{I}) = P ∘ π^{−1}(S_{J}). Then |R_{I}| ≥ c |I|, where c is a constant independent of I and J.

Proof. Without loss of generality, assume that π maps the half fundamental domain below the orthocircular arc containing Î to the upper half plane. As in the proof of Lemma 10.3, let w_{J} be the midpoint of J, and let z_{J} be the point in the upper half plane, directly above w_{J}, such that |z_{J} − w_{J}| = α|J|, where α is a small fixed number independent of J. Let w_{I} be the point in Î such that π(w_{I}) = w_{J}. See Figure 6 in Section 10.

The harmonic measure of S_{J} in the upper half plane, taken from z_{J}, is the same for all J, since S_{J} is always in the same place in E_{J}. The proof of Lemma 10.3, applied to S_{J} instead of to E_{J}, shows that |R_{I}| ≥ c (1 − |w_{I}|), where c is a constant depending on c_{1} and c_{2}. By Lemmas 8.1 and 10.2, 1−|w_{I}| is comparable to |I| with a constant independent of I and J, which proves the lemma.

We have shown that for all standard intervals I ∈ H, if S_{J} is a fixed segment of E_{J}, where J = π(Î), then |R_{I}|/|I| = |P ∘ π^{−1}(S_{J})|/|P ∘ π^{−1}(E_{J})| is uniformly bounded away from zero. The same result holds for non-standard intervals I, if S_{J} is defined so that its length is a fixed multiple of |J| and so that it is always in the same position in E_{J}.

On each interval I in the grid H of subintervals of the unit circle we define a function F which is (δ, η)-suitable for I, where the constants δ and η are independent of I. Recall from Section 6 the definition of (δ, η)-suitable: F has mean value one on I; 0 < δ ≤ F (x) ≤ 1/δ on I; and F ≡ 1 on subintervals I_{l} and I_{r} of I, at the left and right ends respectively of I, such that |I_{l}| ≥ η|I| and |I_{r}| ≥ η|I|.

The idea is as follows. Divide each E_{J} into five segments S_{1}, …, S_{5} whose lengths are prescribed fractions of |E_{J}|. Pull these back to Ĩ via π^{−1}, then project them via P to subintervals R_{j} = P ∘ π^{−1}(S_{j}), 1 ≤ j ≤ 5, of I. In Lemma 11.1 we gave bounds on the distortion in length caused by P ∘ π^{−1}. Each ratio |R_{j}|/|I| is uniformly bounded away from zero and infinity for all I. Define F to be constant on each R_{j}, in such a way that F is large on R_{3}, identically equal to one on R_{1} and R_{5}, and small enough on R_{2} ∪ R_{4} to ensure that F has mean value one on I. See Figure 10. We show that F is uniformly bounded away from zero and infinity for all I.

Figure 10. Graph of F; α = (1 − ε) |I|/|R_{3}|.

Now we define the segments S_{j} and R_{j} for a standard grid interval I. Let Î be the Whitney interval in ⋃_{n} A_{n} such that I = P (Î), and let J = π(Î). Without loss of generality we describe the case when E_{J} is to the right of J. Divide E_{J} into five segments S_{j}, 1 ≤ j ≤ 5, numbered from left to right, as follows. The left half of E_{J} is an open component of \overline{IR} \ K of length 3^{N}|J|. Let S_{3} be the central Whitney interval J_{c} in this open interval. The length of S_{3} is |S_{3}| = \frac{1}{2}(1 − 3^{−N})|E_{J}|.

Let Q be a large integer; its value will be fixed below. Let S_{1} be the interval with the same left endpoint as E_{J} and with length |S_{1}| = \frac{3}{2}3^{−Q}|J|. S_{1} is a union of whole Whitney intervals. Let S_{5} be the interval with the same right endpoint as E_{J} and with length |S_{5}| = 3^{−Q}|J|. S_{5} is a closed construction interval of the Cantor set K. Let S_{2} be the interval between S_{1} and S_{3}, and let S_{4} be the interval between S_{3} and S_{5}.

Now pull back these intervals S_{j} to Ĩ via the appropriate branch of π^{−1}. Then project them to I via P, obtaining five intervals R_{j} = P ∘ π^{−1}(S_{j}), 1 ≤ j ≤ 5, whose union is I. By Lemma 11.1, there are positive constants c_{j}, 1 ≤ j ≤ 5, such that |R_{j}| ≥ c_{j} |I| for all standard intervals I. These constants are less than one; they depend on N and Q but are independent of J and I.

Let ε be a positive number, small enough that (1 − ε)|I|/|R_{3}| > 1; its exact value will be fixed below. Define a step function F on I by

F (x) = \{ \begin{matrix}1, & x ∈ R_{1} ∪ R_{5}; \\ (1 − ε)\frac{|I|}{|R_{3}|}, & x ∈ R_{3}; \\ δ, & x ∈ R_{2} ∪ R_{4}; \end{matrix}  (11.1)

where δ = (ε|I| − |R_{1} ∪ R_{5}|)/|R_{2} ∪ R_{4}| is chosen so that \frac{1}{|I|} ∫_{I} F = 1. See Figure 10.

This function F takes its maximum on R_{3}, where F (x) ≡ (1 − ε)|I|/|R_{3}| ≤ 1/c_{3}; this upper bound is independent of I. The subintervals R_{1} and R_{5} at each end of I, on which F ≡ 1, each have length at least η|I|, where η = min (c_{1}, c_{5}) is independent of I.

If dμ_{n} = F_{n} ⋯ F_{1} dx, and each F_{m} is defined on each I ∈ H_{m−1}, m ≥ 1, according to (11.1), then most of the mass μ_{n−1}(I) is concentrated onto the subinterval R_{3} of I:

\begin{aligned}μ_{n}(R_{3}) & = ∫_{R_{3}} F_{n}(x) dμ_{n−1}(x) \\ & = (1 − ε)\frac{|I|}{|R_{3}|} μ_{n−1}(R_{3}) \\ & = (1 − ε)\frac{|I|}{|R_{3}|} · \frac{|R_{3}|}{|I|} μ_{n−1}(I) \\ & = (1 − ε) μ_{n−1}(I). \end{aligned}  (11.2)

The third step is valid because μ_{n−1} is a constant multiple of Lebesgue measure on I.

To show that the function F defined in (11.1) is (δ_{0}, η)-suitable for I, with constants δ_{0} and η independent of I, it only remains to give a uniform lower bound for the value δ of F on R_{2} ∪ R_{4}. Observe that

δ = \frac{ε|I| − |R_{1} ∪ R_{5}|}{|R_{2} ∪ R_{4}|} = (ε − \frac{|R_{1} ∪ R_{5}|}{|I|}) \frac{|I|}{|R_{2} ∪ R_{4}|} ≥ ε − \frac{|R_{1} ∪ R_{5}|}{|I|}.

We would like to ensure that |R_{1} ∪ R_{5}|/|I| ≤ ε/2, say, by choosing |S_{1} ∪ S_{5}|/|E_{J}| sufficiently small (that is, by choosing Q sufficiently large). Unfortunately, an application of Lemma 11.1 to S_{J} = E_{J} \ (S_{1} ∪ S_{5}) does not guarantee that this can be done, since the lemma does not give a good estimate on the size of the constant c, nor on its behaviour as c_{1} → 0. However, the sharper estimate in Lemma 13.1 below, based on the A_{∞}-equivalence of harmonic measure and arclength on the boundary of a chord-arc domain, does imply that |R_{1} ∪ R_{5}|/|I| goes to zero with |S_{1} ∪ S_{5}|/|E_{J}|. Therefore, we may choose a large Q so that |S_{1} ∪ S_{5}|/|E_{J}| = \frac{5}{2} · \frac{1}{2} · 3^{−Q−N} is sufficiently small that |R_{1} ∪ R_{5}|/|I| ≤ ε/2. Then δ ≥ ε/2.

We have shown that the function F defined in (11.1) is (δ_{0}, η)-suitable for I, where the constants δ_{0} and η depend on ε, N, and Q, but are independent of I, for all standard grid intervals I ∈ H.

In Section 14 we define the function F for non-standard intervals I ∈ H, and show that it is (δ_{0}, η)-suitable for those I.

