DYNAMICS OF QUADRATIC POLYNOMIALS.

I. COMBINATORICS AND GEOMETRY OF THE YOCCOZ PUZZLE.

February 5, 1995

Mikhail Lyubich

Abstract. This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the “principal nest” of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.

§1. Introduction

This is the beginning of a series of notes on dynamics of quadratic polynomials. The forthcoming notes will be concerned with parameter geometry, applications to the complex and real Rigidity Problems, and measurable dynamics. This first part is a detailed version of §§2,3 of [L4].

The goal of this part is to study combinatorics and geometry of the Yoccoz puzzle, with an application to the problem of local connectivity of Julia sets. The puzzle was introduced by Branner and Hubbard [BH] for cubic maps with one escaping critical point and by Yoccoz for quadratics (see [H], [M2]). The main geometric result of these works is the divergence property of moduli of a certain nest of annuli. This implies that the corresponding domains (“puzzle pieces”) shrink to points, which yields local connectivity of the Julia set. The corresponding result in the parameter plane yields rigidity.

The geometric result of Branner-Hubbard and Yoccoz does not contain information on the rate at which the pieces shrink to points. In this work we tackle this problem. We consider a smaller nest V^{0} ⊃ V^{1} ⊃ … called principal, and prove that the annuli moduli of this nest grow at linear rate over a certain combinatorially specified subsequence of levels (Theorem III). This implies that the first return maps g_{n} : V^{n} → V^{n−1} are becoming purely quadratic at exponential rate. We expect this fact to have many applications.

Theorem III also yields a priori bounds for infinitely renormalizable quadratics of “sufficiently big type” (Theorems IV and IV′ ). For real quadratics of “bounded type” the problem of complex bounds was resolved by Sullivan (see [S2], [MvS]). Our result for real quadratics is a kind of complement to Sullivan’s, though there is still one special combinatorial situation, “saddle-node cascades”, which is not covered by either of these results. In the forthcoming notes we will give an appropriate extension of Sullivan’s bounds which pick saddle-node cascades [LY]: Local connectivity of the Julia sets for all real quadratics follows.** As to the non-real situation, to the best of our knowledge the problem of a priori bounds was not settled before for any one quadratic of bounded type (compare Rees [R]).

Supported in part by Sloan Research Fellowship, MSRI and NSF grants DMS-8920768 and DMS-9022140.

An immediate consequence of our a priori bounds is that the puzzle pieces of the “full principal nest” shrink to points, which yields local connectivity of the Julia set. This is a nice property, since such a Julia set has an explicit topological model (see Douady [D2]). Note that a large pool of locally connected examples was already known (Douady - Hubbard, Yoccoz (see [DH1], [H]), Hu - Jiang [HJ], [J], Petersen [P]). Unfortunately there are also counter-examples: Cremer and some infinitely renormalizable quadratics have non-locally connected Julia sets (see [L3], [M2]).

However, the main reason why complex a priori bounds are important is that they provide a key to the rigidity problem and the renormalization theory (compare Sullivan [S2] and McMullen [McM]). These applications will be the subject of the forthcoming notes.

Let us now describe the structure of the paper.

In the next section, §2, we overview the necessary preliminaries in holomorphic dynamics, particularly Douady-Hubbard renormalization and the Yoccoz puzzle.

In §3 we present our approach to combinatorics of the puzzle. The main concepts involved are the principal nest of puzzle pieces, generalized renormalization and central cascades. As we indicated above, the principal nest V^{0} ⊃ V^{1} ⊃ … contains the key combinatorial and geometric information about the puzzle. We describe the combinatorics of this nest by means of generalized renormalizations, that is, appropriately restricted first return maps considered up to rescaling.

It may happen that the quadratic-like map g_{n} : V^{n} → V^{n−1} has ”almost connected” Julia set. This phenomenon often requires a special treatment. Such a map generates a subnest of the principal nest called a central cascade. The number of central cascades in the principal nest is called the height χ(f) of a map f . In other words, χ(f) is the number of different quadratic-like maps among the g_{n} ’s (where ”different” means: ”with different Julia sets”). It will play an important role for our discussion.

In §4 we study the initial geometry of the puzzle. The main result of this section is the construction of an initial annulus V^{0}\V^{1} with definite modulus, provided the hybrid class of a map is selected from a truncated secondary limb (Theorem I).

In §5 we prove the main geometric result of the paper saying that the principal moduli grow linearly with the number of central cascades. We introduce a new geometric parameter (worked out jointly with J. Kahn), the asymmetric modulus, and prove first that it is monotonically non-decreasing when we go down along the principal nest (Theorem II). This already provides us with lower bounds for the principal moduli μ_{m} = mod (V^{m−1}\V^{m}) (which, by the way, implies the Branner-Hubbard-Yoccoz divergence property), and upper bounds on the distortion. We reach these results by means of a purely combinatorial analysis plus the standard Grötzsch inequality.

However this analysis does not always yield the linear growth of moduli. In particular, it is not enough for the basic example called the Fibonacci map. The crucial part of the proof is to gain this extra growth for Fibonacci-like combinatorics. The argument based on the Definite Grötzsch inequality involves estimates of hyperbolic distances between puzzle pieces and analysis of their shapes. The key observation is that sufficiently pinched pieces make a definite extra contribution to the moduli growth (Theorem III).

** This result with a different proof has been also announced by Levin and van Strien [LS].

At the end of this section we prove a priori bounds for infinitely renormalizable quadratics of sufficiently big type (Theorems IV and IV′ ). The meaning of this condition is that certain combinatorial parameters of the renormalized maps R^{n}f are sufficiently big (depending on the truncated secondary limb to which the internal class of R^{n}f belongs). The main such parameter is the above mentioned height, but there are also several others. These conditions together mean roughly that the periods of R^{n}f are sufficiently big. The only difference is a possibility of long “parabolic or Siegel cascades”.

In the next short section, §6, we show that the Julia sets of the quadratics under consideration are locally connected. This follows from shrinking of the puzzle-pieces.

In the last section, §7, we outline the content of the forthcoming notes.

Before finishing let us draw the reader’s attention to the work of Graczyk and Swiatek [GS] containing a related result on the growth of moduli (within a certain nest of domains different from the principal nest) for real quadratics.

Acknowledgement. I would like to thank Jeremy Kahn for a fruitful suggestion, and Curt McMullen, Mary Rees and Mitsuhiro Shishikura for useful comments on the results. I also had useful discussions with John Milnor on combinatorics of external rays. Feedback from the Stony Brook dynamical group during my course in the fall 1994 was very helpful for cleaning up the exposition. Let me also thank Scott Sutherland and Brian Yarrington for help with the computer pictures. The results of this work were obtained in June 1993 during the Warwick Workshop on hyperbolic geometry. I am grateful to the organizers, particularly David Epstein and Caroline Series, for that wonderful time.

§2. Preliminaries: Douady-Hubbard renormalization and Yoccoz puzzle.

2.1. General terminology and notations. Given two subsets V and W of the complex plane, we say that V is strictly contained in W , V ⊂⊂ W , if cl V ⊂ int W .

Let B(z, ϵ) denote the disk of radius ϵ centered at z. Given two sets A and B , let dist (A, B) = inf{dist (z, ζ) : z ∈ A, ζ ∈ B}.

By orb z we denote the forward orbit {f^{n}z}_{n=0}^{∞} of z, and by ω(z) its ω -limit set. Let also orb_{n}z = {f^{m}z}_{m=0}^{n} . By a topological disk we will mean a simply connected region in C . By an annulus we mean a doubly connected region. A horizontal curve in an annulus A is a preimage of a circle centered at 0 by the Riemann mapping A → {z : 1 < |z| < R}.

We assume that the reader is familiar with the basic theory of quasi-conformal maps (see [A]). Quasi-conformal and quasi-symmetric maps will be abbreviated as qc and qs correspondingly.

2.2. Polynomials. By now there are many surveys and books on holomorphic dynamics. The reader can consult [Be], [GC], [M1] for general reference, and [B], [DH1] for the quadratic case. Below we will state the main definitions and facts required for discussion. However we assume that the reader is familiar with the notions of periodic points, and their classification as attracting, neutral, parabolic and repelling.

Let f : C → C be a polynomial of degree d ≥ 2 . The basin of ∞ is the set of points escaping to ∞:

D_{f}(∞) ≡ D(∞) = {z ∈ C : f^{n}z → ∞}.

Its complement is called the filled Julia set:

K(f) = C\D(∞).

The Julia set is the common boundary of K(f) and D(∞) :

J(f) = ∂K(f) = ∂D(∞).

The Fatou set F (f) is defined as C\J(f) . The Julia set (and the filled Julia set) is connected if and only if non of the critical points escape to ∞, that is, all of them belong to K(f) .

Given a polynomial f , there is a conformal map (the Böttcher function)

B_{f} : U_{f} → {z : |z| > r_{f} ≥ 1}

of a neighborhood U_{f} of infinity onto the exterior of a disk such that B_{f}(fz) = (B_{f}z)^{d} and B_{f}(z) ∼ z as z → ∞. There is an explicit dynamical formula for this map:

B_{f}(z) = lim_{n→∞}(f^{n}z)^{1/d^{n}}  (2-1)

with an appropriate choice of the branch of the d^{n} th root.

If the Julia set J(f) is connected then ∂U_{f} contains a critical point b of f . Otherwise B_{f} coincides with he Riemann mapping of the whole basin of infinity D(∞) onto {z : |z| > 1} (in this case r_{f} = 1 ).

The external rays R_{θ} with angle θ and equipotentials E_{h} of level h are defined as the B_{f} -preimages of the straight rays {re^{iθ} : r_{f} < r < ∞} and the round circles {e^{iθ} : 0 ≤ θ ≤ 2π}. They form two orthogonal invariant foliations of D(∞) .

Theorem 2.1 (see [M1], §18, or [H]). If a is a repelling periodic point of f , then there is at least one but at most finitely many external rays landing at a.

Sketch. Let us sketch the geometric construction of these rays (see the above references for the details). Let us linearize f^{p} near the periodic point a: Let ψ : (U, 0) → (V, a) be a local conformal map normalized by ψ′(0) = 1 and such that

ψ(λz) = f^{p}(ψz)  (2-2),

where λ is the multiplier at a. By means of this functional equation ψ can be analytically extended to the whole complex plane. Thus we obtain an entire function ψ : C → C satisfying (2-2).

Let D_{i} be the components of ψ^{−1}D(∞) . It is not hard to see that ψ : D_{i} → D(∞) is a universal covering map. Moreover, the function

h_{i} = log B_{f} ∘ ψ : D_{i} → H  (2-3)

gives the Riemann map of D_{i} onto the upper half-plane H .

The key point is that there are only finitely many components D_{i} which hence invariant under some iterate f^{p} . Then the map h_{i} conjugates z ↦ λ^{p}z on D_{i} to z ↦ d^{p}z on H . Take now the hyperbolic geodesic γ joining 0 and ∞ in H (the vertical ray) and the corresponding hyperbolic geodesic R̂_{i} = h_{i}^{−1}γ in D_{i} . It follows that R̂_{i} is invariant with respect to z ↦ λ^{p}z, and hence lands at 0. Hence the external ray R_{i} = ψR̂_{i} lands at a. ⊔ ⊓

Thus the external rays landing at a are organized in several cycles. The rotation number of these cycles is the same, and is called the combinatorial rotation number ρ(a) of a. Let R(a) denote the union of the external rays landing at a, and

R(ā) = ∪_{k=0}^{p−1}R(f^{k}a)

(where p is the period of a). This configuration (with the external angles marked on the rays) is called the rays portrait of the cycle ā. The class of isotopic portraites is called the abstract rays portrait.

2.3. Quadratic family. Let now f ≡ P_{c} : z ↦ z^{2} + c be the quadratic family. In this case the rays portraites of periodic cycles have quite specific combinatorial properties. The reader can consult [DH1], [At], [GM], [Sch], [M4] for the proofs of the results quated below.

Proposition 2.2 (see [M4]). Let ā = {a_{k}}_{k=0}^{p−1} be a repelling periodic cycle such that there are at least two rays landing at each point a_{k} .

(i) Let S_{1} be the components of C\R(ā) containing the critical value c. Then S_{1} is a sector bounded by two external rays.

(ii) Let S_{0} be the component of C\f^{−1}R(ā) containing the critical point 0. Then S_{0} is bounded by four external rays: two of them lands at a periodic point a_{k} , and two others land at the symmetric point −a_{k} .

(iii) The rays of R(ā) form either one or two cycles under iterates of f .

A particular situation of such kind is the following. Let b̄ = {b_{k}}_{k=0}^{p−1} be an attracting cycle, p > 1 . Let D_{k} be the components of its basin of attraction containing b_{k} . Then the boundaries of D_{k} are Jordan curves, and the restrictions f^{p}|D_{k} are topologically conjugate to the doubling map z ↦ z^{2} of the unit circle. Hence there is a unique f^{p} - fixed point a_{k} ∈ D_{k} . Altogether these points form a repelling periodic cycle ā (whose period may be smaller than p), with at least two rays landing at each a_{k} . The portrait R(ā) will be also called the rays portrait associated to the attracting cycle b̄.

A case of special interest for what follows is the fixed points portraites. There is always a fixed point called β which is the landing point of the invariant ray R_{0} . Moreover, this is the only ray landing at this point, so that it is non-dividing: the set K(f)\{β} is connected.

If the second fixed point called α is also repelling, it is much more interesting. Namely, this point is dividing, so that there are several external rays landing at it. These rays are cyclically permuted by dynamics with some combinatorial rotation number q/p.

The Mandelbrot set M is defined as the set of c ∈ C for which J(P_{c}) is connected, that is, 0 does not escape to ∞ under iterates of P_{c} . If c ∈ C\M , then J(P_{c}) is a Cantor set.

The Mandelbrot set itself is connected (see [DH1], [CG]). This is proven by constructing explicitly the Riemann mapping B_{M} : C\M → {z : |z| > 1}. Namely, let D_{c}(∞) be the basin of ∞ of P_{c} , and B_{c} be the Böttcher function (2-1) of P_{c} . Then

B_{M}(c) = B_{c}(c).  (2-4)

The meaning of this formula is that the “conformal position” of a parameter c ∈ C\M coincides with the “conformal position” of the critical value c in the basin D_{c}(∞) . This relation is the key to the similarity between dynamical and parameter planes.

Using the Riemann mapping B_{M} we can define the parameter external rays and equipotentials as the preimages of the straight rays going to ∞ and round circles centered at 0. This gives us two orthogonal foliations in the complement of the Mandelbrot set.

A quadratic polynomial P_{c} with c ∈ M is called hyperbolic if it has an attracting cycle. The set of hyperbolic parameter values is the union of some components of int M called hyperbolic components. Conjectually all components of int M are hyperbolic. This Conjecture would follow from the MLC Conjecture (Douady & Hubbard [DH1]).

Let H ⊂ int M be hyperbolic component of the Mandelbrot set, and let b̄(c) = {b_{k}(c)}_{k=0}^{p−1} be the corresponding attracting cycle. On the boundary of H the cycle b̄ becomes neutral, and there is a single point c_{H} ∈ ∂H where f^{p}(b_{0}) = 1 [DH1]. This point is called the root of H .

Given a c ∈ H , there is the rays portrait R_{c} associated to the corresponding attracting periodic point. Let θ_{1} and θ_{2} be the external angles of the two rays bounding the sector S_{1} of Proposition 2.2.

Theorem 2.3 (see [DH1], [M4], [Sch]). The parameter rays R_{1}^{H} and R_{2}^{H} with angles θ_{1} and θ_{2} land at the root c_{H} . There are no other rays landing at c_{H} .

The region W_{H} in the parameter plane bounded by the rays R_{i}^{H} and containing H is called the wake (originated from H ). The part of the Mandelbrot set contained in the wake together with the root c_{H} is called the limb L_{H} of the Mandelbrot set originated at H . The root of H is also called the root of wake W_{H} and limb L_{H} .

Recall that for c ∈ H , ā_{c} denotes the repelling cycle associated to the basin of attraction of the attracting cycle b̄_{c} . The dynamical meaning of the wakes is reflected in the following statement.

Proposition 2.4 (see [GM]). The repelling cycle ā_{c} stays repelling throughout the wake W_{H} . The corresponding rays portrait R(a_{c}) preserves its isotopic type throughout this wake.