12. Definition of auxiliary functions X_{i}

Let I_{0} be an interval from the l^{th} layer H_{l} of the grid, such that π(Î_{0}) is not J_{∞}. For each point x ∈ I_{0} and for each i ≥ 0, let I_{i}(x) be the unique interval in H_{l+i} which contains x. The intervals I_{i}(x) are nested: I_{0}(x) ⊃ I_{1}(x) ⊃ I_{2}(x) ⊃ ⋯ ⊃ I_{i}(x) ⊃ ⋯ ∋ x. Let Î_{i}(x) be the Whitney interval in the disc such that P (Î_{i}(x)) = I_{i}(x). Let J_{i}(x) = π(Î_{i}(x)). We have associated to the point x ∈ I_{0} ⊂ ∂ID a sequence {J_{i}(x)}_{i≥0} of Whitney intervals in \overline{IR} \ K.

We now define auxiliary functions X_{i}(x) which keep track of the lengths |J_{i}(x)| of these intervals. Make the convention that

|J_{c}| = |L|/3  (12.1)

when J_{c} is the non-standard central Whitney interval in a component L of [0, 1] \ K. Let

X_{1}(x) = log_{3} [\frac{|J_{1}(x)|}{|J_{0}|}]  (12.2)

for x ∈ I_{0}. For i ≥ 2, let

X_{i}(x) = \{ \begin{matrix}log_{3} [\frac{|J_{i}(x)|}{|J_{i−1}(x)|}] , & if J_{1}(x), …, J_{i−1}(x) ≠ J_{∞}; \\ 1, & otherwise, \end{matrix}  (12.3)

for x ∈ I_{0} and i ≥ 1. Let

S_{k}(x) = ∑_{i=1}^{k} X_{i}(x),  (12.4)

for k ≥ 1. If none of J_{1}(x), …, J_{k−1}(x) is the large interval J_{∞} = \overline{IR} \ (−σ, 1 + σ), then S_{k}(x) = log_{3}(|J_{k}(x)|/|J_{0}|). Note that X_{i} and S_{k} are integer-valued.

Let V (I_{0}) denote the collection of those grid intervals I in I_{0} which satisfy π(Î) = J_{∞}, and which are maximal with respect to this property. In other words, V (I_{0}) is the collection of the maximal grid intervals I in I_{0} which are images of J_{∞} under branches of P ∘ π^{−1}.

Observe that V (I_{0}) = {x ∈ I_{0} | S_{k}(x) → +∞}. For if x is contained in some interval I = I_{N}(x), say, which is in V (I_{0}), then X_{i}(x) = 1 for all i ≥ N, and so S_{k}(x) tends to infinity. And if x is not contained in any interval I in V (I_{0}), then no J_{i}(x) is J_{∞}, and so S_{k}(x) remains bounded above, by log_{3}[(9|J_{0}|)^{−1}], for all k. (Recall that J_{0} = π(Î_{0}) is fixed in this discussion, and that the largest Whitney interval other than J_{∞} is the J_{c} in (\frac{1}{3}, \frac{2}{3}), which has length 1/9 by our convention.)

In the remainder of the paper we show that for appropriate choices of the parameters in the definitions of the functions F_{n}, the measures μ_{n} given by dμ_{n} = F_{n} ⋯ F_{1} dx converge to a measure μ whose restriction to I_{0} is supported on V (I_{0}). In Sections 13 and 14 we show that for all grid intervals I ⊂ I_{0}, the mean of X_{i} on I (where i is such that J_{i−1} = π(Î)) with respect to μ is uniformly large, and the second moment is uniformly small. In Section 15 we conclude that S_{k} → +∞ a.e. (dμ) on I_{0}; that is, μ(V (I_{0})) = μ(I_{0}).

Finally, also in Section 15, we observe that this implies the existence of a doubling measure μ supported on the set S of points which lie in infinitely many grid intervals corresponding to J_{∞}. Heuristically, the grid intervals corresponding to J_{∞} are near preimages Î_{∞} of J_{∞} under π. Each preimage Î_{∞} contains an orbit point g(0), g ∈ G, since the covering map π is normalized so that the orbit of 0 is π^{−1}(∞), and ∞ ∈ J_{∞}. We prove that S is contained in the conical limit set of G, which completes the proof of Theorem 1.2.

13. Estimates EX_{i} ≥ c_{1} and EX_{i}^{2} ≤ c_{2} for standard intervals

The main result of this section (Lemma 13.4) is that on each standard grid interval I ∈ H, the function X_{i} satisfies μ(I)^{−1} ∫_{I} X_{i} dμ ≥ c_{1} and μ(I)^{−1} ∫_{I} X_{i}^{2} dμ ≤ c_{2}, where c_{1} and c_{2} are positive constants independent of I. Here X_{i} = log_{3}(|J_{i}(x)|/|J_{i−1}(x)|) is the auxiliary function, defined in Section 12, for a particle which has not yet reached J_{∞} and which makes its i^{th} jump from J = π(Î). We begin by establishing an estimate, sharper than that in Section 11, on the distortion in length caused by the map P ∘ π^{−1}.

As usual, for a standard grid interval I with I = P (Î), let J = π(Î), and let E_{J} = π(Ĩ) be the tip of the leaf Λ_{J} based at J. See Figure 11.

Figure 11. B is in I; π(B̂) is in E_{J}.

Lemma 13.1. Let I ∈ H_{n−1}, n ≥ 1, be a standard grid interval. Let B ⊂ I be a union of grid intervals from H_{n}. Let B̂ be the subset of Ĩ such that P (B̂) = B. There are positive constants c and α such that

\frac{|B|}{|I|} ≤ c [\frac{|π(B̂)|}{|E_{J}|}]^{α} .  (13.1)

The constants c and α are independent of B, I, and n, but they depend on the large number N such that |E_{J}| = 2 · 3^{N}|J| for standard intervals.

To prove this lemma, we use Lemmas 13.2 and 13.3 below and the following remarks. A Jordan curve Γ is said to be chord-arc if for all points x and y on Γ the Euclidean

length |x − y| of the chord between x and y is comparable, with a uniform constant, to the Euclidean arclength of the shorter arc of Γ between x and y. In other words, there is a constant c > 0 such that

|x − y| ≤ ℓ_{Γ}(x, y) ≤ c |x − y|  (13.2)

for all x, y on Γ. A domain D is a chord-arc domain if its boundary is a chord-arc curve. The leaves Λ_{J} are chord-arc domains with chord-arc constant independent of J. If D is a bounded chord-arc domain, then arclength on its boundary ∂D and harmonic

measure based at any point z in D are A_{∞}-equivalent. This means that there are positive constants c_{1}, c_{2}, α_{1}, and α_{2} such that whenever S is a segment of ∂D and E is a Borel subset of S, then

\frac{|E|}{|S|} ≤ c_{1} [\frac{ω(z, E, D)}{ω(z, S, D)}]^{α_{1}} and \frac{ω(z, E, D)}{ω(z, S, D)} ≤ c_{2} [\frac{|E|}{|S|}]^{α_{2}} .  (13.3)

See [JK]. Moreover, if D is bounded, then for all points z ∈ D which satisfy

dist (z, ∂D) ≥ C diam (D),  (13.4)

the constants c_{1}, c_{2}, α_{1}, and α_{2} depend only on C and the chord-arc constant of ∂D.

In the situation of Lemma 13.1, consider the domain D = π^{−1}(Λ_{J}) whose boundary consists of Î, Ĩ, and two arcs γ_{1} and γ_{2} which project via π to the two semicircles in the boundary of the leaf Λ_{J}. We prove (Lemma 13.2) that ∂D is chord-arc, with a constant independent of I, and (Lemma 13.3) that the inequality (13.4) holds, with a constant independent of I, for the point z_{I} ∈ D defined in Section 10. Then we prove Lemma 13.1.

Lemma 13.2. The boundary ∂D of D is chord-arc, with a chord-arc constant which is independent of I.

Proof. We follow the proof of a similar result by González [G]. The boundary of D consists of Î, γ_{1}, γ_{2}, the orthocircular arcs in Ĩ, and the subset Ĩ ∩ ∂ID of the unit circle. It is sufficient to show that:

1. Wherever two of Î, γ_{1}, γ_{2}, and Ĩ meet, the angle they form is bounded below by some θ_{0} > 0 which is independent of I; and

2. The components of ∂Λ_{J} \K (which are J = π(Î); the semicircles π(γ_{1}) and π(γ_{2}); and the open intervals in E_{J} \ K) are chord-arc, in the hyperbolic metric, with a chord-arc constant which is independent of I.

Then the components of ∂D \ ∂ID are also chord-arc in the hyperbolic metric; they are chord-arc in the Euclidean metric; and the whole boundary ∂D is chord-arc in the Euclidean metric with a constant depending only on the chord-arc constant of Λ_{J}.

1. Î meets γ_{1} and γ_{2} at right angles, because J meets π(γ_{1}) and π(γ_{2}) at right angles and π^{−1} is conformal.

Figure 12. Λ_{J} and D = π^{−1}(Λ_{J}) are chord-arc domains.

Let γ_{1} be the arc in ∂D such that π(γ_{1}) is the smaller semicircle in ∂Λ_{J}, and let A be the orthocircular arc in the lower part of ∂D which meets γ_{1}. See Figure 12. We show that the hyperbolic distance between any two points z_{1} ∈ γ_{1} and z_{2} ∈ A is uniformly bounded away from zero; this implies that γ_{1} and A do not form a cusp at their common endpoint but make a positive angle there. Take a path from π(z_{1}) to π(z_{2}) in Λ_{J}. Let γ be the segment of this path which starts at π(z_{1}) and has length Im(z_{1})/2. Then

d_{hyp}(z_{1}, z_{2}) = d_{hyp}(π(z_{1}), π(z_{2}))

≥ c_{Ω} ∫_{γ} \frac{|dw|}{dist (w, K)}

≥ c_{Ω} \frac{1}{\frac{3}{2} Im π(z_{1})} · \frac{Im π(z_{1})}{2}

≥ c_{Ω}/3.

(13.5)

So the angle formed by γ_{1} and A is bounded below by some θ_{0} > 0, independent of I.

The same reasoning applied to γ_{2} shows that γ_{2} makes a positive angle, uniformly bounded below, with ∂ID.

2. J and the open intervals in E_{J} \ K are geodesic arcs in Ω = \overline{C} \ K, so they are chord-arc in the hyperbolic metric. To show that the semicircle π(γ_{1}) is hyperbolically chord-arc, take points z and ζ in π(γ_{1}). Let γ be the segment of π(γ_{1}) between z and ζ, and let L be the chord between z and ζ. It is enough to consider z and ζ in the part of π(γ_{1}) near (in the Euclidean metric) to an endpoint of π(γ_{1}). Then γ is almost vertical, and

ℓ_{hyp}(γ) ∼ ∫_{γ} \frac{|dw|}{dist (w, K)} ∼ ∫_{γ} \frac{|dw|}{Im w}  (13.6)

is comparable to

ℓ_{hyp}(L) ∼ ∫_{L} \frac{|dw|}{dist (w, K)} ∼ ∫_{L} \frac{|dw|}{Im w}  (13.7)

with a uniform constant.

This completes the proof of Lemma 13.2.

Lemma 13.3. For each standard grid interval I ∈ H, let z_{I} be the point in D, near Î, defined in Section 10. There is a constant C > 0 independent of I such that

dist (z_{I}, ∂D) ≥ C diam (D).  (13.8)

Proof. We compare both sides of (13.8) to 1 − |w_{I}|, where w_{I} is the point in Î ⊂ ∂D, near z_{I}, defined in Section 10. (See Figure 6.)

The hyperbolic distance from z_{I} to ∂D is uniformly bounded away from zero. To see this, consider a geodesic arc from z_{J} = π(z_{I}) to ∂Λ_{J}. Let γ be the segment of this geodesic which has z_{J} as one endpoint and which has Euclidean length α|J|/2. Then

d_{hyp}(z_{I}, ∂D) = d_{hyp}(π(z_{I}), π(∂D))

= d_{hyp}(z_{J}, ∂Λ_{J})

≥ c_{Ω} ∫_{γ} \frac{|dz|}{dist (z, K)}

≥ c_{Ω} · \frac{1}{\frac{3}{2}α|J| + \frac{1}{2}|J|} · \frac{α|J|}{2}

= c (Ω, α).

(13.9)

The hyperbolic ball of radius c (Ω, α) centred at z_{I} contains a Euclidean ball of radius c′(1 − |z_{I}|), centred at z_{I}. Here c′ is independent of I. Since (1 − |z_{I}|) ∼ (1 − |w_{I}|), with a constant independent of I, we conclude that

dist (z_{I}, ∂D) ≥ c (1 − |w_{I}|),  (13.10)

where c is independent of I.

To estimate diam (D), it is sufficient to show that

dist (x, Î) ≤ c (1 − |w_{I}|)  (13.11)

for all x ∈ ∂D. For if x and y are in ∂D, then

|x − y| ≤ dist (x, Î) + dist (y, Î) + |Î|,  (13.12)

and we saw in Section 10 that |Î| ∼ 1 − |w_{I}|. We showed (13.11) for x ∈ Ĩ, in the proof of Lemma 10.5. We prove (13.11) for x ∈ γ_{1}; the same argument works for x ∈ γ_{2}.

Let a and â be the endpoints of γ_{1} which lie in I and Î respectively. Using the fact that γ_{1} is chord-arc,

dist (x, Î) ≤ |x − â|
≤ ℓ_{γ_{1}}(x, â)
≤ ℓ_{γ_{1}}(a, â)
≤ c |a − â|
≤ |I| + |Î| + dist (I, Î).

(13.13)

Now |I| ≤ c (1 − |w_{I}|)) by Lemma 8.1; |Î| ≤ c |I| by Lemma 10.3, and dist (I, Î) ≤ c (1 − |w_{I}|)) by the proof of Lemma 10.5. So we have established (13.11), which together with (13.10) proves Lemma 13.3.

Proof of Lemma 13.1. By Lemma 13.2, Lemma 13.3, and the remarks after the statement of Lemma 13.1, arclength on ∂D and harmonic measure at z_{I} in D are A_{∞}-equivalent with constants depending only on the chord-arc constant of Λ_{J} and the constant C from Lemma 13.3. Let c_{1} and α_{1} be such that

\frac{|E|}{|S|} ≤ c_{1} [\frac{ω(z_{I}, E, D)}{ω(z_{I}, S, D)}]^{α_{1}}  (13.14)

for all Borel subsets E of segments S of ∂D. The leaves Λ_{J} are also chord-arc, and d_{hyp}(z_{J}, ∂Λ_{J}) ≥ c > 0, so we may choose c_{2} and α_{2} independent of I such that

\frac{ω(z_{J}, E, Λ_{J})}{ω(z_{J}, S, Λ_{J})} ≤ c_{2} [\frac{|E|}{|S|}]^{α_{2}}  (13.15)

for all Borel subsets E of segments S of ∂Λ_{J}.

By the distortion estimates in Section 10, |Ĩ| ∼ |I| and |B| ∼ |B̂|. We use (13.14) and (13.15) on the sets B̂ ⊂ Ĩ ⊂ ∂D and π(B̂) ⊂ E_{J} ⊂ ∂Λ_{J}:

\begin{aligned}\frac{|B|}{|I|} & ≤ c \frac{|B̂|}{|Ĩ|} \\ & ≤ c c_{1} [\frac{ω(z_{I}, B̂, D)}{ω(z_{I}, Ĩ, D)}]^{α_{1}} \\ & = c c_{1} [\frac{ω(π(z_{I}), π(B̂), π(D))}{ω(π(z_{I}), π(Ĩ), π(D))}]^{α_{1}} \\ & = c c_{1} [\frac{ω(z_{J}, π(B̂), Λ_{J})}{ω(z_{J}, E_{J}, Λ_{J})}]^{α_{1}} \\ & ≤ c c_{1} [c_{2} (\frac{|π(B̂)|}{|E_{J}|})^{α_{2}}]^{α_{1}} , \end{aligned}  (13.16)

which establishes the right hand inequality in (13.1). A similar argument, again using (13.3), proves the left hand inequality in (13.1). The constants are independent of I and B, but they depend on the N such that |E_{J}| = 2 · 3^{N}|J|, since the chord-arc constant of Λ_{J} depends on this N.