The main cardioid of M is defined as the set of points c for which P_{c} has a neutral fixed point α_{c} , that is, |P′_{c}(α_{c})| = 1 . It encloses the hyperbolic component where P_{c} has an attracting fixed point. In the exterior of the main cardioid both fixed points are repelling.

The limbs attached to the main cardioid are called primary. Let H be a hyperbolic component attached to the main cardioid. The limbs attached to such a component are called secondary. We refer to a truncated limb if we remove from it a neighborhood of its root (Figure 1).

Figure 1: Truncated secondary limbs of the Mandelbrot set.

We refer to a truncated limb if we remove from it a neighborhood of its root (Figure 1).

2.4. Douady-Hubbard polynomial-like maps. The main reference for the following material is [DH2]. Let U′ ⊂⊂ U be two topological disks (with a non-degenerate annulus U\U′ ). A branched covering f : U′ → U is called a DH polynomial-like map (we will sometimes skip “DH” in case this does not cause confusion with “generalized” polynomiallike maps defined below). Every polynomial with connected Julia set can be viewed as a polynomial-like map after restricting it onto an appropriate neighborhood of the filled Julia set. Polynomial-like maps of degree 2 are called (DH) quadratic-like. We will always normalize quadratic-like maps so that the origin 0 will be its critical point.

One can naturally define the filled Julia set of f as the set of non-escaping points:

K(f) = {z : f^{n}z ∈ U′ : n = 0, 1, …}.

The Julia set is defined as J(f) = ∂K(f) . These sets are connected if and only if non of the critical points is escaping.

The choice of the domain U′ and range U of a polynomial-like map is not canonical. Two polynomial-like maps f : U′ → U and g : V′ → V are considered to be the same if K(f) = K(g) and f coincides with g on a neighborhood of K(f) .

Given a polynomial-like map f : U′ → U , we can consider a fundamental annulus A = U′\U . It is certainly not a canonical object but rather depending on the choice of U′ and U . Let

mod (f) = sup mod A,

where A runs over all fundamental annuli of f .

Two polynomial-like maps f and g are called topologically (quasi-conformally, conformally) conjugate if there is a homeomorphism (qc map, conformal isomorphism correspondingly) h from a neighborhood of K(f) to a neighborhood of K(g) such that h ∘ f = g ∘ h.

If there is a qc conjugacy h between f and g with ¯∂h = 0 almost everywhere on the filled Julia set K(f) , then f and g are called hybrid or internally equivalent. A hybrid class H(f) is the space of DH polynomial-like maps hybrid equivalent to f modulo conformal equivalence. According to Sullivan [S1], a hybrid class of polynomial-like maps should be viewed as as infinitely dimensional Teichmüller space. In contrast with the classical Teichmüller theory this space has a preferred point:

Straightening Theorem [DH2]. Any hybrid class H(f) of DH polynomial-like maps with connected Julia set contains a unique (up to affine conjugacy) polynomial.

In particular, any hybrid class of quadratic-like maps with connected Julia set contains a unique quadratic polynomial z ↦ z^{2} + c with c = c(f) ∈ M . So the hybrid classes of quadratic-like maps are labeled by the points of the Mandelbrot set. In what follows we will freely identify a quadratic hybrid class with its label c ∈ M .

Sullivan supplied any hybrid class with the following Teichmüller metric [S1]:

dist_{T} (f, g) = inf log K_{h},

where h runs over all hybrid conjugacies between f and g, and K_{h} denotes the qc dilatation of h. The Teichmüller distance from f to the quadratic P_{c(f)} : z ↦ z^{2} + c(f) in its hybrid class is controlled by the modulus of f :

Proposition 2.5. If mod (f) ≥ μ > 0 then dist_{T} (f, P_{c(f)}) ≤ C with a C = C(μ) depending only on μ. Moreover, C(μ) → 0 as μ → ∞.

This is the reason why the control of moduli of polynomial-like maps is crucial for the renormalization theory (see [S2], [McM]).

Let us finish with the following remark: Given a polynomial-like map with connected Julia set, we can define external rays and equipotentials near the filled Julia set by conjugating it to a polynomial. This definition is certainly not canonical but rather depends on the choice of conjugacy. We will specify the particular choice whenever it matters.

2.5. Douady-Hubbard renormalization. The reverse procedure under the name of tuning is discussed in [DH2], [D] and [M3]. A more general point of view (but which is equivalent to the tuning, after all) is presented in [McM].

Let f : U′ → U be a quadratic-like map. Let ā be a dividing repelling cycle, so that there are at least two rays landing at each point of ā. Let R ≡ R(ā) denote the configuration of rays landing at ā, and let R′ = −R be the symmetic configuration. Let us aslo consider an arbitrary equipotential E . Let now Ω be the component of C\(E ∪ R ∪ R′) containing the critical point 0. By Proposition 2.2, it is bounded by four arcs γ_{i} of external rays ant two pieces of the equipotential E .

Figure 2: A renormalization domain for the Feigenbaum polynomial.

Let p be the period of the above rays, and a ∈ ∂Ω be the periodic point of cycle ā. Let us consider a domain Ω′ ⊂ Ω , the component of f^{−p}Ω attached to a (see Figure 2). If Ω′ ∋ 0 then f^{p} : Ω′ → Ω is a double covering map. A quadratic-like map f is called DH-renormalizable if there is a repelling cycle ā as above such that Ω′ ∋ 0 , and 0 does not escape Ω′ under iterates of f^{p} . We will also say that this renormalization is associated to the periodic point a. We call f immediately DH renormalizable if a can be selected as the dividing fixed point α of f .

Note that the disks Ω′, fΩ′, …, f^{p−1}Ω′ have disjoint interiors. Indeed, otherwise f^{k}Ω′ would be inside Ω for some k < p. But this is impossible since the external rays which bound f^{k}Ω′ are outside of Ω .

In the DH-renormalizable case by means of a “thickening procedure” (see [DH1] or [M2]) one can extract a polynomial-like map f^{p} : V′ → V . Namely, let us consider a little bit bigger domain V ⊃ Ω bounded by arcs of four external rays close to γ_{i} , two arcs of circles going around the points a and a′ = −a, and two arcs of E . Pulling V back by f^{p} , we obtain a domain V′ ⊂⊂ V such that the map f^{p} : V′ → V is quadratic-like. This map considered up to conformal conjugacy is called the DH renormalization of f .

Let now f : z ↦ z^{2} + c_{0} be a quadratic polynomial, c_{0} ∈ M . If it is renormalizable then there is a homeomorphic copy M_{0} ∋ c_{0} of the Mandelbrot set with the following properties (see [DH2, [D]). For z ∈ M′_{0} = M_{0}\{one point} the polynomial P_{c} : z ↦ z^{2} + c is renormalizable. Moreover, there is the parameter analytic extension a_{c} of the periodic point a to a neighborhood of M′_{0} such that the above renormalization of P_{c} is associated to a_{c} . At the parameter value b removed from M_{0} the periodic point a_{c} is becoming parabolic with multiplier one. This parameter value is called the root of M_{0} . The component H of M_{0} corresponding to the component of M enclosed by the main cardioid “gives origin” for the copy M_{0} . Vice versa, any hyperbolic component H of the Mandelbrot set gives origin to a copy of M . In particular, the copies corresponding to the immediate renormalization are attached to the main cardioid.

We will see below that among all renormalizations there is the first one, which we denote Rf (see §3.4). This renormalization corresponds to a maximal copy of the Mandelbrot (that is a copy, which is not contained in any bigger copies except M itself). Let M denote the family of maximal Mandelbrot copies.

Given any sequence M_{0}, M_{1}, … of maximal copies of M , there is an infinitely renormalizable quadratic polynomial P_{b} such that c(R^{m}P_{b}) ∈ M_{m} , m = 0, 1, . . .. Indeed, the sets

C_{N} = {b : c(R^{m}P_{b}) ∈ M_{m}, m = 0, 1, …, N}

form a nest of copies of M whose intersection consists of the desired parameter values.

We say that these infinitely renormalizable quadratics have the same combinatorics determined by the sequence M_{0}, M_{1}, . . .. The MLC problem for these parameter values is equivalent to the assertion that there is only one quadratic with a given combinatorics. In other words, the above copies C_{N} shrink to a point as N → ∞.

2.6. Yoccoz puzzle. Let f : U′ → U be a quadratic-like map with both fixed points α and β repelling. As usual, α denotes the dividing fixed point with rotation number ρ(α) = q/p, p > 1 . Let E be an equipotential sufficiently close to K(f) (so that both E and fE are closed curves). Let R_{α} denote the union of external rays landing at α. These rays cut the domain bounded by E into p closed topological disks Y_{i}^{(0)} , i = 0, …, p−1 , called puzzle pieces of zero depth (Figure 3). The main property of this partition is that f∂(∪Y_{i}^{(0)}) is outside of int (∪Y_{i}^{(0)}) .

Let us now define puzzle-pieces Y_{i}^{(n)} of depth n as the connected components of f^{−n}Y_{k}^{(0)} . They form a finite partition of the the neighborhood of K(f) bounded by f^{−n}E . If the critical orbit does not land at α, then for every depth there is a single puzzle-piece containing the critical point. It is called critical and is labeled as Y^{(n)} ≡ Y_{0}^{(n)} .

Let M(f) denote the family of all puzzle pieces of f of all levels. It is Markov in the following sense:

(i) Any two puzzle pieces are either nested or have disjoint interiors. In the former case the puzzle-piece of bigger depth is contained in the one of smaller depth.

(ii) The image of any puzzle-piece Y_{i}^{(n)} of depth n > 0 is a puzzle piece Y_{k}^{(n−1)} of the previous depth. Moreover, f : Y_{i}^{(n)} → Y_{k}^{(n−1)} is a two-to-one branched covering or a conformal isomorphism depending on whether Y_{i}^{(n)} is critical or not.

We say that f^{k}Y_{i}^{(n)} l -to-one covers a union of pieces ∪_{m,j}Y_{j}^{(m)} if f^{k}|int Y_{i}^{(n)} is l -to-one covering map onto its image, and

f^{k}|Y_{i}^{(n)} ∩ J(f) = ⋃_{m,j} Y_{j}^{(m)} ∩ J(f).

In this case ∪Y_{j}^{(m)} is obtained from f^{k}|Y_{i}^{(n)} by cutting with appropriate equipotentials.

On depth 1 we have 2p − 1 puzzle pieces: one central Y^{(1)} , p − 1 non-central Y_{i}^{(1)} attached to the fixed point α (cuts of Y_{i}^{(0)} by the equipotential f^{−1}E ), and p − 1 symmetric ones Z_{i}^{1} attached to α′ . Moreover, fY^{(1)} two-to-one covers Y_{1}^{(1)} , Y_{i}^{(1)} univalently covers Y_{i+1}^{(1)} , i = 1, …, p − 2 , and fY_{p−1}^{(1)} univalently covers Y^{(1)} ∪_{i} Z_{i}^{(1)} . Thus f^{p}Y^{(1)} cut by f^{−1}E is the union of Y^{(1)} and Z_{i}^{(1)} (Figure 3).

Theorem 2.6. (Yoccoz, 1990). Assume that both fixed points of a polynomial-like map f are repelling, and that f is DH non-renormalizable. Then the following divergence property holds:

∑_{n=0}^{∞} mod (Y^{(n)}\Y^{(n+1)}) = ∞.

Hence diam Y^{(n)} → 0 as n → ∞.

Corollary 2.7. Under the circumstances of the above theorem the Julia set J(f) is locally connected.

The reader can consult [H], [M] or [L2] for a proof (or read §5 of this paper).

The Yoccoz puzzle provides us with the Markov family of puzzle pieces to play with. There are several different ways to do this: by means of the Branner-Hubbard tableaux [BH], or by means of the Yoccoz τ -function (unpublished), or by means of the principal nest and generalized renormalization, as will be described below (compare [L2], [L3]).

§3. Principal nest and generalized renormalization.

In what follows we will always assume that both fixed points of the DH quadratic-like maps under consideration are repelling.

3.1 Principal nest. Given a set W = cl (int W) and a point z such that f^{l}z ∈ int W , let us define the pull-back of W along the orb_{l}z as the chain of sets W_{0} = W, W_{−1} ∋ f^{n−1}z, …, W_{−l} ∋ z such that W_{−k} is the closure of the component of f^{−1}(int W_{−k+1}) containing f^{l−k}z. In particular if z ∈ int W and l > 0 is the moment of first return of orb z back to int W we will refer to the pull-backs corresponding to the first return of orb z to int W .

Let us consider the puzzle pieces of depth 1 as described above: Y^{(1)}, Y_{i}^{(1)} and Z_{i}^{(1)} , i = 1, … p − 1 (Figure 3). If z ∈ Y^{(1)} then f^{p}z is either in Y^{(1)} or in one of Z_{i}^{(1)} . Hence either f^{pk}0 ∈ Y^{(1)} for all k = 0, 1, …, or there is the smallest t > 0 and a ν such that f^{tp}0 ∈ Z_{ν}^{(1)} . Thus either f is immediately DH renormalizable, or the critical point escapes through one of the non-critical pieces, attached to α′ .

In the immediately renormalizable case the principal nest of puzzle pieces consists of just single puzzle piece Y^{0} (which is not too informative). In the escaping case we will construct the principal nest

Y^{0} ⊃ V^{1} ⊃ V^{2} ⊃ …  (3-0)

in the following way. Let t be the first moment when f^{tp}0 ∈ Z_{i}^{(1)} . Then let V^{0} ∋ 0 be the pull-back of Z_{i}^{(1)} along the orb_{tp}0 . Further, let us define V^{n+1} as the pull-back of V^{n} corresponding to the first return of the critical point 0 back to int V^{n} . Of course it may happen that the critical point never returns back to int V^{n} . Then we stop, and the principal nest turns out to be finite. This case is called combinatorially non-recurrent. If the critical point is recurrent in the usual sense, that is ω(0) ∋ 0 , it is also combinatorially recurrent, and the principal nest is infinite.

Let l = l(n) be the first return time of the critical point back to int V^{n−1} . Then the map g_{n} = f^{l(n)} : V^{n} → V^{n−1} is two-to-one branched covering. Indeed, by the Markov property of the puzzle, f^{k}V^{n} ∩ int V^{n−1} = ∅ for k = 1, …, l − 1 , so that the maps f : V^{k} → V^{k−1} are univalent for those k ’s.

Let us call return to level n − 1 central if g_{n}0 ∈ V^{n} . In other words l(n) = l(n + 1) . Let us say that a sequence n, n + 1, n + N − 1 (with N ≥ 1 ) of levels (or corresponding puzzle pieces) of the principal nest form a (central) cascade if the returns to all levels n, …, n + N − 2 are central, while the return to level n + N − 1 is non-central. In this case

g_{n+k}|V^{n+k} = g_{n+1}|V^{n+k}, k = 1, …, N.

and g_{n+1}0 ∈ V^{n+N−1}\V^{n+N} . Thus all maps g_{n+1}, …, g_{n+N} are the same as quadraticlike. We call the number N of levels in the cascade its length. Note that a cascade of length 1 consists of a single non-central level. Let us call the cascade maximal if the return to level n − 1 is non-central. Clearly the whole principal nest is the union of disjoint maximal cascades. The number of such cascades is called the height χ(f) of f . In other words, χ(f) is the number of different quadratic-like maps among the g_{n} ’s. (If f is immediately renormalizable set χ(f) = −1 .)

Remark. Given a quadratic polynomial f : z ↦ z^{2} + c, the principal nest determines a specific way to approximate c by superattracting parameter values. Namely, one should perturb c in such a way that the critical point becomes fixed under g_{n} , while the combinatorics on the preceeding levels keeps unchanged. The number of points in this approximating sequence is equal to the height χ(f) . This resembles ”internal addresses” of Dierk Schleicher [Sch] but turns out to be different.

3.2. Initial Markov partition. Let P_{i} be a finite or countable family of topological disc with disjoint interiors, and g : ∪P_{i} → C be a map such that the restrictions g|P_{i} are branched coverings onto their images. This map is called Markov if gP_{i} ⊃ P_{j} whenever int gP_{i} ∩ int P_{j} ≠ ∅. Let us call it a standard Markov map if all restrictions g|P_{i} are one-to-one onto their images.

A Markov map is called Bernoulli (with range D) if gP_{i} ⊃ D ⊃ ∪P_{j} for all i. Similarly we can define a standard Bernoulli map.