Lemma 13.4. There are positive constants c_{1} and c_{2} such that for all n ≥ 1, for every standard grid interval I ∈ H_{n−1},

EX_{i} = \frac{1}{μ(I)} ∫_{I} X_{i}(x) dμ ≥ c_{1};  (13.17)

and

EX_{i}^{2} = \frac{1}{μ(I)} ∫_{I} X_{i}^{2}(x) dμ ≤ c_{2}.  (13.18)

Here X_{i}(x) = log_{3}(|J_{i}(x)|/|J|) is the auxiliary function for a particle which makes its i^{th} jump from J = π(Î) and which has not yet reached J_{∞}.

Proof. Let L be the component of [0, 1] \ K which contains J = π(Î).

We begin by slightly modifying some definitions, in order to simplify the calculations. In each component L′ of E_{J} \ K, we amalgamated the 2N Whitney intervals in the centre of L′ into a single interval J_{c} = J_{−N} ∪ ⋯ ∪ J_{N}. We now temporarily assume that we have retained the individual intervals J_{−N}, …, J_{N} instead. We also change the definitions of the region R_{3} and of the function F_{n}. Set S_{3} = J_{−1} ∪ J_{1} in the component L′ of [0, 1] \ K which is the left half of E_{J}, and let R_{3} = P ∘ π^{−1}(S_{3}). (In Section 11, we had S_{3} = J_{−N} ∪ ⋯ ∪ J_{N} ⊂ L′. The regions R_{1} and R_{5} are unchanged; the regions R_{2} and R_{4} become larger since they now include J_{−N} ∪ ⋯ ∪ J_{−2} and J_{2} ∪ ⋯ ∪ J_{N} respectively. )

Define

F_{n}(x) = \{ \begin{matrix}1, & x ∈ R_{1} ∪ R_{5}; \\ (1 − ε)\frac{|I|}{|R_{3}|}, & x ∈ R_{3}; \\ δ, & x ∈ R_{2} ∪ R_{4}, \end{matrix}  (13.19)

using the new definition of R_{3}. (At the end of the proof we show that the estimates with these new definitions imply the estimates for the original functions F_{n} and X_{i}.)

With these assumptions, we begin with (13.17). Let I ∈ H_{n−1} be a standard interval. Let J = π(Î). For each point x in I, let I_{i}(x) be the interval in H_{n} which contains x. I, n, and i are fixed throughout the proof. Let J_{i}(x) = π(Î_{i}(x)). Then

X_{i}(x) = log_{3} [\frac{|J_{i}(x)|}{|J|}] .  (13.20)

Notice that to compute EX_{i} or EX_{i}^{2}, we need only deal with F_{n}. The functions F_{l} for l ≠ n are irrelevant, since for l < n F_{l} is constant on I, and for l > n F_{l} has mean value one on each subset of I where X_{i} is constant.

Let

\begin{aligned}B_{k} & = \{x ∈ I | X_{i}(x) = N − k\} \\ & = \{x ∈ I | |J_{i}(x)| = 3^{N−k}|J|\} \\ & = \{x ∈ I | |J_{i}(x)| = 2^{−1} · 3^{−k}|E_{J}|\}, \end{aligned}  (13.21)

for k ≥ 1. These B_{k}’s are exactly the sets on which the integrand X_{i} in (13.17) is constant. The union of the B_{k}’s is I. We estimate (13.17) by integrating over each B_{k} separately (see (13.27) below); before that we need some preliminaries.

Let B̂_{k} be the subset of Ĩ such that P (B̂_{k}) = B_{k}. Then π(B̂_{k}) is the union of all Whitney intervals in E_{J} of length 2^{−1} · 3^{−k}|E_{J}|.

We count the number of Whitney intervals of this length in E_{J}. The segment E_{J} contains one component of \overline{IR} \ K of length 2^{−1}|E_{J}|, and it contains 2^{j} components of \overline{IR} \ K of length 2^{−1} · 3^{−j−1}|E_{J}|, for each j ≥ 0. Within each component L of \overline{IR} \ K there are two Whitney intervals of size 3^{−l}|L|, for each l ≥ 1. Therefore E_{J} contains 2^{k} Whitney intervals of length 2^{−1} · 3^{−k}|E_{J}| = 3^{N−k}|J|, for each k ≥ 1. Hence

|π(B̂_{k})| = 2^{k} · 2^{−1} · 3^{−k}|E_{J}| = \frac{1}{2} (\frac{2}{3})^{k} |E_{J}|,  (13.22)

for k ≥ 1.

Therefore, by Lemma 13.1, there are positive constants c and α, independent of I but dependent on N, such that

\frac{|B_{k}|}{|I|} ≤ c [\frac{|π(B̂_{k})|}{|E_{J}|}]^{α} = c [\frac{1}{2} (\frac{2}{3})^{k}]^{α} = c β^{k},  (13.23)

where β = (2/3)^{α} is strictly less than one.

We also estimate the sizes of B_{k} ∩ R_{1} and B_{k} ∩ R_{5}, for k ≥ 1. The sets R_{j} and S_{j}, 1 ≤ j ≤ 5, were defined in Section 11. S_{1} = π(R̂_{1}) is the segment at one end of E_{J}, contained in E_{J} \ K, such that |S_{1}| = \frac{3}{2} · 3^{−Q}|J|. Q is a large integer, independent of I. The largest Whitney interval in S_{1} has length 3^{−Q}|J| = 2^{−1} · 3^{−Q−N}|E_{J}|. The segment S_{1} contains exactly one Whitney interval of length 2^{−1} · 3^{−k}|E_{J}|, in other words one Whitney interval which lies in π(B̂_{k}), for each k ≥ Q + N. By Lemma 13.1,

\begin{aligned}\frac{|B_{k} ∩ R_{1}|}{|I|} & ≤ c [\frac{|π(B̂_{k} ∩ R̂_{1})|}{|E_{J}|}]^{α} \\ & = c [\frac{|π(B̂_{k}) ∩ S_{1}|}{|E_{J}|}]^{α} \\ & = c [\frac{2^{−1} · 3^{−k} |E_{J}|}{|E_{J}|}]^{α} \\ & = c (3^{−α})^{k}, \end{aligned}  (13.24)

for k ≥ Q + N. The left hand side is zero when 0 ≤ k ≤ Q + N − 1.

S_{5} = π(R̂_{5}) is the segment at the other end of E_{J} such that |S_{5}| = 3^{−Q}|J|. S_{5} is a closed construction interval of the Cantor set K. The largest Whitney intervals in S_{5} are two intervals of length 3^{−2} · 3^{−Q}|J| = 2^{−1} · 3^{−Q−N−2}|E_{J}|. S_{5} contains 2^{k−Q−N−1} Whitney intervals of size 2^{−1} · 3^{−k}|E_{J}|, for each k ≥ Q + N + 2. By Lemma 13.1,

\begin{aligned}\frac{|B_{k} ∩ R_{5}|}{|I|} & ≤ c [\frac{|π(B̂_{k}) ∩ S_{5}|}{|E_{J}|}]^{α} \\ & = c [\frac{2^{k−Q−N−1} · 2^{−1} · 3^{−k}|E_{J}|}{|E_{J}|}]^{α} \\ & = c (2^{−Q−N})^{α}(2/3)^{kα} \\ & = c (2^{−α})^{Q+N}β^{k}, \end{aligned}  (13.25)

for k ≥ Q + N + 2. The left hand side vanishes for 0 ≤ k ≤ Q + N + 1.

The set B_{1} is exactly R_{3}, and so F_{n} ≡ (1−ε) |I|/|B_{1}| on B_{1}. Also, on I, μ_{n−1} is given by dμ_{n−1} = (μ_{n−1}(I)/|I|) dx. Therefore

\begin{aligned}μ(B_{1}) & = μ_{n}(B_{1}) \\ & = ∫_{B_{1}} F_{n}(x) dμ_{n−1}(x) \\ & = (1 − ε) \frac{|I|}{|B_{1}|} \frac{μ_{n−1}(I)}{|I|} |B_{1}| \\ & = (1 − ε) μ_{n−1}(I) \\ & = (1 − ε) μ(I). \end{aligned}  (13.26)

After these preliminaries we can estimate EX_{i}:

\begin{aligned}EX_{i} & = \frac{1}{μ(I)} ∫_{I} log_{3} [\frac{|J_{i}(x)|}{|J|}] dμ(x) \\ & = ∑_{k=1}^{∞} (N − k) \frac{μ(B_{k})}{μ(I)} \\ & = (N − 1) \frac{μ(B_{1})}{μ(I)} + ∑_{k=2}^{∞} (N − k) \frac{μ(B_{k})}{μ(I)} \\ & ≥ (N − 1) (1 − ε) + ∑_{k=N}^{∞} (N − k) \frac{μ(B_{k})}{μ(I)} \\ & ≥ (N − 1) (1 − ε) − ∑_{k=N}^{∞} k \frac{μ(B_{k})}{μ(I)} \\ & = (N − 1) (1 − ε) − ∑_{k=N}^{∞} k \frac{μ_{n}(B_{k})}{μ_{n−1}(I)} \\ & ≥ (N − 1) (1 − ε) − ∑_{k=N}^{∞} \frac{k}{μ_{n−1}(I)} ∫_{B_{k}} F_{n}(x) dμ_{n−1}(x). \end{aligned}  (13.27)

The last series converges. To show this, we split each B_{k} into three pieces, B_{k} ∩ (R_{2} ∪ R_{4}), B_{k} ∩ R_{1}, and B_{k} ∩ R_{5}, and estimate the sums over the three types of terms.

First, since F_{n} ≡ δ on R_{2} ∪ R_{4},

\begin{aligned}∑_{k=N}^{∞} \frac{k}{μ_{n−1}(I)} ∫_{B_{k}∩(R_{2}∪R_{4})} & F_{n}(x) dμ_{n−1}(x) \\ & = ∑_{k=N}^{∞} k δ \frac{μ_{n−1}(B_{k} ∩ (R_{2} ∪ R_{4}))}{μ_{n−1}(I)} \\ & = δ ∑_{k=N}^{∞} k \frac{|B_{k} ∩ (R_{2} ∪ R_{4})|}{|I|} \\ & ≤ δ ∑_{k=N}^{∞} k \frac{|B_{k}|}{|I|} \\ & ≤ δ c (N) ∑_{k=N}^{∞} k β^{k} \\ & ≤ δ \frac{c (N)}{(1 − β)^{2}} , \end{aligned}  (13.28)

which can be made arbitrarily small by choosing ε small (which implies δ small) in the definition of F_{n}. Here the second equality holds because, on I, μ_{n−1} is a constant multiple of Lebesgue measure, and the second last line holds by (13.23).

Similarly, since F_{n} ≡ 1 on R_{1},

\begin{aligned}∑_{k=N}^{∞} \frac{k}{μ_{n−1}(I)} ∫_{B_{k}∩R_{1}} & F_{n}(x) dμ_{n−1}(x) \\ & = ∑_{k=N}^{∞} k \frac{μ_{n−1}(B_{k} ∩ R_{1})}{μ_{n−1}(I)} \\ & = ∑_{k=N}^{∞} k \frac{|B_{k} ∩ R_{1}|}{|I|} \\ & = ∑_{k=N+Q}^{∞} k \frac{|B_{k} ∩ R_{1}|}{|I|} \\ & ≤ c (N) ∑_{k=N+Q}^{∞} k (3^{−α})^{k} , \end{aligned}  (13.29)

which can be made arbitrarily small by choosing Q large. Here the last line holds by (13.24). Finally, since F_{n} ≡ 1 on R_{5},

\begin{aligned}∑_{k=N}^{∞} \frac{k}{μ_{n−1}(I)} ∫_{B_{k}∩R_{5}} & F_{n}(x) dμ_{n−1}(x) \\ & = ∑_{k=N}^{∞} k \frac{μ_{n−1}(B_{k} ∩ R_{5})}{μ_{n−1}(I)} \\ & = ∑_{k=N}^{∞} k \frac{|B_{k} ∩ R_{5}|}{|I|} \\ & = ∑_{k=Q+N+2}^{∞} k \frac{|B_{k} ∩ R_{5}|}{|I|} \\ & ≤ c(N) (2^{−α})^{Q+N} ∑_{k=Q+N+2}^{∞} k β^{k} \\ & ≤ \frac{c(N) (2^{−α})^{Q+N}}{(1 − β)^{2}} , \end{aligned}  (13.30)

which decreases to zero as Q increases. Here the second last inequality holds by (13.25).

Once N is fixed, we can make (13.28), (13.29), and (13.30) arbitrarily small by choosing ε sufficiently small and Q sufficiently large. Therefore we can make the last term in (13.27) arbitrarily small. So for some positive constant c_{1} depending on N, we can ensure that EX_{i} ≥ c_{1} for all standard intervals I in H_{n−1}, for all n ≥ 1.

We can estimate the second moment more simply. We showed in Section 11 that there is an upper bound 1/δ_{0} for F_{n} on I, independent of I and n. Then

\begin{aligned}EX_{i}^{2} & = \frac{1}{μ(I)} ∫_{I}\{log_{3} [\frac{|J_{i}(x)|}{|J|}]\}^{2} dμ(x) \\ & = ∑_{k=1}^{∞} (N − k)^{2} \frac{μ(B_{k})}{μ(I)} \\ & = ∑_{k=1}^{∞} (N − k)^{2} \frac{μ_{n}(B_{k})}{μ_{n−1}(I)} \\ & = ∑_{k=1}^{∞} (N − k)^{2} \frac{1}{μ(I)} ∫_{B_{k}} F_{n}(x) dμ_{n−1}(x) \\ & ≤ \frac{1}{δ_{0}} ∑_{k=1}^{∞} (N − k)^{2} \frac{μ_{n−1}(B_{k})}{μ_{n−1}(I)} \\ & = \frac{1}{δ_{0}} ∑_{k=1}^{∞} (N − k)^{2} \frac{|B_{k}|}{|I|} \\ & ≤ \frac{1}{δ_{0}} ∑_{k=1}^{∞} (N^{2} + k^{2}) \frac{|B_{k}|}{|I|} \\ & ≤ \frac{c(N)}{δ_{0}} ∑_{k=1}^{∞} (N^{2} + k^{2}) β^{k} \\ & ≤ \frac{c(N)}{δ_{0}} [\frac{N^{2}}{(1 − β)} + \frac{1}{(1 − β)^{3}} ] , \end{aligned}  (13.31)

using (13.23) in the second last line. In the sixth line we used the fact that μ_{n−1} is a constant multiple of Lebesgue measure on I. Again, once N, Q, and ε are fixed, this estimate gives a uniform upper bound c_{2} on EX_{i}^{2} for all standard intervals I ∈ H_{n−1}, for all n ≥ 1.

It remains to prove the same estimates for our original functions X_{i} and F_{n}. Write Y_{i} and G_{n} for the modified versions of X_{i} and F_{n} defined at the start of the proof and used in the calculations (13.19)–(13.31) above.

To estimate EX_{i}^{2}, define Z_{i}(x) = N^{2} + k^{2} on B_{k}, k ≥ 1, where B_{k} = {x ∈ I | Y_{i}(x) = N − k} as above. A comparison of Y_{i}, X_{i}, and Z_{i} shows that on each B_{k},

X_{i}^{2} ≤ (N + k)^{2} ≤ 2 (N^{2} + k^{2}) = 2 Z_{i}.  (13.32)

The calculation (13.31) shows that the mean value of Z_{i}G_{n} on I is at most c_{2}. Also, F_{n} ∼ G_{n} with constants independent of I, since they are both bounded above and below by positive constants independent of I. Therefore

\begin{aligned}EX_{i}^{2} & = \frac{1}{|I|} ∫_{I} X_{i}^{2}F_{n} dx \\ & ≤ c \frac{1}{|I|} ∫_{I} 2 Z_{i}G_{n} dx \\ & ≤ 2 c c_{2}, \end{aligned}  (13.33)

for all I.