We know that f^{p}Y^{(1)} two-to-one covers Y^{(1)} itself and the puzzle pieces Z_{i}^{(1)} attached to α′ . If f^{p}0 ∈ Y^{(1)} (central return) then the pull back of Y^{(1)} by this map is the critical piece Y^{(1+p)} , while each Z_{i}^{(1)} has two univalent pull-backs Z_{j}^{(1+p)} (we count them by j in an arbitrary way) (see Figure 3).

Figure 3: Initial partition ( p = 3 , t = 2 ).

Now, f^{p}Y^{(1+p)} two-to-one covers all these puzzle pieces. If we again have a central return, that is f^{p}0 ∈ Y^{(1+p)} , then the pull-back will give us one critical piece Y^{(1+2p)} , and 4(p − 1) off-critical Z_{j}^{(1+2p)} .

Repeating this procedure t times (where f^{tp}0 ∈ Z_{i}^{(1)} ), we obtain the initial central nest

Y^{(1)} ⊃ Y^{(1+p)} ⊃ … ⊃ Y^{(1+tp)},  (3-1)

and a family of non-critical puzzle pieces Z_{j}^{(1+sp)} , 0 ≤ s ≤ k − 1 . Moreover

f^{p}0 ∈ Z_{ν}^{(1+tp)}  (3-2)

for some ν .

Let us say that a set D is partitioned by pieces W_{i} rel F (f) if the int W_{i} are disjoint, and D ∩ J(f) = ∪W_{i} ∩ J(f) .

Thus we have partitioned Y^{(0)} rel F (f) by the pieces Z_{i}^{(1+sp)} , 0 ≤ s ≤ t, and Y^{(1+tp)} . Let us look closer at this last piece. Its image under f^{p} two-to-one covers all above puzzle pieces of depth 1 + tp. The pull-back of Z_{ν}^{(1+tp)} from (3-2) gives us exactly V^{0} ∋ 0 , the first puzzle-piece in the principal nest. The pull-backs of other pieces Z_{j}^{(1+tp)} give some non-critical pieces Z_{j}^{(1+(t+1)p)} . Finally, we have two univalent pull-backs Q_{1} and Q_{2} of Y^{(1+tp)} . Altogether these pieces give a partition of Y^{(1+tp)} rel F (f) .

To understand how the critical point returns back to V^{0} we need to partition further Q_{1} ∪ Q_{2} . To this end let us iterate the standard Bernoulli map f^{p}|Q_{1} ∪ Q_{2} with range Q_{1} ∪ Q_{2} ∪ V^{0} ∪ Z_{j}^{(1+(t+1)p)} . So take a point z ∈ Q_{1} ∪ Q_{2} and push it forward by iterated f^{p} until it escapes Q_{1} ∪Q_{2} (or iterate forever if it does not escape). It can escape through the piece V^{0} or through a piece Z_{j}^{(1+(t+1)p)} . In any case pull the corresponding piece back to this point. In such a way we will obtain the partition

Q_{1} ∪ Q_{2} = ⋃_{k>0} ⋃_{i} X_{i}^{kp} ⋃_{k>t+1} ⋃_{j} Z_{j}^{(1+kp)} ∪ K rel F (f),

where X_{i}^{k} denote the pull-backs of V^{0} under f^{kp} (for k = 0 we have just one piece X_{0}^{0} ≡ V^{0} ), Z_{j}^{1+kp} denote the pull-backs of Z_{j}^{(1+(t+1)p)} under f^{(k−t−1)p} , and K denote the residual set of non-escaping points.

Altogether we have constructed the initial Markov partition:

Y^{(0)}\K = V^{0} ⋃_{k>0} ⋃_{i} X_{i}^{kp} ⋃_{k>0} ⋃_{j} Z_{j}^{(1+kp)} rel F (f).  (3-3)

It is convenient (to restrict the number of iterates in the following considerations) to pass here to the following Markov map:

G : ⋃_{k,i} X_{i}^{k} ⋃_{k,j} Z_{j}^{(1+kp)} → C  (3-4).

Observe that for any j there is an i such that f^{sp}Z_{j}^{(1+sp)} univalently covers Z_{i}^{(1)} . Moreover f^{p−i}Z_{i}^{(1)} univalently covers Y^{(0)} . Let us set G|Z_{j}^{1+sp} = f^{ps+(p−i)} . The image of each piece Z_{j}^{(1+sp)} under this map univalently covers Y^{(0)} . Similarly let us set G|V^{0} = f^{tp+p−i} , where f^{tp}0 ∈ Z_{i}^{(1)} . The image of this piece two-to-one covers Y^{(0)} . Finally G|W_{i}^{k} = f^{kp} for k > 0 . These pieces are univalently mapped onto V^{0} .

3.3. Non-degenerate annulus. Yoccoz has shown that if f is non-renormalizable then in the nest Y^{(1)} ⊃ Y^{(2)} … there is a non-degenerate annulus Y^{(n)}\Y^{(n+1)} . However the modulus of this annulus is not under control. We will construct a different non-degenerate annulus whose modulus we will be able to control.

Let A^{n} = V^{n}\V^{n+1} .

Proposition 3.1. Let f be a DH quadratic-like map which is not immediately DH renormalizable. Then all annuli A^{n} are non-degenerate.

Proof. Observe first that V^{0} is strictly inside Y^{(0)} , that is, the annulus Y^{(0)}\V^{0} is non-degenerate. Indeed, V^{0} is the pull-back of Z_{i}^{(1)} which is strictly inside Y^{(0)} . As the iterates of ∂Y^{(0)} stay outside int Y^{(0)} , V^{0} may not touch ∂Y^{(0)} .

For the same reason all other pieces Z_{j}^{(1+kp)} and X_{i}^{k} of the initial Markov partition are strictly inside Y^{(0)} as well.

Let us consider the orbit of the critical point 0 under iterates of the map G from (3-4) until it returns back to V^{0} . It first goes through the Z -pieces of the initial Markov partition, then at some moment l ≥ 1 lands at a X_{i}^{s} , and the latest at the next moment returns back to V^{0} .

Since the map G : V^{0} ∪ Z_{j}^{(1+kp)} → C is Bernoulli with range Y^{(0)} , there is a piece P ⊂ V^{0} , such that G^{l}P two-to-one covers Y^{(0)} . Clearly V^{1} is the pull-back of X_{i}^{s} by G^{l} . Since X_{i}^{s} ⊂⊂ Y^{(0)} , we conclude that V^{1} ⊂⊂ V^{0} .

Now it is easy to see that all annuli A^{n} are non-degenerate as well. Indeed, it follows that the orbit of ∂V^{1} stays away from V^{1} . Hence V^{2} cannot touch ∂V^{1} , for otherwise there would be a point on ∂V^{1} which returns back to V^{1} . So A^{2} is no-degenerate. Now we can proceed inductively. ⊔ ⊓

3.4. DH renormalization and central cascades. In this section quadratic-like maps and renormalization are understood in the sense of Douady and Hubbard.

Proposition 3.2. A quadratic-like map is renormalizable if and only if it is either immediately renormalizable, or the principal nest V^{0} ⊃ V^{1} ⊃ … ends with an infinite cascade of central returns. Thus the height χ(f) is finite if and only if f is renormalizable.

Proof. Let the principle nest ends with an infinite central cascade V^{m−1} ⊃ V^{m} ⊃ … Then l_{m} = l_{m+1} = … = l, and g ≡ g_{m} = f^{l}|V^{m} . Hence ∩V^{k} consists of all points which never escape V^{m} under iterates of g, that is, ∩V^{k} = K(g) . Since 0 ∈ ∩V^{k} , K(g) is connected, and g : V^{m} → V^{m−1} is quadratic-like (take into account that V^{m} ⊂⊂ V^{m−1} by Proposition 3.1).

Take now the non-dividing fixed point b of g. Let us show that b is dividing for the big Julia set K(f) . To this end let us consider the configuration of the full external rays whose segments bound V ≡ V^{m} . They divide the plane into the central component containing V , and the family S = {S_{i}}_{i=1}^{t} of sectors. As g maps V onto a bigger piece, g(∂S_{i}) ⊂ S_{σ(i)} for any sector S_{i} ∈ S .

Let i_{1}, …, i_{r} be a set of indeces which are cyclically permuted by the map σ : {1, …, t} → {1, …, t}. Then there is a sequence of sectors S_{i_{1}} ≡ T_{1}, …, T_{r}, … with the following properties:

• Each T_{k} is bounded by two external rays, and does not contain 0; • The angle of each T_{k} at infinity is smaller than π;

• T_{k} ⊃ S_{i_{l}} with l ≡ k mod r;

• g(∂T_{k+1}) = ∂T_{k} , k = 1, 2, …

• T_{k} ⊂ T_{k+r} , while T_{k+1}, …, T_{k+r} are pairwise disjoint. Then for any k we have a nest of sectors, T_{k} ⊂ T_{k+r} ⊂ T_{k+2r} …, converging to a

sector T̃_{k} . These sectors also satisfy all the above properties, and moreover T̃_{k} = T̃_{k+r} . Hence g cyclically permut the vertices of these sectors, which therefore form a cycle of period r. Moreover, the points of this cycle are clearly dividing.

Let us finally note that actually r must be equal to 1, so that this cycle actually coincides with the fixed point b. Indeed, the opposite situation would contradict Proposition 2.2 (ii).

So we have a dividing periodic point b. Let Ω′ ⊂ Ω be the corresponding domains constructed in §2.5. Then Ω′ ⊃ K(g) and hence Ω′ ∋ 0 . It follows that g : Ω′ → Ω is a double covering. Moreover, g^{n}0 ∈ K(g) ⊂ Ω′ , n = 0, 1, . . .. Thus f is renormalizable.

Assume now that f is renormalizable. Let a be the corresponding periodic point, and Ω′ be the region bounded by rays landing at a and arcs of an equipotential, such that Rf = f^{l} : Ω′ → Ω is a double covering.

Then the fixed point α may not lie in int Ω′ , for otherwise int fΩ′ would intersect int Ω′ . Hence α does not cut the filled Julia set K(Rf) . But then the preimages of α don’t cut K(Rf) either. Hence given a puzzle-piece Y_{i}^{(n)} , either K(Rf) is contained in Y_{i}^{(n)} , or K(Rf) ∩ int Y_{i}^{(n)} = ∅. In particular V^{m} ⊃ K(Rf) . But then f^{l}0 ∈ V^{m} for all m, so that the first return times to V^{m} are uniformly bounded. But then this nest must end up with a central cascade. ⊔ ⊓

The above discussion shows that there is a well-defined first renormalization Rf with the biggest Julia set, and it can be constructed in the following way. If f is immediately renormalizable, then Rf is obtained by thickening Y^{(1)} → Y^{(0)} . Otherwise the principal nest ends up with the infinite central cascade V^{m−1} ⊃ V^{m} ⊃ . . .. Then Rf = g_{m} : V^{m} → V^{m−1} .

The internal class c(Rf) of the first renormalization belongs to a maximal copy M_{0} of the Mandelbrot set, that is, a copy which is not contained in any other one except the whole M .

3.5. Return maps and Koebe space. Let f be a DH quadratic-like map, and let V ∈ M(f) be a puzzle piece.

Lemma 3.3. Let z be a point whose orbit passes through int V . Let l be the first positive moment of time for which f^{l}z ∈ int V . Let U ∋ z be the puzzle piece mapped onto V by f^{l} . Then f^{l} : U → V is either a univalent map or two-to-one branched covering depending on whether U is non-critical or otherwise.

Proof. Let U_{k} = f^{k}U, k = 0, 1, … l. Since f^{k}z ∉ int V for 0 < k < l, by the Markov property of the puzzle, U_{k} ∩ int V = ∅ for those k ’s. Hence f : U_{k} → U_{k+1} is univalent for k = 1, …, l − 1 , and the conclusion follows. ⊔ ⊓

Let z ∈ int V be a point which returns back to int V , and let l > 0 be the first return time. Then there is a puzzle piece V (z) ⊂ V containing z such that f^{l}V (z) = V .

It follows that the first return map A_{V} f to int V is defined on the union of disjoint open puzzle pieces int V_{i} . Moreover, if

f^{m}∂V ∩ V = ∅, m = 1, 2, …  (3-4)

then it is easy to see that the closed pieces V_{i} are pairwise disjoint and are contained in int V . Indeed, otherwise there would be a boundary point ζ ∈ ∂V whose orbit would return back to V , despite (3-4).

Somewhat non-rigorously, we will call the map

A_{V} f : ⋃ V_{i} → V  (3-5)

the first return map to V . (Note: it perhaps may happen that a point z ∈ ∂V returns back to V but does not belong to ∪V_{i} .) Let V_{0} denote the critical (“central”) puzzle piece (provided the critical point returns back to V ). Now Lemma 3.3 immediately implies:

Lemma 3.4. The first return map f_{V} univalently maps all non-critical pieces V_{i} onto V , and maps the critical piece V_{0} onto V as a double branched covering.

Thus the first return map A_{V} f is Bernoulli, and is standard Bernoulli on ⋃_{i≠0} V_{i} . Let us now improve Lemma 3.3.

Lemma 3.5. Let z be a point whose orbit passes through the central domain int V_{0} of the first retun map (3-5), and l ≥ 0 be the first moment when f^{l}z ∈ V_{0} . Then there is a puzzle piece Ω ∋ z mapped univalently by f^{l} onto V .

Proof. Let s be the first moment when f^{s}z ∈ V . Then f^{l}z = f_{V}^{k} (f^{s}z) for some k ≥ 0 . Moreover, (A_{V} f)^{r}(f^{s}z) ∉ int V_{0} for r < k.

Since the return map is standard Bernoulli outside of the central piece, there is a piece X ⊂ V containing f^{s}z which is univalently mapped by (A_{V} f)^{k} onto V . On the other hand, by Lemma 3.3 there is a domain D ∋ z which is univalently mapped by f^{s} onto V . Hence the domain D ∩ f^{−s}X is univalently mapped by f^{l} onto V . ⊔ ⊓

Let us now consider the principal nest of f :

Y^{0} ⊃ V^{0} ⊃ V^{1} ⊃ ….

Let

g_{n} : ∪V_{i}^{n} → V^{n−1}  (3-6)

be the first return map to V^{n−1} , where V_{0}^{n} ≡ V^{n} ∋ 0 . We will call them the principal return maps. We will also let g_{0} ≡ f .

Let Φ : z ↦ z^{2} be the quadratic map. Since f : U′ → U is a double covering with the critical point at 0, it can be decomposed as h ∘ Φ where h is a univalent map with range U .

Corollary 3.6. For n ≥ 1 the pieces V_{i}^{n} are pairwise disjoint, and the annuli V^{n−1}\V_{i}^{n} are non-degenerate. For n ≥ 2 the map g_{n}|V_{i}^{n} can be decomposed as h_{n,i} ∘ Φ where h_{n,i} is a univalent map with range V^{n−2} .

Proof. As V^{n} ⊂ int V^{n−1} (Proposition 3.1), and

f^{m}(∂V^{n}) ∩ int V^{n−1} = ∅,

condition (3-4) is satisfied here, and the first statement follows.

Take a piece V_{i}^{n} , and let g_{n}|V_{i}^{n} = f^{l} . By Lemma 3.5 the map f^{l−1} : (fV_{i}^{n}) → V^{n−1} can be extended to a univalent map with range V^{n−2} , and the second statement also follows. ⊔ ⊓

3.6. Bernoulli scheme associated to a central cascade. In the case of a central cascade we need a more precise analysis of the Koebe space. Let us consider a central cascade C ≡ C^{m+N} :

V^{m} ⊃ V^{m+1} ⊃ … ⊃ V^{m+N−1} ⊃ V^{m+N},  (3-7)

where g_{m+1}0 ∈ V^{m+N−1}\V^{m+N} . Set g = g_{m+1}|V^{m+1} . Then g : V^{k} → V^{k−1} is a double branched covering, k = m + 1, … m + N .

Let us consider the first return map g_{m+1} : V_{i}^{m+1} → V^{m} as in (3-6). Let us pull the pieces V_{i}^{m+1} back to the annuli A^{k} = V^{k−1}\V^{k} by iterates of g, k = m + N, …, m + 1 . We obtain a family W(C) of pieces W_{j}^{k} . By construction, W_{j}^{k} ⊂ A^{k} and g^{k−m+1} univalently maps W_{j}^{k} onto some V_{i}^{m+1} ≡ W_{i}^{m+1} .