Finally, we estimate EX_{i}. Let G_{b} (respectively F_{b}) be the maximum value of G_{n} (respectively F_{n}). Then G_{b} ∼ F_{b} with constants independent of I, by a comparison of harmonic measures in Λ_{J}. Let

I_{+} = {x ∈ I | F_{n}(x) = F_{b}}  (13.34)

and

I_{s} = {x ∈ I | F_{n}(x) = δ′},  (13.35)

where δ′ is the minimum value of F_{n} on I. In the next calculation we neglect the region where Y_{i} ≡ 1. Then, since X_{i} ≥ Y_{i} on I,

\begin{aligned}EX_{i} & = \frac{1}{|I|} ∫_{I} X_{i}F_{n} dx \\ & ≥ \frac{1}{|I|} ∫_{I} Y_{i}F_{n} dx \\ & ≥ \frac{1}{|I|} ∫_{I_{+}} Y_{i}F_{b} dx − \frac{1}{|I|} ∫_{I_{s}} Y_{i}F_{n} dx \\ & ≥ c \frac{1}{|I|} ∫_{I_{+}} Y_{i}G_{b} dx − \frac{1}{|I|} ∫_{I_{s}} Y_{i}F_{n} dx \\ & ≥ c \frac{N − 1}{2} − δ′ \frac{1}{|I|} ∫_{I_{s}} Y_{i} dx; \end{aligned}  (13.36)

by (13.27). We showed above that \frac{1}{|I|} ∫_{I_{s}} Y_{i} dx is bounded above by a constant independent of I. Therefore, by choosing δ′ small enough in the definition of F_{n}, we may ensure that EX_{i} is bounded below by a positive constant independent of I.

This completes the proof of Lemma 13.4.

14. Estimates EX_{i} ≥ c_{1} and EX_{i}^{2} ≤ c_{2} for non-standard intervals

Let I ∈ H be any non-standard grid interval such that J_{c} = π(Î) is the non-standard central interval in some component L of [0, 1] \ K. In this section we prove (Lemma 14.3) the estimates μ(I)^{−1} ∫_{I} X_{i} dμ ≥ c_{1} and μ(I)^{−1} ∫_{I} X_{i}^{2} dμ ≤ c_{2}, where c_{1} and c_{2} are positive constants independent of I. Here X_{i}(x) = log_{3}(|J_{i}(x)|/|J_{c}|) is the auxiliary function for a particle which makes its i^{th} jump from J_{c} = π(Î) and which has not yet reached J_{∞}. We begin with an estimate, analogous to Lemma 13.1, on the distortion in length caused by the map P ∘ π^{−1}.

Lemma 14.1. Let I ∈ H_{n−1}, n ≥ 1, be a non-standard grid interval such that J_{c} = π(Î) is the central interval in some component L of [0, 1] \ K. Fix a number λ > 1, and let q be any positive integer. Let B and T be unions of grid intervals from H_{n} such that B ⊂ T ⊂ I and π(T̂) ⊂ {w | λ^{q} ≤ dist (w, J_{c}) ≤ λ^{q+1}}. There are constants c and α, independent of q, B, T, I, and n, such that

\frac{|B|}{|T |} ≤ c [\frac{|π(B̂)|}{|π(T̂)|}]^{α} .  (14.1)

The proof is essentially the same as for standard intervals, but we need a stronger version of A_{∞}-equivalence because ∂Λ_{J} is unbounded. Namely, if Ω is a chord-arc domain, and z ∈ Ω, then there are positive constants c_{1}, c_{2}, α_{1}, and α_{2} such that

\frac{|E|}{|S|} ≤ c_{1} [\frac{ω(z, E, Ω)}{ω(z, S, Ω)}]^{α_{1}} and \frac{ω(z, E, Ω)}{ω(z, S, Ω)} ≤ c_{2} [\frac{|E|}{|S|}]^{α_{2}}  (14.2)

whenever S is a segment of ∂Ω satisfying

S ⊂ {w | λ^{q} ≤ dist (w, z) ≤ λ^{q+1}}  (14.3)

for any positive integer q and for a fixed λ > 1, and E is a Borel subset of S. The constants c_{1}, c_{2}, α_{1}, and α_{2} depend on the chord-arc constant of ∂Ω and on the constant λ, but not on q.

This follows from the case of bounded chord-arc domains. By dilations and translations we may assume that z = i, and that −i is at least distance ε from Ω. Consider the Möbius transformation τ : w ↦ (w + i)^{−1}. This maps Ω to a bounded chord-arc domain. Harmonic measure is invariant under Möbius transformations. Also, for each positive integer q, the map τ scales everything in the annulus in (14.3), in particular E and S, by approximately the same factor. The estimates in (14.2) follow.

Consider the domain D = π^{−1}(Λ_{J_{c}}), where I is a non-standard interval as above. The analogues of Lemmas 13.2 and 13.3 hold, and Lemma 14.1 follows in the same way as Lemma 13.1 follows from Lemmas 13.2 and 13.3.

Notice that, although the interval I is no longer standard, arclength and harmonic measure are still A_{∞}-equivalent on ∂D, since D is a bounded chord-arc domain. Therefore, for any segment S of ∂D and Borel subset E of S, we still have

\frac{|E|}{|S|} ≤ c_{1} [\frac{ω(z_{J}, π(E), Λ_{J_{c}})}{ω(z_{J}, π(S), Λ_{J_{c}})}]^{α_{1}} ,  (14.4)

and the analogous inequality in the other direction, both with constants independent of I. We will frequently use this fact, combined with direct estimates of harmonic measures in Λ_{J_{c}}. We refer to such an argument as a comparison of harmonic measure on the domain side.

We will need the next lemma, which follows immediately from our decomposition of \overline{IR} \ K into Whitney intervals.

Lemma 14.2. Let A be a subinterval of [0, 1] of length 3^{−l}, l ≥ 0. Then

a) A contains a Whitney interval of length at least 3^{−2}|A|; and

b) A meets at most 2^{k} Whitney intervals of length 3^{−k}|A|, for k ≥ 1.

The main result of this section is:

Lemma 14.3. There are positive constants c_{1} and c_{2} such that for all n ≥ 1, for every non-standard grid interval I ∈ H_{n−1} such that π(Î) = J_{c} is the central Whitney interval in some component L of [0, 1] \ K,

EX_{i} = \frac{1}{μ(I)} ∫_{I} X_{i}(x) dμ ≥ c_{1};  (14.5)

and

EX_{i}^{2} = \frac{1}{μ(I)} ∫_{I} X_{i}^{2}(x) dμ ≤ c_{2}.  (14.6)

Here X_{i}(x) = log_{3}(|J_{i}(x)|/|J_{c}|) is the auxiliary function for a particle which has not yet reached J_{∞} and which makes its i^{th} jump from J_{c} = π(Î).

Proof. Let J_{c} be the central Whitney interval, J_{c} = J_{−N} ∪⋯∪J_{N}, in some component L of [0, 1]\K. Make the convention that |J_{c}| = |L|/3. The tips of the leaves for the standard intervals in L cover K_{l} and K_{r}, the closed construction intervals of K immediately to the left and right of L. So the tip E_{J_{c}} of the leaf for J_{c} is \overline{IR} \ (K_{l} ∪ L ∪ K_{r}). See Figure 13.

To simplify the calculations, we again assume, as in the proof of Lemma 13.4, that in each component L of E_{J_{c}} \ K we have not amalgamated the 2N central Whitney intervals but have retained them individually. As before, the result follows from this simplified version.

Figure 13. A non-standard central interval J_{c}.

We divide E_{J_{c}} into subsets V_{j}, j ≥ 1, of length |V_{j}| = 3^{j+10}|J_{c}|. Let each V_{j} have two components of equal length, one on each side of J_{c}. Put the components of V_{1} as close as possible to J_{c}, that is, on each side of and adjacent to K_{l} ∪ L ∪ K_{r}. Put the components of V_{2} on each side of and adjacent to V_{1} ∪ K_{l} ∪ L ∪ K_{r}, and so on. For each j ≥ 1, let T_{j} be the subset of I such that π(T̂_{j}) = V_{j}.

Fix j ≥ 1, and for each k ≥ 1 define the subset B_{k} of T_{j} by

\begin{aligned}B_{k} & = \{x ∈ T_{j} | |J_{i}(x)| = 3^{−k} |V_{j}|\} \\ & = \{x ∈ T_{j} | |J_{i}(x)| = 3^{−k} · 3^{j+10}|J_{c}|\} \\ & = \{x ∈ T_{j} | X_{i}(x) = j + 10 − k\}. \end{aligned}  (14.7)

By Lemma 14.2, the set π(B̂_{k}) meets at most 2^{k} Whitney intervals of size 3^{−k}|V_{j}|, so

|π(B̂_{k})| ≤ (\frac{2}{3})^{k} |V_{j}|.  (14.8)

By Lemma 14.1,

\frac{|B_{k}|}{|T_{j}|} ≤ c [\frac{|π(B̂_{k})|}{|V_{j}|}]^{α} = c [(\frac{2}{3})^{k}]^{α} = c β^{k},  (14.9)

where β = (2/3)^{α} is strictly less than one, and c and β are independent of j.