Let us define a standard Bernoulli map G ≡ G^{m+N} :

G : ⋃_{W(C)} W_{j}^{k} → V^{m}  (3-8)

as follows: G|W_{j}^{k} = f_{m+1} ∘ g^{k−m+1} (see Figure 4).

Remark. This Bernoulli map is similar to the initial Bernoulli map constructed in §3.5. Actually in the initial construction we deal with the central cascade (3-1) with degenerate annuli.

Lemma 3.7. Let us consider the central cascade (3-7). Let z be a point whose orbit passes through int V^{m+N} , and l be the first moment for which f^{l}z ∈ V^{m+N} . Then there is a piece Ω ∋ z which is univalently mapped by f^{l} onto V^{m} .

Proof. Let s be the first moment for which f^{s}z ∈ V^{m} . Then f^{l}z = G^{k}(f^{s}z) where G is the Bernoulli map (3-8). Now repeat the argument of Lemma 3.5 just using G instead of the first return map. ⊔ ⊓

Corollary 3.8. Let us consider the central cascade (3-7). Then the map g_{m+N+1} : V^{m+N+1} → V^{m+N} can be represented as h_{m+N+1}(z^{2}) where h_{m+N+1} is a univalent map with range V^{m} .

Proof. Repeat the argument for Corollary 3.6 using Lemma 3.7 instead of 3.5. ⊔ ⊓

3.7. Generalized polynomial-like maps and renormalization. Let {U_{i}} be a finite or countable family of topological discs with disjoint interiors strictly contained in a topological disk U . We call a map g : ∪U_{i} → U a (generalized ) polynomial-like map if g : U_{i} → U is a branched covering of finite degree which is univalent on all but finitely many U_{i} .

Figure 4: Bernoulli scheme associated to a central cascade.

Then we can define the filled Julia set K(g) as the set of all non-escaping points, and the Julia set J(g) as its boundary. The DH polynomial-like maps correspond to the case of a single disk U_{0} .

Let us say that a polynomial-like map g is of finite type if its domain consists of finitely many disks U_{i} .

Generalized Straightening Theorem. Any generalized polynomial-like map of finite type is qc conjugate to a polynomial with the same number of non-escaping critical points.

Lemma 3.9. A generalized polynomial-like map with non-escaping critical point has a connected Julia set/(filled Julia set) if and only if it is DH polynomial-like.

Let us call a (generalized) polynomial-like map a (generalized) quadratic-like if it has a single (and non-degenerate) critical point. In such a case we will always assume that 0 is the critical point, and count the discs U_{i} in such a way that U_{0} ∋ 0 . In what follows we will deal exclusively with quadratic-like maps, namely with the principal sequence g_{n} of the first return maps (3-6).

Given a V_{j}^{n+1} , let l be its first return time back to V^{n} , that is, g_{n+1}|V_{j}^{n+1} = g_{n}^{l} . Then

g_{n}^{k}V_{j}^{n+1} ⊂ V_{i(k)}^{n}, k = 0, 1, …, l,

with i(0) = i(s) = 0 . Moreover, g_{n}^{k}V_{j} ⊂⊂ V_{i(k)}^{n} for k < s. The sequence 0 = i(0), i(1), …, i(s) = 0 is called the itinerary of V_{j}^{n+1} through the domains of previous level.

Philosophically the dynamical renormalization is the first return map to an appropriate piece of the phase space considered up to rescaling. In our setting let us define the n -fold generalized renormalization T^{n}f of f as the first return map g_{n} restricted to the union of puzzle pieces V_{i}^{n} meeting the critical set ω(0) , and considered up to rescaling. In the most interesting situations these maps are of finite type (compare [L3]):

Lemma 3.10. If f is a DH renormalizable quadratic-like map, then all maps T^{n}f are of finite type.

Proof. In the tail of the principal nest the maps g_{n} are DH quadratic-like, and their domains consist just of one component. So we should take care only of the initial piece of the cascade.

Let us take the renormalization Rf = f^{l} : V^{t+1} → V^{t} with t ≥ n. Since 0 is non-escaping under iterates of Rf , we have the following property: the first return time of any point f^{k}0 back to V^{t+1} is at most l. All the more, the return time to any bigger domain V^{n} ⊃ V^{t} is bounded by l. Hence the components of f^{−t}V^{n} , t = 0, 1, …, l − 1 , cover the whole postcritical set. For sure there is only finitely many such components. But the domain of T^{n}f is the union of some of these components (which are inside V^{n} ). ⊔ ⊓

3.8. Return graph. Let I^{n} be the family of puzzle pieces V_{i}^{n} intersecting ω(0) , that is, the pieces in the domain of the generalized renormalization

T^{n}f : ⋃_{I^{n}} V_{i}^{n} → V^{n−1}.

Let us consider a graded graph Υ_{f} whose vertices of level n are the pieces V_{j}^{n} ∈ I^{n} , n = 0, 1, . . .. Let us take a vertex V_{j}^{n} ∈ I^{n} , and let i(1), …, i(t) = 0 be its itinerary through the pieces of the previous level under the iterates of g_{n} (see the previous section). Then for n ≥ 1 join V_{j}^{n+1} with V_{i}^{n} by k edges, provided the symbol i appears in the above itinerary k times. This means that the piece V_{j}^{n+1} under iterates of g_{n} passes through V_{i}^{n} k times before the first return back to V^{n−1} . As to the top level, let us join each V_{j}^{1} with V^{0} by the number of edges equal to the first return time of V_{j}^{1} back to V^{0} under iterates of f = g_{0} .

Remark. A similar graph in the real one-dimensional setting was introduced by Marco Martens [Ma]. The above graph is not exactly the same as Marten’s graph, as the latter is related to the iterates of f itself rather than the renormalized maps.

Note that for any vertex V_{j}^{n+1} with n > 0 there is exactly one edge joining it to the critical vertex V_{0}^{n} of the previous level. Note also that by Lemma 3.10 in the DH renormalizable case the number of vertices on a given level is finite. In any case there are clearly only finitely many edges leading from a V_{j}^{n+1} to the previous level.

By a path in the graph Υ_{f} we mean a connected sequence of edges such that no two of them join the same two levels, up to reversing the order of the sequence. So we don’t endow the paths with orientation, and can go along them either strictly upwards or strictly downwards.

Diverse combinatorial data can be easily read off this graph. For example, given n ≥ m, the number of paths joining V_{j}^{n+1} to V_{i}^{m} is equal to the number of times which the g_{m} -orbit of V_{j}^{n+1} passes through V_{i}^{m} before the first return back to V^{n} . Hence the return time of V_{j}^{n+1} back to V^{n} under iterates of g_{m} is equal to the total number of paths in Υ_{f} leading from V_{j}^{n+1} up to level m. Let g_{m} − time(V_{j}^{n+1}) denote this return time.

Let now the map f be DH renormalizable, and t be a renormalization level in the principal nest, that is, g_{t+1} : V^{t+1} → V^{t} is a quadratic-like map with non-escaping critical point. Then there is a single vertex V^{t+1} at level t + 1 , and below it the return graph is just the “vertical path” through the critical vertices. Let per(f) ≡ g_{0} − time(V^{t+1}) denote the renormalization period, that is, the return time of V^{t+1} back to V^{t} under iterates of the original map f = g_{0} .

By the above discussion, per(f) is equal to the total number of paths in the graph Υ_{f} (joining the top vertex V^{0} to the bottom vertex V^{t+1} ). It follows that the per(f) is bounded if and only if the DH-level t is bounded, and all return times g_{m} − time(V_{i}^{m+1}) are bounded for 1 ≤ m ≤ t and any i. For example, the “if” statement means: If t ≤ T and g_{m}−time(V_{i}^{m+1}) ≤ R, then per(f) ≤ P (T, R) . Indeed, in this case the total number of paths in the graph is bounded by R^{T} .

Let us now take a closer look at the central cascades. Let us consider a central cascade (3-7) and a non-pre-critical piece V_{j}^{m+N+1} , that is g_{m+N}V_{j}^{m+N+1} ≠ V_{0}^{m+N} . Then let us denote by G_{m+N} − time(V_{j}^{m+N+1}) the first return time of g_{m+N}V_{i}^{m+N+1} back to V^{m+N} under the iterates of the map G_{m+N} from (3-8).

This time can be expressed in terms of the following reduced graph Υ_{f}^{r} . This graph is obtained from the Υ_{f} by removing from all central cascades (3-7) the edges leading from the non-critical pieces V_{j}^{k+1} , j ≠ 0 , to the critical piece V_{0}^{k} , k = m + N − 1, …, m. Then G_{m+N} − time(V_{j}^{m+N+1}) is equal to the number of paths in the reduced graph leading from V_{j}^{m+N+1} up to level m.

Let us define the reduced period per_{r}(f) as the total number of paths in the reduced graph. This means that we don’t count the moments of time when the orbit goes through the intermediate levels of central cascades. Clearly the reduced period of f is bounded if and only if the height χ(f) and all G − times are bounded.

Let us finally define one more combinatorial notion, the rank. Let V_{j}^{n+1} and V_{i}^{n} be two Υ_{f} -adjacent puzzle pieces, and let γ denote an edge joining them. Let D^{n} ⊃ V_{i}^{n} be a puzzle piece contained in V^{n−1} . Consider the first moment t for which g_{n}^{t}V_{j}^{n+1} ⊂ V_{i}^{n} . The piece D^{n+1} ⊃ V_{j}^{n+1} in V^{n} such that g_{n}^{t}D^{n+1} = D^{n} will be called the pull-back of D^{n} along the edge γ . More generally, let us define the pull-back of D^{n} along a path γ leading from V_{i}^{n} ⊂ D^{n} downwards by composing the pull-backs along the edges.

Given a piece V_{k}^{n} , let rankV_{k}^{n} be the number of non-central levels in the shortest path leading from V_{k}^{n} down to a critical piece. For a piece D^{n} ⊂ V^{n−1} as above, let rank(D^{n}) denote the minimum of ranks of puzzle pieces V_{k}^{n} contained in D^{n} . Note that rank(D^{n}) = 0 iff D^{n} is critical.

By a path through D^{n} we will mean a path through a piece V_{i}^{n} ⊂ D^{n} .

Lemma 3.11. Let γ be the shortest path leading from D^{n} down to a central piece V_{0}^{n+t} , and let D^{n+t} be the pull-back of D^{n} along this path. Then

V^{n+t} ⊂ D^{n+t} ⊂ V^{n+t−1},

and the map D^{n+t} → D^{n} is a double branched covering.

Proof. Follows from the definitions. ⊔ ⊓

3.9. Full principal nest. Let f be a DH renormalizable, but not immediately, quadraticlike map. Then its principal nest

Y^{(0,0)} ⊃ V^{0,0} ⊃ V^{0,1} ⊃ … ⊃ V^{0,t(0)} ⊃ V^{0,t(0)+1} ⊃ …

ends up with an infinite cascade of central returns (we mark this nest with two labels for the reason which will become clear in a moment). Let us select a level t(0) of this cascade, so that the return map Rf = g_{0,t(0)+1} : V^{0,t(0)+1} → V^{0,t(0)} is DH quadratic-like. We will call such a level DH. (The particular choice of DH renormalizable levels in what follows will depend on the geometry).

If Rf is not immediately DH renormalizable, let us cut the puzzle piece V^{0,t(0)+1} by the external rays, and construct its short principal nest:

Y^{(1,0)} ⊃ V^{1,0} ⊃ V^{1,1} ⊃ … ⊃ V^{1,t(1)} ⊃ V^{1,t(1)+1} ⊃ …

If Rf is DH renormalizable, then this nest also ends up with an infinite central cascade. Then select a DH level t(1) + 1 , and pass to the next short nest.

If f is infinitely DH renormalizable but non of the renormalizations are immediate, then in such a way we construct the full principal nest

\begin{matrix}Y^{(0,0)} ⊃ V^{0,0} ⊃ V^{0,1} ⊃ … ⊃ V^{0,t(0)} ⊃ V^{0,t(0)+1} ⊃ \\ Y^{(1,0)} ⊃ V^{1,0} ⊃ V^{1,1} ⊃ … ⊃ V^{1,t(1)} ⊃ V^{1,t(1)+1} ⊃ … \\ Y^{(m,0)} ⊃ V^{m,0} ⊃ V^{m,1} ⊃ … ⊃ V^{m,t(m)} ⊃ V^{m,t(m)+1} ⊃ … \end{matrix} .  (3-9)

Here Y^{(m,0)} is the first critical Yoccoz puzzle piece for the m -fold DH renormalization R^{m}f , while the pieces V^{m,n} form the corresponding short principal nest. Moreover, for m > 1 , Y^{(m,0)} is obtained by cutting V^{m−1,t(m−1)+1} with the external rays of R^{m}f : V^{m−1,t(m−1)+1} → V^{m−1,t(m−1)} .

The annuli A^{m,n} = V^{m,n−1}\V^{m,n} will be called the principal annuli.

3.10. Big type: special families of Mandelbrot copies. Assume that we associated to any quadratic-like map its “combinatorial type” τ(f) , which depends only on the hybrid class c(f) and is constant over any maximal copy of the Mandelbrot set. (Keep in mind the height function χ(f) or the period per(f) .) Thus we can use the notation τ(M′) .

Let S ⊂ M be a family of maximal copies of the Mandelbrot set. Let us call it τ -special if it satisfies the following property: For any truncated secondary limb L there is a τ_{L} such that M contains all copies M′ ⊂ L of the Mandelbrot set with τ(M′) ≥ τ_{L} .

Let f be an infinitely DH renormalizable quadratic-like map. Let us say that it is of S -type if all internal classes c(R^{n}f) belong to copies M′ from S .

§4. Initial geometry.

4.1. Geometry of rays. Let L ≡ L_{b} be a limb of the Mandelbrot set with root at b. By L^{tr} = L_{b}^{tr} we denote a truncated limb, that is, L with a neighborhood of the root removed.

We say that a pre-periodic point is repelling if the corresponding periodic point is. Given a repelling periodic or pre-periodic point a of P_{c} , let R(a) be the union of the segments of rays landing at a up to the equipotential level 1 together with point a. We will use the Hausdorff topology on the space of configurations of curves. We will often suppress the label c unless it may cause confusion. Let B_{c} denote the Böttcher function (2-1) of P_{c} .

Lemma 4.1 (see [GM]). Let a_{c} be a periodic point of P_{c} continuously depending on c ∈ L_{b} which stays repelling on the unrooted limb L_{b}\{b}. Then the rays configuration R(a_{c}) depends continuously on c ∈ L_{b}\{b}.

Proof. By [GM] the external angles of the rays landing at a_{c} are the same through the limb. So R(a_{c}) is the B^{−1} -image of the fixed configuration of segments in the annulus {z : 1 < |z| < e} with the point a_{c} added.

Let us go back to the construction of the external rays in §2.2. It is easy to see from the explicit formulas for the Rimann map and the linearizing coordinate that the map (2-3) h_{i}^{−1} : H → D_{i} depends continuously (in the compact-open topology) on c ranging over the unrooted limb. Let I = {iy : 1 ≤ y ≤ 2^{p}} be a fundamental segment of the vertical geodesic in H . It follows that the curves Ĵ = h_{i}^{−1}I depend smoothly on c. Hence the curves

R̂_{i} = ⋃_{n=0}^{∞} λ^{−n}Ĵ

depend continuously on c in the Hausdorff topology. Then the rays R_{i} = ψ(R̂_{i}) depend continuously on c as well. ⊔ ⊓

Given a configuration C_{0} of finitely many parametrized curves and point in C , let us consider the space Teich (C_{0}) of all configurations qc equivalent to C_{0} modulo conformal equivalence. There is a natural Teicmüller distance on this space:

dist_{T} (C_{1}, C_{2}) = inf log K_{h},

where h runs over all qc equivalences between C_{1} and C_{2} .

We say that configurations of some family have bounded geometry if they stay bounded Teichmüller distance from a reference configuration C_{0} whose curves are smooth and intersect transversally.

Let us consider a family of two topological nested disks D_{1} ⊂ D_{2} with Γ_{i} = ∂D_{i} , and A = D_{2}\D_{1} . The statement that mod (A) > ϵ with an ϵ > 0 uniform over the family will be freely expressed in the following ways: “The annulus A has a definite modulus”, or “ D_{1} is well inside D_{2} ”, or “There is a definite space in between Γ_{1} and Γ_{2} .”

A statement like “If f has a definite modulus then a certain quantity is bounded” means: “If mod (f) > ϵ then there is a bound on that quantity depending only on ϵ ”.

Lemma 4.2. Under the circumstances of Lemma 4.1 the configuration R(a_{c}) has a bounded geometry when c ranges over the truncated limb L_{b}^{tr} of the Mandelbrot set.

Proof. Let us consider the configuration I of intervals obtained by rotating the interval [0, 1] by the cyclic group Z_{p} of order p. We will show that the configurations R(α_{c}) of rays stay bounded Teichmüller distance from I .

Let us use the notations from the proof of the previous lemma. Since the configuration R̂ of rays continuously depends on c ranging over the truncated limb L^{tr} , we can find a smooth Jordan curve γ enclosing 0 and smoothly depending on c, which intersects each infinite ray R̂_{i}^{∞} at only one point, namely the endpoint of R̂_{i} , and this intersection is transversal. The curves γ_{−N} = λ^{−N}γ clearly satisfy the same property. But for sufficiently big N (uniform over the truncated limb) γ_{−N} lies strictly inside γ with a definite space in between.

Let us consider the annulus bounded by γ and γ_{−N} with the segments of the rays R_{i} in between. It follows from the above that this configuration C stays bounded Teichmüller distance from the standard one C_{0} : the round annulus {re^{iθ} : 1/2 ≤ r ≤ 1} with p equally spaced straight intervals inside.

So there a K -qc map h : C → C_{0} with dilatation K depending only on the truncated limb, which conjugates z ↦ λ^{N}z to z ↦ 2z on the inner boundaries of the configurations. Pulling this map back by linear dynamics, we obtain a desired K -qc equivalence between the configuration R̂ and I .

Finally, it is easy to see that the linearizing map ψ is univalent in a neighborhood of 0 whose size is uniform over the truncated limb. Hence there is a uniform l such that ψ is univalent on R_{−l} ≡ λ^{−l}R̂. Hence the configuration R = f^{l}ψ(R_{−l}) has bounded geometry. ⊔ ⊓

Corollary 4.3. Let us consider a limb L_{b} of M and a finite set A_{c} of periodic or preperiodic points which stay repelling through the unrooted limb L_{b}^{tr} . Let C_{c} be the union of configuration R(A_{c}) of rays landing at points of A_{c} cut at level 1, together with A_{c} and several equipotentials (whose levels don’t depend on c). Then the configuration C_{c} has bounded geometry when c ranges over a truncated limb L_{b}^{tr}

Proof. It follows from Lemma 4.2 that near A_{c} the configuration C_{c} has bounded geometry (where “near” is uniform over the truncated limb). On the other hand, it clearly has bounded geometry outside a uniform neighborhood of A_{c} , since ϕ_{c}^{−1} is a normal family of maps. ⊔ ⊓

Let Y_{c}^{(n)} ≡ Y_{c}^{(n)}(f) denote the configuration of cutting curves for the Yoccoz puzzle of depth n for a map f . Now we immediately conclude:

Corollary 4.4. For any given n the Yoccoz configuration Y^{(n)}(P_{c}) has bounded geometry when c ranges over a truncated primary limb L_{b} .

If we consider a DH quadratic-like map f with mod (f) > ϵ, then we can K(ϵ) -qc conjugate it to a quadratic polynomial, and transfer the net of rays and equipotentials from this quadratic. In what follows we always assume that the choice of the net is made in this way.

Corollary 4.5. If f is a quadratic-like map with a definite modulus and internal class c(f) ranging over a primary truncated limb L_{b} of M , then Yoccoz configuration Y^{(n)}(f) has a bounded geometry for any given n.

4.2. Fundamental domain near the fixed point. The goal of this subsection is to construct a combinatorially defined fundamental domain with bounded geometry near the fixed point α. It is where the secondary limbs condition comes into the scene.

Let f be a DH quadratic-like map whose α -fixed point has rotation number q/p. This map has a single periodic point γ ∈ int Y^{(1)} of period p. Let γ′ be the symmetric point. Consider the family R(γ′) of rays landing at γ′ . Let D = D(f) be the component of Y^{(1)}\R(γ′) attached to the fixed point α (see Figure 5). Then f^{p}D univalently covers the component of Y^{(0)}\R(γ) attached to α. Note that ∂D ∩ ∂(fD) is contained in the union of two rays landing at α.

Figure 5: Fundamental domain near the fixed point.

Hence there is a univalent branch of f^{−p} which fixes α and maps D inside itself. It in now easy to see that f^{−pn}D shrink to α as n → ∞. So we can select Q = Q(f) = D\f^{−p}D as a fundamental domain for f^{p} near α: any trajectory which starts near α must pass through Q = Q(f) . Now Corollary 4.3 yields:

Lemma 4.6. Geometry of the fundamental domain Q(f) is bounded if c(f) ranges over a truncated secondary limb and f has a definite modulus.

4.5. Modulus of the first annulus.

Lemma 4.7. Let P_{c} be a quadratic polynomial with c outside the main cardioid but not immediately renormalizable. If c ranges over a truncated secondary limb L_{b} , then all pieces X of the initial Markov partition (3-3) are well inside Y^{(0)} : mod (Y^{(0)}\X) > ν(L_{b}) > 0.

Proof. Take a little ϵ > 0 . Then find an N and δ ∈ (0, ϵ) such that the equipotential E_{1/2^{N}} does not intersect the δ -neighborhood of ∂Y^{(0)}\B(α, ϵ) (for all c in the truncated limb).

It follows from Proposition 3.1 and Corollary 4.4 that the statement is true for all pieces of depth ≤ N .

Any other piece X is enclosed by the equipotential E_{1/2^{N}} (where E ≡ E_{1} is the outer-most equipotential of external radius 1). Hence if dist (X, {α}) > ϵ then dist (X, ∂Y^{(0)}) ≥ δ. As diam X is uniformly bounded, we conclude that X is well inside Y^{(0)} .

Assume now that dist (X, {α}) < ϵ. Then X intersects the domain D = D(f) . Since ∂D ∩ ∂X = ∅, X ⊂ D. Let us consider the iterates f^{pk}X , k = 0, 1, … until the last moment l such that f^{pl}X ⊂ D. At this moment f^{pl}X must intersect the fundamental domain Q. Since their boundaries don’t intersect, we conclude that f^{pl}X ⊂ Q.

Let us consider domain Q′ ⊂ Q obtained by truncating Q with the equipotential f^{−p}E . This domain has a bounded geometry since the fundamental domain Q does (Lemma 4.6). Hence Q′ is well inside f^{p}D. Moreover, f^{pl}X ⊂ Q′ since all puzzle pieces inside Y^{1} are enclosed by the equipotential f^{−p}E (see Figure 3). Hence f^{pl}X is well inside fD as well.

On the other hand, if part of the puzzle-piece f^{pl}X lies in Q\Q′ , it belongs to one of the pieces of depth n ≤ 2p, of the initial partition (3-2). As we have noted above, this piece is well inside fD. Hence f^{pl}X is also well inside fD.

We conclude that there is always a definite space around f^{pl}X in f^{p}D. Pulling this space back by iterates of the univalent branch f^{−p}f^{p}D → D, we obtain a definite space around X in D. ⊔ ⊓

Remember that A^{n} = V^{n−1}\V^{n} are the the principal annuli.

Theorem I. Let f be a quadratic-like map with internal class c(f) ranging over a truncated secondary limb L_{b}^{tr} . If mod (f) ≥ R > 0 then

mod (A^{1}) ≥ C(R) ν(L_{b}^{tr}) > 0

where C(R) > 0 monotonically depends on R > 0 and C(R) ↗ 1 as μ ↗ ∞.

Proof. Let us go through the proof of Proposition 3.1. We found an l and a puzzle piece P ⊂ V^{0} such that G^{l}P two-to-one covers Y^{(0)} , where G is the Markov map (3-4). Let G^{l}0 ∈ X_{i}^{s} where X_{i}^{s} is a puzzle-piece of the initial partition (3-3). Then V^{1} is the pull-back of X_{i}^{s} by G^{l}|P . But by Lemma 4.7 X_{i}^{s} is well inside Y^{(0)} . Hence V^{1} is well inside V^{0} . ⊔ ⊓

§5 Increasing of moduli.

5.1. Statement of the results. Let f be a Douady-Hubbard quadratic-like map. Let us consider its principal nest Y^{0} ⊃ V^{0} ⊃ V^{1} ⊃ …, and the corresponding nest of annuli A^{n} = V^{n}\V^{n−1} . Let us call their moduli μ_{n} = mod (A^{n}) the principal moduli of g_{1} .

In this section we will prove the central result of the paper:

Theorem III. Let n(k) counts the non-central levels in the principal nest {V^{n}}. Then

mod A^{n(k)+1} ≥ Bk,

where the constant B depends only on the first modulus μ_{1} = mod A_{1} .

On the way to this result we prove a priori moduli and distortion bounds along the principal nest (Theorem II). Note that already this result yields the divergence property of the Yoccoz Theorem (see §2).

Theorem III will also imply a priori bounds for infinitely renormalizable quadratics of sufficiently big height:

Theorem IV. There is a χ -special family S of the Mandelbrot copies with the following property. Let f be an infinitely renormalizable quadratic of S -type then, and A^{n,m} be its principal annuli. Then there is an Q = Q(χ) → ∞ as χ → ∞, such that mod (R^{n}f) ≥ mod A^{n,1} ≥ Q, n = 0, 1, . . ..

At the end of this section we will describe other combinatorial factors which yield big space. This is summarized in Theorem IV′ which loosely says that if the periods of R^{n}f are sufficiently big and there are no “parabolic” or “Siegel cascades” in the principal nests then there are a priori bounds.

Recall that g_{n} : ∪V_{i}^{n} → V^{n−1} denotes the principal sequence of return maps (3-6), and M ≡ M(f) denotes the full family of all puzzle pieces (§2.6). Let V^{n} ⊂ M denote the family of all pieces V_{i}^{n} of level n.

5.2. First estimates. Let us start with a lemma which partly explains the importance of the principal nest: the principal moduli control the distortion of the first return maps (see the Appendix for the definition of the distortion). Let us consider the decomposition:

g_{n}|V^{n} = h_{n} ∘ Φ,  (5-0)

where Φ is a purely quadratic map and h_{n} is a diffeomorphism of ΦV^{n} onto V^{n−1} .

Lemma 5.1. Let D ∈ M be a puzzle piece such that f^{l}D = V^{n} , while f^{k}D ∩ V^{n} = ∅, k = 0, …, l − 1 . If μ_{n} ≤ μ̄ then the distortion of f^{l} on D is O(exp(−μ_{n−1})) with a constant depending on μ̄. Hence the distortion of h_{n} is O(exp(−μ_{n−2})) .

Proof. This follows from Lemma 3.5, Corollary 3.6 and the Koebe Theorem (see the Appendix). ⊔ ⊓

Let us fix a level n > 0 , denote V^{n−1} = Δ, V_{i} = V_{i}^{n} , g = g_{n} , A = A^{n} = Δ\V_{0} , μ = μ_{n} , and mark the objects of the next level n + 1 with prime. Thus Δ′ ≡ V ≡ V_{0} , and g′ : ∪V′_{i} → Δ′ . (We restore the index n whenever we need it).

Lemma 5.2. Let D′ ⊂ Δ′ be a puzzle piece such that g^{i(k)}D′ ⊂ V_{i(k)}, k = 0, 1, …, l with i(k) ≠ 0 for 0 < k < l. Then

mod(Δ′\D′) ≥ \frac{1}{2} ∑_{k=1}^{l} mod(Δ\V_{i(k)}).

Proof. Let us consider the following nest of topological disks:

Δ′ = W_{1} ⊃ … ⊃ W_{l} ⊃ W_{l+1} ≡ D′,

where W_{k} is the pullback of Δ under g^{k} , k = 1, … l (which has itinerary 0 = i(0), i(1), …, i(k − 1) through the pieces of level n). Then g^{k} is a two-to-one branched covering of the annulus W_{k}\W_{k+1} over the annulus Δ\V_{i(k)} . Hence

mod (W_{k}\W_{k+1}) = \frac{1}{2}mod (Δ\V_{i(k)}), (1 ≤ k ≤ l).

But by the Grötzsch inequality

mod (Δ′\D′) ≥ ∑_{k=1}^{l} mod (W_{k}\W_{k+1}),

and the desired estimate follows. ⊔ ⊓

This lemma immediately yields:

Corollary 5.3. Given a puzzle piece V′_{j} , we have

mod(Δ′\V′_{j}) ≥ \frac{1}{2}μ.

Moreover, if the return to level n is non-central, that is g0 ∈ V_{i} with an i ≠ 0 , then

mod (Δ′\V′_{j}) ≥ \frac{1}{2}(μ + mod (Δ\V_{i})).

So, a definite principal modulus on some level produces a definite space on the next level.

5.3. Isles and asymmetric moduli. Let {V_{i}}_{i∈I} ⊂ V^{n} be a finite family of disjoint puzzle pieces consisting of at least two pieces (that is |I| ≥ 2 ) and containing a critical puzzle piece V_{0} . Let us call such a family admissible. We will freely identify the label set I with the family itself.

Given a puzzle piece D, let I|D denote the family of puzzle pieces of I contained in D. Let D be a puzzle piece containing at least two pieces of family I . For V_{i} ⊂ D set

R_{i} ≡ R_{i}(I|D) ⊂ D\ ⋃_{j∈I|D} V_{j}

be an annulus of maximal modulus enclosing W_{i} but not enclosing other pieces of the family I . Such an annulus exists by the Montel Theorem (see Figure 6). We will briefly call it the maximal annulus enclosing V_{i} in D (rel the family I ).

Let us define the asymmetric modulus of the family I in D as

σ(I|D) = ∑_{i∈I} \frac{1}{2^{1−δ_{i0}}} modR_{i}(R(I|D),  (5-1)

where δ_{ji} is the Kronecker symbol. So the critical modulus is supplied with weight 1, while the non-critical moduli are supplied with weights 1/2 (if D is a non-critical island then all weights are actually 1/2).

Remark. A real analogue of this parameter, “the asymmetic Poincaré length”, appeared in [L3]. Its complex counterpart for the Fibonacci map was suggested by Jeremy Kahn. A general notion involving admissible familes and isles is given by the author.

Figure 6: Annulus R_{i} .

Take a level n − 1 which is not DH renormalizable, that is, the family V^{n} consists of more than one piece. Let D = V^{n−1} , and let {V_{i}^{n}}_{i∈I} be an admissible subfamily of V^{n} . Then set σ_{n}(I) ≡ σ(I|V^{n−1}) and

σ_{n} = min_{I} σ_{n}(I),  (5-2)

where I runs over all admissible subfamilies of V^{n} .

The principal moduli μ_{n} and the asymmetric moduli σ_{n} are the main geometric parameters of the renormalized maps g_{n} . Again, in what follows the label n will be suppressed as long as the level is not changed.

Let {V′_{i}}_{i∈I′} be an admissible subfamily of V′ . Let us organize the pieces of this family in isles in the following way. A puzzle piece D′ ⊂ Δ′ is called an island (for family I′ ) if

• D′ contains at least two puzzle pieces of family I′ ;

• There is a t ≥ 1 such that g^{k}D′ ⊂ V_{i(k)}, k = 1, … t − 1, with i(k) ≠ 0 , while g^{t}D = Δ.

Given an island D′ , let ϕ_{D′} = g^{t} : D′ → Δ . This map is either a double covering or a biholomorphic isomorphism depending on whether D′ is critical or not. In the former case, D′ ⊃ V′_{0} (for otherwise D′ ⊂ V′_{0} contradicting the first part of the definition of isles).

We call a puzzle piece V′_{j} ⊂ D′ ϕ_{D′} -pre-critical if ϕ_{D′}(V′_{j}) = V_{0} . There are at most two pre-critical pieces in any D′ . If there are actually two of them, then they are noncritical and symmetric with respect to the critical point 0. Thus in this case D′ contains also the critical puzzle piece V′_{0} .

Let D′ = D(I′) be the family of isles associated with I′ . Let us consider the asymmetric moduli σ(I′|D′) as a function on this family. This function is clearly monotone:

σ(I′|D′) ≥ σ(I′|D′_{1}) if D′ ⊃ D′_{1},  (5-3)

and superadditive:

σ(I′|D′) ≥ σ(I′|D′_{1}) + σ(I′|D′_{2}),

provided D′_{i} are disjoint subisles in D′ .