In fact, Lemma 14.1 in the form stated above may not apply, since B_{k} and T_{j} need not be unions of whole grid intervals. However, the difficulty is only that at either end of T_{j} there may be grid intervals which meet T_{j} (and hence some B_{k}) but are not contained in T_{j} (or B_{k}). The distortion estimates in Section 10, and the fact that grid intervals are comparable in size to their neighbours, imply that Lemma 14.1 can be extended to cover this case.

We first prove the estimate on EX_{i}^{2}. In estimating X_{i}^{2} we can neglect the function F_{n}, since the F_{n} we use will be bounded above by a constant independent of I. So it is enough to find a bound independent of I for

\frac{1}{|I|} ∫_{I} X_{i}^{2} dx = ∑_{j=1}^{∞} \frac{|T_{j}|}{|I|} \frac{1}{|T_{j}|} ∫_{T_{j}} \{log_{3} [\frac{|J_{i}(x)|}{|J_{c}|}]\}^{2} dx.  (14.10)

First, for each j,

\begin{aligned}\frac{1}{|T_{j}|} ∫_{T_{j}} \{log_{3} [\frac{|J_{i}(x)|}{|J_{c}|}]\}^{2} & = ∑_{k=1}^{∞} \{j + 10 − k\}^{2} \frac{|B_{k}|}{|T_{j}|} \\ & ≤ c ∑_{k=1}^{∞} \{j^{2} + (k − 10)^{2}\} β^{k} \\ & ≤ c′j^{2} + c′′. \end{aligned}  (14.11)

Also, |T_{j}|/|I| decays exponentially in j:

\begin{aligned}\frac{|T_{j}|}{|I|} & ≤ c \frac{|T̂_{j}|}{|Ĩ|} \\ & ≤ c c_{1} [\frac{ω(z_{I}, T̂_{j}, π^{−1}(Λ_{J_{c}}))}{ω(z_{I}, Ĩ, π^{−1}(Λ_{J_{c}}))}]^{α_{1}} \\ & = c c_{1} [\frac{ω(z_{J}, V_{j}, Λ_{J_{c}})}{ω(z_{J}, E_{J_{c}}, Λ_{J_{c}})}]^{α_{1}} \\ & ≤ c [ω (z_{J}, V_{j}, U)]^{α_{1}}, \end{aligned}  (14.12)

with constants independent of j. Now π·ω(z_{J}, V_{j}, U) is the angle subtended at z_{J} by V_{j}. By elementary trigonometry, this angle is less than c 3^{−j}, with c independent of j. Therefore

\frac{|T_{j}|}{|I|} ≤ c 3^{−α_{1}j}.  (14.13)

Hence

\frac{1}{|I|} ∫_{I} X_{i}^{2} dx ≤ c ∑_{j=1}^{∞} 3^{−α_{1}j}(c′j^{2} + c′′).  (14.14)

The right hand side is finite and independent of I.

It remains to prove the estimate on EX_{i}. Fix ε > 0, and j ≥ 2. By Lemma 14.2, B_{2} is not empty. (For those V_{j} ⊄ [0, 1], B_{2} may be empty; part a) of Lemma 14.2 need not hold as stated because of the amalgamation of the large Whitney intervals far from [0, 1] to form J_{∞}. However our argument can be modified to cover this case.) Let B = B_{1} ∪ B_{2}, and define F_{n} on T_{j} by

F (x) = F_{n}(x) = \{ \begin{matrix}(1 − ε) \frac{|T_{j}|}{|B|}, & x ∈ B; \\ δ, & x ∈ T_{j} \ B, \end{matrix}  (14.15)

where δ = ε|T_{j}|/|T_{j} \ B| is chosen so that F has mean value one on T_{j}. A comparison of harmonic measure shows that |T_{j}|/|B| is uniformly bounded for all j and I. Therefore F is uniformly bounded above and below by positive constants independent of j and I.

On B, X_{i} ≥ j + 8. Then

\begin{aligned}\frac{1}{μ(T_{j})} ∫_{T_{j}} X_{i} dμ & = \frac{1}{|T_{j}|} ∫_{T_{j}} X_{i}F dx \\ & ≥ (j + 8)(1 − ε) + δ ∑_{k=3}^{∞} (j + 10 − k) \frac{|B_{k}|}{|T_{j}|} \\ & ≥ (j + 8)(1 − ε) − δ ∑_{k=j+10}^{∞} (k − j − 10) \frac{|B_{k}|}{|T_{j}|} \\ & ≥ (j + 8)(1 − ε) − δ c(N) ∑_{k−j=10}^{∞} (k − j − 10) β^{k}. \end{aligned}  (14.16)

The sum is finite and independent of j. Hence, by choosing δ (or equivalently ε) small enough in the definition of F, we may ensure that the mean of X_{i}F with respect to μ on T_{j} is at least some positive constant c_{1}, independent of j.

To ensure that F ≡ 1 on suitable intervals at each end of I, we define F slightly differently on T_{1}. Specifically, let I_{l} and I_{r} be subintervals at each end of I. Then π(Î_{l}) and π(Î_{r}) are subintervals of V_{1}, one on each side of J_{c} and as close as possible to J_{c}; in other words, adjacent to K_{l} ∪ L ∪ K_{r}. Choose the lengths of I_{l} and I_{r} so that

\frac{|π(Î_{l})|}{|T_{1}|} = \frac{|π(Î_{r})|}{|T_{1}|} = \frac{3}{2} · 3^{−Q},  (14.17)

for some large integer Q. The harmonic measure of π(Î_{l})∪π(Î_{r}) in Λ_{J_{c}} as seen from z_{J} is a positive constant independent of I, since the domains Λ_{J_{c}} for different I are all Euclidean dilations of each other. By a comparison of harmonic measure, there is an η > 0 such that |I_{l}| and |I_{r}| ≥ η |I| for all I. Now let

F (x) = \{ \begin{matrix}(1 − ε)\frac{|T_{1}|}{|B|}, & x ∈ B; \\ 1, & x ∈ I_{l} ∪ I_{r}; \\ δ, & x ∈ T_{1} \ (B ∪ I_{l} ∪ I_{r}). \end{matrix}  (14.18)

Again, we choose δ so that F has mean value one on T_{1}. Then

δ = (ε − \frac{|I_{l} ∪ I_{r}|}{|T_{1}|}) \frac{|T_{1}|}{|T_{1} \ (B ∪ I_{l} ∪ I_{r})|} ≥ ε/2,  (14.19)

say, if Q is chosen large enough that |I_{l} ∪ I_{r}|/|T_{1}| ≤ ε/2. Therefore F is bounded above and below on T_{1} by positive constants independent of I.

Notice that with this new definition of F on T_{1}, we can still make the mean value of X_{i}F on T_{1} greater than c_{1}. For, just as for standard intervals, harmonic measure estimates show that |B_{k} ∩ (I_{l} ∪ I_{r})|/|T_{1}| decays exponentially in k, and so the contribution from I_{l} ∪ I_{r} to the mean value goes to zero as Q goes to infinity.

This completes the proof of Lemma 14.3.

15. μ is supported on L_{c}(G)

Define the functions F_{n} as in Section 11 on all standard intervals I ∈ H_{n−1}, n ≥ 1, and as in Section 14 on all non-standard intervals I ∈ H_{n−1}, n ≥ 1, such that J_{c} = π(Î) is the non-standard central Whitney interval in any component of [0, 1] \ K. Define F_{n} ≡ 1 on the remaining grid intervals I ∈ H_{n−1}, n ≥ 1; these are exactly those I ∈ H such that π(Î) = J_{∞}. We have shown that the first two types of these functions are (δ, η)-suitable for all I ∈ H_{n−1}, n ≥ 1, where δ and η are constants independent of I and n. This is also true when F_{n} ≡ 1 on I ∈ H_{n−1}. Therefore, by Lemma 6.3, the measures μ_{n} defined by dμ_{n} = F_{n}(x) ⋯ F_{1}(x) dx converge to a doubling measure μ on the circle.

Consider the random walk on the tree discussed in Section 4, where the vertices in the tree are the Whitney intervals Î in the boundary arcs ⋃_{n} A_{n} of the half fundamental domains in our tiling of the disc; V is the subset of vertices such that π(Î) = J_{∞} (in other words, those intervals which contain orbit points g(0), g ∈ G); and the probabilities of jumps between adjacent vertices are determined by the functions F_{n} defined above.