Let us call an island D′ innermost if it does not contain any other isles of the family D(I′) . As this family is finite, innermost isles exist.

5.4. Non-decreasing of moduli.

Lemma 5.4. Let I′ be an admissible family of puzzle pieces. Let D′ be an innermost island associated to the family I′ , and let J′ = I′|D. Let i(j) is defined for j ∈ J′ by the property ϕ_{D′}(V′_{j}) ⊂ V_{i(j)} , and let I = {i(j) : j ∈ J′} ∪ {0}. Then {V_{i}}_{i∈I} is an admissible family of puzzle pieces, and

σ(I′|D′) ≥ \frac{1}{2} ((|J′| − s)μ + s modR_{0} + ∑_{j∈J′, i(j)≠0} modR_{i(j)}) ,  (5-4)

where s = #{j : i(j) = 0} is the number of ϕ_{D′} -pre-critical pieces, and R_{i} are the maximal annuli enclosing V_{i} in Δ rel family I .

Proof. Let ϕ ≡ ϕ_{D′} . Let us show first that the family I is admissible. This family is finite since J′ ⊂ I′ is finite. The critical puzzle piece belongs to I by definition. So the only property to check is that |I| ≥ 2 . But otherwise J′ would consist of two pre-critical puzzle pieces. But then D′ would be critical, and thus should have also contained the critical piece V′_{0} , which is a contradiction.

Let us observe next that

mod (V_{i(j)}\ϕV′_{j}) ≥ μ if i(j) ≠ 0.  (5-5)

Indeed, in this case g^{m}(ϕV′_{j}) = V_{0} for some m > 0 . Let W ⊂ V_{i(j)} be the pull-back of Δ under g^{m} . Then the annulus W \ϕV′_{j} is univalently mapped by g^{m} onto the annulus Δ\V_{0} . Hence mod (W \ϕV′_{j}) = mod (Δ\V_{0}) = μ, and (5-5) follows.

Given a i ∈ I , let us consider a topological disk Q_{i} = R_{i} ∪ V_{i} ⊂ Δ (“filled annulus R_{i} ”). By the Grötzsch inequality and (5-5),

mod (Q_{i(j)}\ϕV_{j}) ≥ mod R_{i(j)} + (1 − δ_{0,i(j)})μ.  (5-6)

For a j ∈ J′ , let us consider an annulus B_{j} ⊂ D′ , the component of ϕ^{−1}R_{i(j)} enclosing V′_{j} . This annulus does not enclose any other pieces V′_{k} ∈ J′ , k ≠ j . Indeed, otherwise the inner component of C\B′_{j} would be an island contained in D′ , despite the assumption that D′ is innermost.

Let us now consider a topological disk P′_{j} obtained by filling the annulus B′_{j} . As it contains a single puzzle piece V′_{j} of family J′ , the annulus P′_{j}\V′_{j} does not go around any other puzzle piece V′_{k} ∈ J′ , k ≠ j . Hence

mod R′_{j} ≥ mod (P′_{j}\V′_{j}),  (5-7)

where R′_{j} ⊂ D′ is the maximal annulus enclosing V′_{j} rel J′ . Moreover ϕ : P′_{j} → Q_{i(j)} is univalent or double covering depending on whether j ≠ 0 or j = 0 . Hence

mod (P′_{j}\V′_{j}) ≥ \frac{1}{2^{δ_{j0}}} mod (Q_{i(j)}\ϕV_{j}).  (5-8)

Inequalities (5-6)-(5-8) yield

mod R′_{j} ≥ \frac{1}{2^{δ_{j0}}} (mod R_{i(j)} + (1 − δ_{0,i(j)})μ).  (5-9)

Summing up estimastes (5-9) over J′ with weights 1/2^{1−δ_{j0}} , we obtain the desired inequality (5-5). ⊔ ⊓

Corollary 5.5. For any island D′ of the family D′ the following estimates hold:

σ(I′|D′) ≥ \frac{1}{2}μ and σ(I′|D′) ≥ σ(I) ≥ σ.

Proof. By monotonicity (5-3), it is enough to check the case of an innermost island D′ . Let us use the notations of the previous lemma. Since the family I is admissible, it contains a non-critical piece. Hence |J′| is always strictly greater than the number s of pre-critical pieces in D′ , and (5-4) implies the first of the above inequality.

Furthermore, as μ ≥ mod(R_{0}) and |J′| ≥ 2 , the right-hand side in (5-4) is bounded from below by

\frac{1}{2} (|J′| mod(R_{0}) + ∑_{i∈I,i≠0} mod(R_{i})) ≥ σ(I).

(Note that σ(I) makes sense since I is admissible). Finally σ(I) ≥ σ, and the second inequality follows. ⊔ ⊓

Let us fix a “big” integer quantifier N_{∗} > 0 . We say that a level n is in the “tail of a cascade” if all levels n − 1, n − N_{∗} belong to a cascade (note that level n − 1 itself may be non-central). Cascades of length at least N_{∗} we call “long”.

Theorem II. Given a generalized quadratic-like map g_{1} , we have the following bounds of the geometric parameters within its principal nest:

• The asymmetric moduli σ_{n} grow monotonically and hence stay away from 0 on all levels (until the first DH renormalizable level): σ_{n} ≥ σ̄ > 0 .

• The principal moduli μ_{n} stays away from 0 (that is, μ_{n} ≥ μ̄ > 0 ) everywhere except for the case when n − 1 is in the tail of a long cascade (the bound μ̄ depends on the choice of N_{∗} ).

• The non-critical puzzle-pieces V_{i}^{n} are well inside V^{n−1} (that is, mod (V^{n−1}\V_{i}^{n} ≥ μ̄ > 0 ) except for the case when V_{i}^{n} is pre-critical and n − 2 is the last level of a long cascade.

• The distortion of h_{n} from (5-0) is uniformly bounded on all levels by a constant K̄ . All bounds depend only on the first principal modulus μ_{1} and (as μ̄ is concerned) on the choice of N_{∗} ). Proof. The first assertion follows from the second inequality of Corollary 5.5. Together with Corollary 5.3 it implies the second one (note that the second inequality of this corollary implies that μ′ ≥ σ/2 in the non-central case). One more application of Corollary 5.3 yields the next assertion.

Let us check the last statement. If n − 2 is not in the tail of a central cascade, then μ_{n−1} ≥ μ̄ by the second statement, and the desired follows from Lemma 5.1.

Let n − 2 be in the tail of a central cascade V^{m} ⊃ … ⊃ V^{n−2} ⊃ . . .. If this is not the last level of this cascade then g_{n}|V^{n} = g_{m+2}|V^{n} , so that h_{n} is just a restriction of the map h_{m+2} with bounded distortion.

Finally, if n − 2 is the last level of a central cascade, then by Corollary 3.8 h_{n} can be extended to a univalent map with range V^{m} , and the Koebe Theorem implies the distortion bound. ⊔ ⊓

5.5. Linear growth of moduli. Our goal is to prove that σ′ ≥ σ + a with a definite a > 0 (that is, dependent only on mod A_{0} ) at least on every other level, except for the tails of long cascades and a couple of the following levels. (Theorem II shows the reason why these tails play a special role: In the tails the principal moduli become tiny which slows down the growth rate of asymmetric moduli.)

Clearly it is enough to show that for any innermost island D′

σ(I′|D′) ≥ σ + a  (5-10)

with a definite a > 0 . The analysis will be split into a tree of cases.

Case I. An island with at least three puzzle pieces.

Proposition 5.6. If an innermost island D′ contains at least three puzzle-pieces V′_{j}, j ∈ J′ , then

σ(J′|D′) ≥ σ(I) + \frac{1}{2}μ.

Proof. Let us split off (1/2)μ in (5-4) and estimate all other μ ’s by mod (R_{0}) . This estimates the right-hand side by

\frac{1}{2}μ + \frac{|J | − 1}{2}mod(R_{0}) + \frac{1}{2} ∑_{i∈I,i≠0} mod(R_{i}),

which immediately yields what is claimed. ⊔ ⊓

Hence under the circumstances of Proposition 5.6 we observe a definite growth of the asymmetric modulus provided level n − 1 is not in the tail of a long cascade. Indeed then by Theorem II μ is bounded away from 0, and (5-10) follows.

Case II. An island with two puzzle pieces. The further analysis needs some preparation in the geometric function theory summarized in the Appendix.

Suppose we have an innermost island D′ containing two puzzle-pieces V′_{j}, j ∈ J′ . Let ϕ ≡ ϕ_{D′} and let ϕV′_{j} ⊂ V_{i} with i = i(j) . Fix a quantifier L_{∗} > 0 . When we say that something is “big”, this means that it is at least C(L_{∗}) where C(L_{∗}) → ∞ as L_{∗} → ∞. Similarly “small” means an upper bound by ϵ(L_{∗}) → 0 as L_{∗} → ∞. The sign ≈ will mean an equality up to an small (in the above sense) error, while the sign ≻ will mean the inequality up to a small error.

Subcase (i). Assume that there is a non-critical puzzle-piece V_{i(j)} whose Poincaré distance in Δ from the critical point is less than L_{∗} . Then by Lemma A.1

μ ≥ mod(R_{0}) + α  (5-11)

with a definite α = α(L_{∗}) > 0 . But observe that when we passed from Lemma 5.4 to Corollary 5.5 we estimated μ by mod (R_{0}) . Using the better estimate (5.10), we obtain a definite increase of σ.

Subcase (ii). Assume now that the hyperbolic distance in Δ from any non-critical puzzle piece V_{i(j)} , from the critical point is at least L_{∗} . Let also n − 2 don’t belong to the tail of a long cascade (for the sake of linear growth it is enough to prove definite growth on such levels). Then V_{0} may not belong to any non-trivial island together with some non-critical piece V_{i(j)} . Indeed, by Theorem II all puzzle pieces of level n − 1 are well inside V^{n−2} . But then by Lemma 5.2 all non-trivial isles of level n are well inside of V^{n−1} ≡ Δ . (The quantifier L_{∗} should be chosen bigger than the a priori bound on the hyperbolic diameters of the isles).

Subcase (ii-a). Assume that both V_{i(j)} are non-critical. Then by Corollary 5.5 σ(J′|D′) is estimated by σ_{n}(I) where the family I consists of three puzzle pieces: two pieces V_{i(j)} and the central puzzle piece V_{0} .

If puzzle pieces V_{i(j)}, j ∈ J′ , don’t belong to the same non-trivial island, then by Proposition 5.6 σ(I) ≥ σ_{n−1} + a with a definite a > 0 , and we are done.

Otherwise the puzzle pieces V_{i(j)} belong to an island W . Since by Lemma 5.2 W is well inside of Δ , it stays on the big Poincaré distance from the critical point (namely, on distance L_{∗} − O(1) ). Hence mod (R_{0}) ≈ μ, and

σ(I) ≥ σ(I|W) + mod(R_{0}) ≻ σ_{n−1} + μ

where μ ≡ μ_{n} is bounded away from 0, since level n − 1 is not in the tail of a long cascade. So we have gained some extra growth, and can pass to the next case.

Below we will restore labels n and n + 1 since many levels will be involved in the consideration.

Subcase (ii-b). Let one of the puzzle pieces V_{i}^{n} be critical. So we have the family I^{n} of two puzzle-pieces V_{0}^{n} and V_{1}^{n} . Remember that we also assume that the hyperbolic distance between these pieces is at least L. Hence, V^{n−1} is the only island containing both of them, so that g_{n−1}V_{0}^{n} and g_{n−1}V_{1}^{n} belong to different puzzle-pieces of level n−1 . For the same reason we can assume that one of these puzzle-pieces is critical. Denote them by V_{0}^{n−1} and V_{1}^{n−1} . Then one of the following two possibilities on level n − 2 can occur:

1) Fibonacci return when g_{n−1}V_{0}^{n} ⊂ V_{1}^{n−1} and g_{n−1}V_{1}^{n} = V_{0}^{n−1} (see Figure 7);

2) Central return when g_{n−1}V_{0}^{n} = V_{0}^{n−1} and g_{n−1}V_{1}^{n} ⊂ V_{1}^{n−1} .

We can assume that one of these schemes occur on several previous levels n−3, n−4, ... as well (otherwise we gain an extra growth by the previous considerations). To fix the idea, let us first consider the following particular case, which plays the key role for the whole theorem.

Fibonacci cascade. Assume that on both levels n − 1 and n − 2 the Fibonacci returns

Figure 7: Fibonacci scheme.

occur. Let us look more carefully at the estimates of Lemma 4. In the Fibonacci case we just have:

mod(R_{1}^{n}) ≥ mod(R_{0}^{n−1}),  (5-12)
mod(R_{0}^{n}) ≥ \frac{1}{2}mod(Q_{1}^{n−1}\g_{n−1}V_{0}^{n}),  (5-13)

where Q_{i}^{n} = V_{i}^{n} ∪ R_{i}^{n} . Applying g_{n−2} we see that

mod(Q_{1}^{n−1}\g_{n−1}V_{0}^{n}) ≥ mod(Q_{0}^{n−2}\V_{0}^{n−1}).  (5-14)

But since V_{1}^{n−2} is hyperbolically far away from the critical point,

mod(Q_{0}^{n−2}\V_{0}^{n−1}) ≈ mod(V_{0}^{n−3}\V_{0}^{n−1}).  (5-15)

By the Grötzsch Inequality there is an a ≥ 0 such that

mod(V_{0}^{n−3}\V_{0}^{n−1}) = μ_{n−1} + μ_{n−2} + a.  (5-16)

Clearly

μ_{n−1} ≥ mod(R_{0}^{n−1}).  (5-17)

Furthermore, let P_{1}^{n−1} ⊂ V^{n−2} be the pull-back of Q_{0}^{n−2} by g_{n−2} . Since ∂P_{1}^{n−1} is hyperbolically far away from V_{1}^{n−1} , we have:

μ_{n−2} ≥ mod(R_{0}^{n−2}) = mod(P_{1}^{n−1}\V_{1}^{n−1}) ≈ mod (V^{n−2}\V_{1}^{n−1}) ≥ mod(R_{1}^{n−1}). (1-18)

Combining estimates (5-13) through (5-18) we get

mod(R_{0}^{n}) ≻ \frac{1}{2}(mod(R_{0}^{n−1}) + mod(R_{1}^{n−1}) + a).  (5-19)

We see from (5-12) and (5-19) that we need to check that the constant a in (5-16) is definitely positive. Assume that this is not the case, that is, for any δ > 0 we can find a level n in the Fibonacci cascade as above such that a < δ. Set Γ_{n} = ∂V^{n} . Then by the Definite Grötzsch Inequality (see the Appendix), the width (Γ_{n−2}) in the annulus T = V^{n−3}\V^{n−1} is at most ξ(δ) with ξ(δ) → 0 as δ → 0 . Since Γ_{n−2} is well inside of T , we conclude by the Koebe Distortion Theorem that Γ_{n−2} is contained in a narrow neighborhood of a curve γ with a bounded geometry. Hence there is a k = k(δ) → 0 as δ → 0 and and ϵ = ϵ(δ, k) > 0 such that the curve Γ_{n−2} is not (k, ϵ) -pinched.

On the other hand, the hyperbolic distance from the puzzle piece V_{1}^{n−1} to the critical point 0 in V^{n−2} is at least L_{∗} . Hence by Lemma A.4 it must be located Euclideanly very close to Γ_{n−2} relatively the Euclidean distance to the critical point (that is, the relative distance is at most β(δ) → 0 as δ → 0 ). Hence the critical value g_{n−1}0 is also very close to Γ_{n−2} relatively the distance to the critical point, that is

\frac{dist (g_{n}0, Γ_{n−2})}{dist (g_{n}0, 0)} ≤ ϵ(L_{∗}),

where ϵ(L_{∗}) → 0 as L_{∗} → ∞.

By the last statement of Theorem II, g_{n−1} is a quadratic map up to a bounded distortion. Hence the curve Γ_{n−1} which is the pull-back of Γ_{n−2} by g_{n−1} must have a huge eccentricity around the critical point. But then by Lemma 7.2 the width of Γ_{n−1} in V^{n−2}\V^{n} is also big, which by the above considerations gives a definite linear growth on the next level. Remark. The actual shape of a deep level puzzle-piece for the Fibonacci cascade is shown on Figure 8. There is a good reason why it resembles the filled Julia set for z ↦ z^{2} − 1 (see [L5]). As the geodesic in V_{0}^{n−1} joining the puzzle-pieces V_{0}^{n} and V_{1}^{n} goes through the pinched region, the Poincaré distance between these puzzle-pieces is, in fact, big.