In this section, we first use the estimates EX_{i} ≥ c_{1} and EX_{i}^{2} ≤ c_{2} established in Sections 13 and 14 to show that a particle starting from any vertex v reaches V (v) with probability one. Here V (v) is the set of vertices w ∈ V below v such that there are no other vertices from V between w and v. We conclude that the doubling measure μ is supported on the set S of points which lie in infinitely many grid intervals I such that π(Î) = J_{∞}. Finally, we show that the conical limit set of G contains this set S, which establishes Theorem 1.2.

Let v be a vertex in the tree, and I_{0} ∈ H_{s} the corresponding grid interval. Let V (I_{0}) be the collection of maximal grid intervals I ⊂ I_{0} such that π(Î) = J_{∞}. We show that the restriction of μ to I_{0} is supported on V (I_{0}).

Recall that

X_{i}(x) = \{ \begin{matrix}log_{3} [\frac{|J_{i}(x)|}{|J_{i−1}(x)|}] , & if J_{1}(x), …, J_{i−1}(x) ≠ J_{∞}; \\ 1, & otherwise, \end{matrix}  (15.1)

that S_{k}(x) = ∑_{i=1}^{k} X_{i}(x), and that V (I_{0}) = {x ∈ I_{0} | S_{k}(x) → +∞}. It is enough to show that S_{k}(x) → +∞ almost surely on I_{0} with respect to μ.

Let E_{I}X_{i} denote the mean value of X_{i} on I with respect to μ. We know that |E_{I}X_{i}| < ∞ for all grid intervals I ∈ H. This is because |E_{I}X_{i}| ≤ E_{I}|X_{i}| ≤ E_{I}X_{i}^{2} < c_{2}, since X_{i} is integer-valued. Define new functions X̃_{i}(x) on I_{0} which have mean zero on each I ∈ H_{s+i−1} which lies in I_{0}:

\begin{aligned}X̃_{1}(x) & = X_{1}(x) − E_{I_{0}}X_{1}; \\ X̃_{i}(x) & = ∑_{{I∈H_{s+i−1} \atop I⊂I_{0}}} (X_{i}(x) − E_{I}X_{i}) χ_{I}(x), \end{aligned}  (15.2)

for i ≥ 1, and let

\begin{aligned}S̃_{k}(x) & = ∑_{i=1}^{k} X̃_{i}(x) \\ & = ∑_{i=1}^{k} (X_{i}(x) − E_{I_{i−1}}X_{i}) \end{aligned}  (15.3)

where x ∈ I_{i−1}, i ≥ 1.

Then {S̃_{k}} is a martingale with respect to μ.

Also, EX̃_{i}^{2} is bounded by EX_{i}^{2}, which is bounded by c_{2}. Therefore, by the strong law of large numbers for martingales [Fel], S̃_{k}/k → 0 almost surely on I_{0} with respect to μ.

Now let x be a point in I_{0} such that S̃_{k}(x)/k → 0, and let I_{i} ∈ H_{s+i} be the interval containing x, for i ≥ 0. Then

\begin{aligned}\frac{S_{k}(x)}{k} & = \frac{1}{k} ∑_{i=1}^{k} X_{i}(x) \\ & = \frac{1}{k} ∑_{i=1}^{k} (X̃_{i}(x) + E_{I_{i−1}}^{dμ}X_{i}) \\ & ≥ \frac{1}{k} S̃_{k}(x) + c_{1} \\ & ≥ c_{1}/2, \end{aligned}  (15.4)

for all k sufficiently large that |S̃_{k}(x)/k | ≤ c_{1}/2. Therefore S_{k}(x) ≥ c_{1}k/2 for all such k, and so S_{k}(x) → +∞. Thus S_{k} → +∞ almost surely with respect to μ on I_{0}. Since V (I_{0}) is exactly the set of points x in I_{0} such that S_{k} → +∞, this implies that μ(V (I_{0})) = μ(I_{0}).

Since a particle starting from any vertex v reaches V (v) with probability one, it is now clear that a particle starting anywhere in the tree will pass through infinitely many vertices in V, with probability one. (Note that we have now defined the only jump probabilities which were missing; these are the probabilities of the jumps from vertices in the subset V, and they are determined by the functions F_{n} ≡ 1 on the corresponding grid intervals.) Thus we have shown that the doubling measure μ is supported on the set S of points in the circle which lie in infinitely many grid intervals I = P (Î) such that π(Î) = J_{∞}.

It remains to show that S is contained in the conical limit set of G. We show that there is a uniform constant l such that for each orbit point g_{j}(0) ≠ 0, the Whitney interval Î_{j} which contains g_{j}(0) projects to a grid interval P (Î_{j} ) which lies in the spherical cap Cap (g_{j}(0), l) on g_{j}(0). Then each point x ∈ S lies in infinitely many of these caps, and therefore in the conical limit set.

By definition, the tip of the leaf based at J_{∞} is E_{J_{∞}} = [1/9, 8/9]. For each g_{j} ∈ G\{id} let A_{j} be the orthocircular arc in ⋃_{n} A_{n} which contains g_{j}(0), and let Ω_{j} be the half fundamental domain below A_{j}. Let F_{j} be the fundamental domain consisting of Ω_{j} and the half fundamental domain above A_{j}. Let Ê_{j} be the segment of ∂Ω_{j}, below g_{j}(0), such that π(Ê_{j}) = E_{J_{∞}}. The endpoints of Ê_{j} lie in ∂ID. Then P (Î_{j} ) = E_{j} is the arc of ∂ID below Ê_{j} (with the same endpoints as Ê_{j}). See Figure 14.

Figure 14. P (Î_{j} ) = E_{j} lies in Cap = Cap (g_{j}(0), l); E = E_{J_{∞}}.

Lemma 15.1. There is a constant l > 0 such that for all j,

P (Î_{j} ) ⊂ Cap (g_{j}(0), l).  (15.5)

Proof. By definition,

Cap (g_{j}(0), l) = {z ∈ ∂ID | dist (z, g_{j}(0)) ≤ l (1 − |g_{j}(0)|)}.

Let z_{j} = g_{j}(0). Let b_{j} ∈ ∂ID be the endpoint of P (Î_{j} ) which is furthest from z_{j}. We show that |z_{j} − b_{j}|/(1 − |z_{j}|) ≤ c.

If b_{j} satisfies |b_{j} − z_{j}| ≤ 10 (1 − |z_{j}|), we are done. If not, let B_{j} be the component of π^{−1}([0, 1] \ E_{J_{∞}}) which has b_{j} as an endpoint; and let B = π(B_{j}). See Figure 14 for one possible configuration.

First, |z_{j} − b_{j}| ≤ c dist (z_{j}, B_{j}). For if a is any point in B_{j} ∩ ∂ID, then |z_{j} − b_{j}| ≤ |z_{j} − a|, while if a is in B_{j} \ ∂ID then a lies in one of the orthocircular arcs in ∂F_{j}, and |z_{j} − a| ≥ c |z_{j} − b_{j}|.

By Beurling’s Lemma (see also (10.12)),

dist (z_{j}, B_{j}) ≤ c \frac{dist (z_{j}, ∂F_{j})}{ω(z_{j}, B_{j}, F_{j})^{2}}

≤ c \frac{dist (z_{j}, ∂F_{j})}{ω(π(z_{j}), π(B_{j}), π(F_{j}))^{2}}

= c \frac{dist (z_{j}, ∂F_{j})}{ω(∞, B, \overline{C} \ K)^{2}}

≤ c dist (z_{j}, ∂F_{j}) ≤ c (1 − |z_{j}|).

(15.6)

We have shown that each point in ∂ID which is at the end of a path which passes through infinitely many orbit points is actually a conical limit point of G; in particular, that the orbit points on the path not only accumulate at the endpoint but lie in some non-tangential cone based at the endpoint. Therefore the set S lies in the conical limit set of G.

This completes the proof of Theorem 1.2.

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Peter W. Jones, Department of Mathematics, Yale University, New Haven, CT 06520-8283 E-mail address: jones@math.yale.edu.

Lesley Ward, Department of Mathematics, Rice University, Houston, TX 77251-1892 E-mail address: lesley@math.rice.edu