Now it is the time to look closer at central cascades.

Central cascades. Let N ≥ 2 , n = m + N , and let us consider a nest C^{m+N} of puzzle pieces

V^{m} ⊃ V^{m+1} ⊃ … ⊃ V^{m+N−1} ⊃ V^{m+N} ⊃ D^{m+N}  (5-20)

satisfying the following properties (see Figure 8):

• The return on level m − 1 is non-central: g_{m}0 ∉ V_{0}^{m} ;

• Central returns occur on levels m, m + 1, … m + N − 2 , that is g_{m+1}0 ∈ V^{m+N−1} ;

• D^{m+N} is an island with a family I^{m+N+1} of two puzzle pieces inside. Let ϕ_{m+N} ≡ ϕ_{D^{m+N}} denote the corresponding double covering D^{m+N} → V^{m+N−1} ;

• One of the puzzle pieces ϕ_{m+N}V_{0}^{m+1} , ϕ_{m+N}V_{1}^{m+1} is critical.

Though the logic of our argument so far allowed to assume that N ≤ N_{∗} , the following argument will require consideration of arbitrary big N . So let N be arbitrary integer ≥ 2 .

Figure 8: Fibonacci puzzle piece (below) vs the Julia set of z ↦ z^{2} − 1 (above) .

We would like to analyze when

σ(I^{m+N+1}|D^{m+N}) ≥ σ_{m+1} + a  (5-21)

with a definite a > 0 . To this end we need to pass from level m + N all way up to level m.

Let us consider the family W(C^{m+N}) of puzzle pieces W_{i}^{k}, k = m + 1, ..., m + N, i ≠ 0 , the pull-backs of the V_{i}^{m+1} ≡ W_{i}^{m+1} to the annuli A^{k} (compare with the initial Markov partition in §3.2). Given a W_{i}^{k} ⊂ A^{k} , there is a V_{j}^{m+1} ⊂ A^{m+1} , j ≠ 0 , such that g_{m+1}^{m+1−k}W_{i}^{k} = V_{j}^{m+1} . Hence g_{m} ∘ g_{m+1}^{m+1−k}W_{i}^{k} = V^{m} . So we can define a Bernoulli map

G : ∪W_{i}^{k} → V^{m}  (5-22)

Figure 9: Central cascade

(with Fibonacci returns on top and bottom)

by letting G|W_{i}^{k} = g_{m} ∘ g_{m+1}^{m+1−k}|W_{i}^{k}. Let V_{∗}^{m+N+1} ⊂ D^{m+N} be a non-pre-critical piece of the family I^{m+N+1} , and

ϕV_{∗}^{m+N+1} ⊂ V_{1}^{m+N} ⊂ W_{i}^{m+N} for some i ≠ 0 . Then the return map g_{m+N+1} : V_{∗}^{m+N+1} → V^{m+N} can be decomposed as G^{l} ∘ ϕ for an appropriate l ≥ 1 . Since the map G is Bernoulli with range V^{m} ,

mod (W_{i}^{m+N}\ϕV_{∗}^{m+N+1}) ≥ mod (V^{m}\V^{m+N})  (5-23).

Let Γ^{k} = ∂V^{k} , and

w_{k} = width (Γ^{k}|V^{k−1}\V^{k+1}).  (5-24)

For k ∈ [m + 1, m + N] let V_{1}^{k} denote the puzzle piece of level k containing g_{m+1}^{∘(k−m−N)}ϕV_{∗}^{m+N+1} , and I^{k} denote the family of two puzzle pieces: V_{0}^{k} and V_{1}^{k} . Moreover, let R_{i}^{k} ⊂ V^{k−1} denote an annulus of maximal modulus going around V_{i}^{k} but not going around the other piece of family I^{k} , i = 0, 1 .

By the Definite Grötzsch Inequality and the second part of Theorem II, there is an a = a(w_{m+1}) such that

mod (V^{m}\V^{m+N}) ≥ ∑_{k=m+N}^{m+1} mod A^{k} + a =

∑_{k=0}^{N−1} \frac{1}{2^{k}} mod(A^{m+1}) + a ≥ (2 − \frac{1}{2^{N−1}} )mod R_{0}^{m+1} + a.

(5-25)

Let S_{0}^{m+N} and S_{1}^{m+N} denote the pull-backs of the annuli R_{0}^{m+1} and R_{1}^{m+1} by the map g_{m+1}^{∘(N−1)} : V^{m+N−1} → V^{m} . Then

mod S_{0}^{m+N} ≥ \frac{1}{2^{N−1}} mod R_{0}^{m+1} and mod S_{1}^{m+N} ≥ mod R_{1}^{m+1}.  (5-26)

Note that the inner boundary of S_{1}^{m+N} coincides with the outer boundary of W_{i}^{m+N}\ϕV_{∗}^{m+N+1} . Let Q_{1}^{m+N} denote the union of these two annuli. This annulus goes around ϕV_{∗}^{m+N+1} but not around V_{0}^{m+N} . Now estimates (5-23), (5-25), (5-26) yield

mod S_{0}^{m+N} + mod Q_{1}^{m+N} ≥ 2mod R_{0}^{m+1} + mod R_{1}^{m+1} + a ≥ 2σ(I^{m+1}) + a.

Finally, pulling S_{0}^{m+N} and Q_{1}^{m+N} back by ϕ_{m+N} to the island D^{m+N} we obtain:

σ(I^{m+N+1}|D^{m+N}) ≥ \frac{1}{2}(mod S_{0}^{m+N} + mod Q_{1}^{m+N}) ≥ σ(I^{m+1}) + a/2.  (5-27)

So we come up with the following statement:

Statement 5.7. There is an increasing function a : R_{+} → R_{+} , a(0) = 0 , such that for the cascade C^{m+N} estimate (5-21) holds with a = a(w_{m+1}) , where w_{m+1} = width (Γ^{m+1}|V^{m}\V^{m+2}) .

Let us fix a quantifier w_{∗} which distinguishes “small width” w from a “definite” one. For further analysis let us go several levels up. Let m − 1 − l be the highest non-central level preceding m − 1 , l ≥ 1 . We are going to study when

σ(I^{m+N+1}|D^{m+N}) ≥ σ_{m−l} + a  (5-28)

with a definite a > 0 . We cannot now assume that l is bounded, so we face a possibility of a long cascade C^{m−1} : V^{m−l} ⊃ … ⊃ V^{m−1} . Set g = g_{m−l+1} ; then g0 ∈ V^{m−1} .

Assume first that m−2 is not the last piece of a long cascade (in particular, this is the case when central return occurs on level m − 2 , that is, l ≥ 2 ). Then by the third part of Theorem II all non-central pieces of level m are well inside V^{m−1} : mod (V^{m−1}\V_{j}^{m}) ≥ μ̄. Hence g_{m}V_{0}^{m+1} and g_{m}V_{1}^{m+1} belong to different pieces of level m. Indeed, otherwise the hyperbolic distance between V_{0}^{m+1} and V_{1}^{m+1} in V^{m} would be bounded by a constant L(μ̄) . But according to our assumption this distance is at least L_{∗} . So this situation is impossible if L_{∗} was a priori selected bigger than L(μ̄) .

For the same reason the pieces g^{k} ∘ g_{m}V_{i}^{m+1} , i = 0, 1 , also belong to different pieces V_{j}^{m−k} for 0 ≤ k ≤ l − 3 . Indeed, assume they belong to the same piece V_{j}^{m−k} . Clearly this piece is non-central, that is j ≠ 0 . Then it is contained in a piece W_{j}^{m−k} of the Bernoulli family W(C^{m−1}) associated to the central cascade C^{m−1} . Hence g_{m}V_{0}^{m+1} and g_{m}V_{1}^{m+1} belong to W_{j}^{m} , the pull-back of W_{j}^{m−k} by g^{k} . As mod (W_{j}^{m}\g_{m}V_{i}^{m+1}) ≥ μ̄, the hyperbolic distance between V_{0}^{m+1} and V_{1}^{m+1} in V^{m} is at most L(μ̄) contradicting our assumptions.

Let us show now that (5-28) holds if both g_{m}V_{i}^{m+1} are non-central. Indeed let us then consider the family I^{m} of three pieces: two pieces of level m containing g_{m}V_{i}^{m+1} and the central piece V^{m} . Let I^{m−k} denote the family of puzzle pieces of level m − k containing the pieces of g^{k}I^{m} . By the previous two paragraphs, I^{m−k} consists of three puzzle pieces. Then by Corollaries 5.5 and 5.7,

σ(I^{m+1}) ≥ σ(I^{m}) ≥ … ≥ σ(I^{m−l+2}) ≥ σ(I^{m−l+1}) + \frac{1}{2}μ̄,

and we are done.

Thus let us assume that the Fibonacci return occurs on level m − 1 . In this case let I^{m−k} denote the family of two puzzle pieces V_{0}^{m−k} and V_{1}^{m−k} containing g^{k} ∘g_{m}V_{i}^{m+1} , i = 1, 2 , k ≤ l − 1 .

Note that in order to have (5-28) it is enough to to have a definite increase of the σ(I^{m−k}) in the beginning of the cascade C^{m−1} . By Statement 5.7 applied to this cascade this is the case if width (Γ^{m−l+1}|V^{m−l}\V^{m−l+2}) ≥ w_{∗}. So assume that the opposite inequality holds. Similarly, because of Subcase (i), we can assume that the hyperbolic distance from V_{1}^{m−l+2} to 0 in V^{m−l+1} is at least L_{∗} .

It follows from Lemma A.4 from the Appendix that the piece V_{1}^{m−l+2} stays Euclidean distance at most ϵ diam Γ^{m−l+1} from Γ^{m−l+1} where ϵ = ϵμ̄(w_{∗}, L_{∗}) → 0 as w_{∗} → 0 , L_{∗} → ∞ (for a fixed μ̄ > 0 ). It follows that the Euclidean distance from V_{1}^{m−l+2} to Γ^{m−l+1} is relatively small as compared with its distance from Γ^{m−l} and Γ^{m−l+2} . More precisely, there is a δ = δμ̄(w_{∗}, L_{∗}) with the same properties as ϵ above such that for any z ∈ V_{1}^{m−l+2} ,

dist (z, Γ^{m−l+1}) ≤ δ dist (z, ∂(V^{m−l}\V^{m−l+2})  (5-29)

Take z_{0} ∈ V_{1}^{m−l} , and let r = dist (z_{0}, ∂(V^{m−l}\V^{m−l+2}) . Note that the disk B(z_{0}, r) can be univalently pulled by g^{l−3} to the annulus V^{m−3}\V^{m−1} . By the Koebe Distortion Theorem and (5-29), for any ζ ∈ V_{1}^{m−3}

dist (ζ, Γ^{m−2}) ≤ Cδdist (ζ, ∂(V^{m−3}\V^{m−1})) ≤ Cδ dist (ζ, 0)

with an absolute constant C . All the more,

dist (ζ, Γ^{m−2}) ≤ Cδ diam V^{m−2},

so that Γ^{m−2} has a big eccentricity about V_{1}^{m−2} (that is, this eccentricity is at least e(w_{∗}, L_{∗}) , where e(w_{∗}, L_{∗}) → ∞ as w_{∗} → 0 , L_{∗} → ∞).

Pulling Γ^{m−2} back by g_{m+1}∘g_{m}∘g_{m−1} , we conclude that Γ^{m+1} has a big eccentricity about 0. Hence it has big width in the annulus V^{m}\V^{m+2} , and Statement 5.7 yields the desired.

Let us summarize the information which will be useful in what follows:

Statement 5.8. If the width w_{m−l+1} is at most w_{∗} and the Poincaré distance from V_{1}^{m−l+2} to 0 in V^{m−l+1} is at least L_{∗} , then the eccentricity Γ^{m} about the origin is at least e(w_{∗}, L_{∗}) , where e(w_{∗}, L_{∗}) → ∞ as w_{∗} → 0 and L_{∗} → ∞.

Let us assume now that m − 2 is the last piece of a long cascade

C^{m−2} : V^{m−2−t} ⊃ … V^{m−2}, t ≥ N_{∗}.

Then non-central return occurs on level m − 2 . We will show that

σ_{(}I^{m+N+1}|D^{m+N}) ≥ σ_{m−2−t} + a  (5-30)

with a definite a > 0 .

Let D^{m} ⊂ V^{m} be the island containing V_{0}^{m} and V_{1}^{m} , and ϕ_{m} : D_{m} → V^{m−1} be the corresponding two-to-one map. Note that in the case under consideration this island may be non-trivial and still the Poincaré distance between V_{0}^{m+1} and V_{1}^{m+1} be big (since the pre-critical puzzle pieces in V^{m−1} are not well inside V^{m−1} ). Moreover the map ϕ_{m} is not necessarily bounded perturbation of the quadratic map. These are the circumstances which make this case special.

As m − 2 is a non-central level, μ_{m+1} ≤ μ̄, and by the previous considerations we are done unless

• The return on level m − 1 is Fibonacci, that is ϕ_{m}V_{1}^{m+1} = V_{0}^{m} and ϕ_{m}V_{0}^{m+1} ⊂ V_{1}^{m} for some puzzle piece V_{1}^{m} ;

• The hyperbolic distance between the puzzle pieces V_{0}^{m} and V_{1}^{m} is at least L_{∗} .

• The return on level m − 2 is also Fibonacci: g_{m−1}V_{1}^{m} = V_{0}^{m−1} and g_{m−1}V_{0}^{m} = V_{1}^{m−1} for some puzzle piece V_{1}^{m−1} .

Let V_{0}^{k} and V_{1}^{k} be the pieces containing the corresponding push forwads of V_{0}^{m−1} and V_{1}^{m−1} along the cascade C^{m−1} , m − 1 ≤ k ≤ m − t − 1 . Then (5-30) follows unless • The width w_{m−t−3} is at most w_{∗} , and the distance between V_{0}^{m−t−4} and V_{1}^{m−t−4} in V^{m−t−3} is at least L_{∗} .

But then by Statement 5.8 applied to the cascade C^{m−2} the eccentricity of Γ^{m−1} about 0 is at least e = e(w_{∗}, L_{∗}) . As g_{m} is a bounded perturbation of the quadratic map, by Lemma A.5 the curve Γ^{m} is ( 0.1, ϵ) -pinched, where ϵ = ϵμ̄(e) → 0 as e → ∞. (Note that the pinched region is not necessarily around V_{1}^{m} , since ϕ_{m} may differ from g_{m} ). Applying Lemma A.5 again, we conclude that the curve Γ^{m+1} is ((10C)^{−1}, C\sqrt{ϵ}) - pinched. By Lemma A.3 Γ^{m+1} has a definite width inside V^{m}\V^{m+2} . Now Statement 5.7 yields (5-30).

5.6. Other factors yielding big space. Theorem III ensures that after many central cascades we will observe a big principal modulus. However, there are other combinatorial factors which yield the same effect. Altogether they are quite close to a “big renormalization period”, except that “parabolic or Siegel cascades” may interfere.

Big return time implies big modulus. This section will rely on combinatorial considerations of §3.8. Recall that I^{n} denotes the family of puzzle pieces V_{i}^{n} intersecting ω(0) . Given two Υ_{f} -adjacent puzzle pieces V_{j}^{n+1} ∈ I^{n−1} and V_{i}^{n} ∈ I^{n} , let t be the first return time of V_{j}^{n+1} back to V_{i}^{n} under iterates of g_{n} . Then we will use the notation mod (V_{j}^{n+1} → V_{i}^{n}) for mod (V_{i}^{n}\g_{n}^{t}V_{j}^{n+1}). If i ≠ 0 then

mod (V_{j}^{n+1} → V_{i}^{n}) ≥ μ_{n}.  (5-31)

Lemma 5.9. Let D^{n} ⊃ V^{n−1} be a puzzle piece containing at least one piece of I^{n} . Let γ be the shortest path leading from D^{n} down to some central piece V_{0}^{n+t} , and let D^{n+t} be the pull-back of D^{n} along this path. Then

mod (D^{n+t}\V^{n+t}) ≥ \frac{μ̄}{2}rank(D^{n}).

Proof. Let D^{n+k} ⊂ V_{j(k)}^{n+k}, k = 0, 1, …, t, be the pull-back of D^{n} along the path γ . By Lemma 3.11, all edges of this pull-back except the last one are univalent, and the last one is a double covering. Hence

mod (D^{n+t}\V^{n+t}) ≥ \frac{1}{2} ∑_{k=1}^{t} mod (V_{j(k)}^{n+k} → V_{j(k−1)}^{n+k−1}),

and by (5-31) and Theorem II the right-hand side of this inequality is estimated from below by the right-hand side of the desired inequality. ⊔ ⊓

Lemma 5.10. Let n − 1 be not in the tail of a central cascade. Assume that for a puzzle piece V_{j}^{n+1} ∈ I^{n+1} , g_{n}−time(V_{j}^{n+1}) ≥ r. Then there is a level m such that μ_{m} ≥ L(r) , where L(r) → ∞ as r → ∞.

Proof. Let M > 0 . We need to find a level m with μ_{m} ≥ M , provided r is sufficiently big. If rank(V_{j}^{n+1}) > M/μ̄, then Lemma 5.9 yields the desired. So let us assume that

rank(V_{j}^{n+1}) ≤ N ≡ M/μ̄ + 1.  (5-32)

Let 0 = i(0), i(1), …, i(r) = 0 be the itinerary of V_{j}^{n+1} through the pieces of the previous level. Let us consider the nest of puzzle pieces

D_{r−1} ⊃ … ⊃ D_{0} ≡ V_{i}^{n},  (5-33)

where g_{n}^{r−k}D_{k} = V_{i(r−k)}^{n} . Then

D_{k+1}\D_{k} ≥ μ̄/2, k = 1, … r − 1.  (5-34)

Let us now pull the pieces D_{k} down along the shortest path γ joining V_{j}^{n+1} with a critical vertex V^{n+t} . Denote the corresponding pull-backs by D_{k}^{n+l} . If this pull-back turns out to be univalent then by (5-34)

μ_{n+t} ≥ mod (D_{r}\V_{j}^{n+1}) ≥ rμ̄/2,

which is greater than M for sufficiently big r. Otherwise let us consider the first level n + s where D_{r−1}^{n+s} hits the critical point. Let us find such an l that 0 ∈ D_{l+1}^{n+s}\D_{l}^{n+s} .

If l > A ≡ 2M/μ̄ then it follows from (5-33) that μ_{n+s} ≥ M , and we are done. Otherwise by (5-34) mod (D_{l}\V_{j}^{n+1}) ≥ (r − A)μ̄. Let us now repeat the same procedure with D_{l}^{n+s} instead of D_{r−1} . Note that rank(D_{l}^{n+s}) ≤ rank(D_{r−1}) − 1 , since the pullback of D_{r−1} along the first central cascade is univalent. Hence this procedure can be repeated at most N times, and the principal modulus at the end will be at least M , provided (r − NA)μ̄ ≥ 2M . ⊔ ⊓

Let us improve this lemma in the case of central cascades. Below we will use notions from §3.8.

Lemma 5.11. Let us consider a central cascade (3-7) such that m − 1 is not in the tail of a central cascade. Assume that G_{m+N} − time(V_{j}^{m+N+1}) ≥ r for some non-pre-critical puzzle piece V_{j}^{m+N+1} . Then there is a level m such that μ_{m} ≥ M(r) , where M(r) → ∞ as r → ∞.

Proof. The argument is the same as for the previous lemma except one modification: To construct a nest (5-33), use the Bernoulli map G_{m+N} instead of g_{m+N} . ⊔ ⊓

Note that if V^{m−1} ⊃ V^{m} … ⊃ V^{m+N} is a central cascade then the first condition is satisfied for the sub-cascade V^{m} … ⊃ V^{m+N} .

Parabolic and Siegel cascades. We will show that we usually will observe a big principal modulus after just one long central cascade. Let us consider a central cascade:

V^{m} ⊃ ... ⊃ V^{m+N−1} ⊃ V^{m+N},  (5-35)

where g_{m+1}0 ∈ V^{m+N−1}\V^{m+N} . The double covering g_{m+1} : V^{m+1} → V^{m} can be viewed as a small perturbation of a quadratic-like map g_{∗} with a definite modulus and with non-escaping critical point.

To make this precise, let us consider the space Q of double coverings g : U′ → U , with 0 ∈ U′ ⊂⊂ U and g′(0) = 0 , modulo affine conjugacy. Let us supply it with the Catathéodory topology (see [McM]). Convergence in this topology means Carathéodory convergence of the domains and the ranges, and uniform convergence of the maps on compact subsets.

Given a μ > 0 , let Q(μ) denote the set of double coverings g ∈ Q with mod (g) ≥ μ. By Theorem II, the return maps g_{m+1} : V^{m+1} → V^{m} of the principal nest belong to Q(μ̄) .

Compactness Lemma (see [McM]). The set Q(μ) is Carathéodory compact.

Let Q_{N}(μ) denote the space of double coverings g : U′ → U from Q(μ) such that g^{n}0 ∈ U, n = 0, 1, … N . Note that Q_{∞}(μ) is the space of DH quadratic-like maps with modulus at least μ.

As ⋂_{N} Q_{N}(μ) = Q_{∞}(μ) , for any neighborhood U ⊃ Q_{∞}(μ) , there is an N such that Q_{N}(μ) ⊂ U . In this sense any double map g ∈ Q_{N}(μ) is close to some quadratic-like map g_{∗} . In particular, this concerns the above return map g_{m+1} generating the cascade (5-35) of big length N . Moreover, since g_{m+1} has an escaping fixed point, the neighborhood of g_{∗} containing g_{m+1} also contains a quadratic-like map with hybrid class c(g_{∗}) ∈ ∂M .

If we have a sequence of maps f_{n} ∈ Q converging to a map g_{∗} ∈ ∂M , we also say that the f_{n} -central cascades converge to the g_{∗} -cascade.

Let us say that the principal nest is minor modified if a piece V^{m} is replaced by a piece Ṽ^{n} ⊂ V^{n} such that cl V_{i}^{n+1} ⊂ Ṽ^{n} for all pieces V_{i}^{n+1} ∈ I^{n+1} .

Lemma 5.12. Let g_{∗} be a DH quadratic-like map with c(g_{∗}) ∈ ∂M which does not have neither parabolic points, nor Siegel disks. Let g_{m+1} be the return map of the principal nest generating cascade (5-35). Take an arbitrary big M > 0 . If g_{m+1} is sufficiently close to g_{∗} (depending on a priori bound μ̄ from Theorem II) then the principal nest can be minor modified in such a way that μ̃_{n} ≥ M for some n > m + N .

Proof. Take a big number e > 0 .

By the above assumptions, the Julia set J(g_{∗}) has empty interior. If g_{m+1} is sufficiently close to g_{∗} then Γ^{m+N−1} = ∂V^{m+N−1} is close in the Hausdorff metric to the Julia set J(g_{∗}) . Hence Γ^{m+N−1} has an eccentricity at least e with respect to any point z ∈ V^{m+N−1} .

As the g_{m} are quadratic maps up to bounded distortion (Theorem II), the curves Γ_{m+N} , Γ_{m+N+1} and Γ_{m+N+2} also have big eccentricity with respect to any enclosed point. Moreover, by the same theorem, there is a definite space in between these two curves. Hence by Lemma A.2, mod (V^{m+N+1}\V^{m+N+3}) is at least M(e) where M(e) → ∞ as e → ∞.

Let us assume that non-central return occurs on level m+N+1 : g_{m+N+2}0 ∈ V_{i}^{m+N+2} with i ≠ 0 . As the map g_{m+N+2} : V_{i}^{m+N+2} → V^{m+N+1} is quadratic up to bounded distortion, the curve Γ_{i}^{m+N+2} = ∂V_{i}^{m+N+2} has a big eccentricity e′ about any enclosed point (that is, e′ can be made arbitrary big by a sufficiently big choice of e, depending on a priori bound μ̄). By Lemma A.2

mod (V^{m+N+1}\g_{m+N+2}V_{0}^{m+N+3}) ≥ M(e),

where M(e) → ∞ as e → ∞. Hence mod A^{m+N+3} ≥ M(e)/2, and we are done.

Let the central return occurs on level m+N+1 but this is not yet a DH-renormalizable level. Then the corresponding central cascade is finite. Let m + N + T be the last level of this cascade. Then by Statement 5.8 and Lemma A.2, μ_{m+N+T +2} ≥ M(e) , where M(e) → ∞ as e → ∞.

Assume finally that m + N + 1 is a DH-renormalizable level. Then let us take a horizontal curve Γ ⊂ A^{m+N+2} which divides this annulus into two subannuli of moduli at least μ̄/2 . Let Γ′ ⊂ A^{m+N+3} be its pull-back by g_{m+N+3} , and Ã be the annulus bounded by Γ and Γ′ . Then by Lemma A.2 mod(Ã) ≥ M(e) with M(e) as above. As this is a minor modification of the nest, we are done. ⊔ ⊓

5.7. Proof of Theorem IV. Let us fix a Q > 0 . Take a truncated secondary limb L ≡ L_{b}^{tr} , and find q = C(Q) ν(L) from Theorem I. Note that q(L, Q) ≥ ν(L)/2 for sufficiently big Q (independently of L). Let us now select all copies M′ of the Mandelbrot set with the height χ(M′) ≥ Q/B , where B = B(q) is the constant from Theorem III. Taking the union of all these copies over all truncated limbs, we obtain a desired special family M.

Let us now consider an infinitely renormalizable DH quadraticlike map f of S -type with mod (f) ≥ Q (to start with, take a quadratic polynomial). Then by Theorem I, mod (A^{1}) ≥ q. Hence by Theorem III, mod (Rf) ≥ B χ(f) ≥ Q.

By induction, mod (R^{n}f) ≥ R for all n. ⊔ ⊓

5.8. Variations. Let us now improve Theorem IV by taking into account not only the height but also the other factors yielding big space.

Theorem IV′ . Let f be an infinitely renormalizable quadratic polynomial, and let P_{m} : z ↦ z^{2} + c_{m} be the straightened R^{m}f . Assume that

• All c_{m} are selected from a finite number of truncated secondary limbs L_{i} , i = i, … s;

• The set A ⊂ Q of accumulation points of the central cascades of P_{m} (of lengths growing to ∞) does not contain parabolic or Siegel maps;

• per(R^{m}f) ≥ p.

Then lim inf_{n→∞} mod (R^{n}f) ≥ Q(p) , where the function Q(p) depends on the choice of the limbs and the accumulation set A, and Q(p) → ∞ as p → ∞. Proof. By Theorem II the top modulus of the central cascades of P_{m} is bounded from below by some μ̄. Hence the set A ⊂ Q(μ̄) is compact. By Lemma 5.12, for any Q there is a neighborhood U ⊃ A such that: If f ∈ U is renormalizable then mod Rf > Q.

As A is the accumulation set for the central cascades of the P_{m} , there is an N such that all but finitely many of these cascades of length ≥ N belong to U . Hence if the principal nest of P_{m} contains a cascade of length ≥ N then mod (R(P_{m})) ≥ Q (for sufficiently big m).

Further, by Theorems I and III, there is a χ such that if the height χ(P_{m}) ≥ χ then mod (R(P_{m})) ≥ Q. Let us also find a T such that if for some cascade the return time from Lemma 5.11 is at least T , then mod (R(P_{m})) ≥ Q.

It is easy to see that there is a p such that: If per(P_{m}) ≥ p then either P_{m} has a central cascade of length at least N , or χ(P_{m}) ≥ χ, or one of the above return times is at least T . In any case mod (R(P_{m})) ≥ Q.

Now the same argument as for Theorem IV yields a priori bounds. ⊔ ⊓

§6. Local connectivity of the Julia sets.

In this section we will show that the Julia sets of quadratic polynomials from Theorems IV and IV′ are locally connected. This follows from the moduli bounds in the full principal nest, which make puzzle pieces shrink to points (compare Yoccoz (see [H]), and Hu-Jiang [HJ], [J]). I thank J. Kahn and C. McMullen for useful discussions of this issue.

Theorem V. Let S be a special family of Mandelbrot copies from Theorem V. Let f be an infinitely renormalizable quadratic of S -type. Then the Julia set J(f) is locally connected.

Proof. Let us consider the full principal nest (3-9). Let f_{m} ≡ R^{m}f and J_{m} ≡ J(f_{m}) . It follows from Theorem IV and the Grötcsz inequality that

mod (Y^{0,0}\J_{m}) ≥ ∑_{k=0}^{m−1} mod A^{m,0} ≥ ϵm → ∞ as m → ∞.

Hence the “little” Julia sets J_{m} shrink down to the critical point. Let us take an δ > 0 , and find an m such that J_{m} is contained in the B(0, δ) .

Let us now inscribe into B(0, δ) a domain bounded by equipotentials and external rays of the original map f (compare Hu and Jiang [HJ], [J]). Let α_{m} denote the dividing fixed point of the Julia set J_{m} , and α′_{m} = −α_{m} be the symmetric point. Let us consider a puzzle piece P_{m}^{0} ∋ 0 bounded by any equipotential and four external rays of the original map f landing at α_{m} and α′_{m} . This is a “degenerate” domain of the renormalized map f_{m} (see §2.5). By definition of the renormalized Julia set, the preimages P_{m}^{k} ≡ f_{m}^{−k}P_{m}^{0} shrink down to J_{m} . Hence there is a puzzle piece P_{m}^{l} contained in the δ -neighborhood of the critical point. As J(f) ∩ P_{m}^{l} is clearly connected, the Julia set J(f) is locally connected at the critical point.

Let us now prove local connectivity at any other point z ∈ J(f) . This is done by a standard spreading of the local information near the critical point around the whole dynamical plane. Let f_{m} : U′_{m} → U_{m} , where U_{m} ≡ V^{m,t(m)} , U′_{m} ≡ V^{m,t(m)+1} are domains from the principal nest (3-9). Let us consider two cases.

Case (i). Let the orbit of z accumulates on all Julia sets J_{m} . Find an l = l(m) such that P_{m}^{l} ⊂ U′_{m} , and then take the first moment k = k(m) ≥ 0 such that f^{k}z ∈ P_{m}^{l} . Let us consider the pull-back V_{m} ⊃ Q_{m}^{l} ∋ z of U_{m} ⊃ P_{m}^{l} along the orbit orb_{k}(z) = {z, ..., f^{k}z}. By Lemma 3.3, the pull-back of the puzzle piece P_{m}^{l} is univalent. Moreover, by construction of the principal nest, U_{m}\J_{m} does not intersect the critical set ω(0) . Hence the pull-back of U_{m} along the orb_{k}(z) is also univalent.

Let Γ_{m} ⊂ U_{m}\U′_{m} be a horizontal curve in the annulus U_{m}\U′_{m} which divides it into two sub-annuli of modulus at least ϵ/2 . By the Koebe Theorem, it has a bounded eccentricity about 0 (with a bound depending on ϵ). Applying Koebe again, we conclude that its pull-back γ_{m} along orb_{k}(z) has a bounded eccentricity about z. Since the inner radius of this curve about z tends to 0 as m → ∞ (follows from the fact that the sufficiently high iterates of any disk in J(f) cover the whole J(f) ), the diam γ_{m} → 0 as well. All the more, the diam Q_{m}^{l} → 0 as m → ∞. As Q_{m}^{l} ∩ J(f) are connected, the Julia set is locally connected at z.

Case (ii). Assume now that the orbit of z does not accumulate on some J_{m} . Hence it accumulates on some point a ∉ ω(0) . Let us consider the puzzle associated with the periodic point α_{m} (so that the initial configuration consists of a certain equipotential and the external rays landing at α_{m} ). Since the critical puzzle pieces shrink to J_{m} , the puzzle pieces Y_{i}^{l} of sufficiently big depth l containing a are disjoint from ω(0) (there are several such pieces if a is a preimage of α_{m} ). Take such an l, and let X be the union of these puzzle pieces. It is a closed topological disk disjoint from ω(0) whose interior contains a. Hence there is a simply connected neighborhood V ⊃ X still disjoint from ω(0) . Consider now the moments k_{i} → ∞ when the orbit of z lands at int X , and pull V ⊃ X back to z. As the pull-backs of V are univalent, the pull-backs of X shrink to z (by the Koebe argument as above). It follows that J(f) is locally connected at z. ⊔ ⊓

§7. Forthcoming notes.

Let us briefly outline the content of the forthcoming notes. They will be mostly based on already existing preprints:

• Rigidity of quadratics of sufficiently big height (§4 of [L4]).

• Parapuzzle geometry: the moduli in the principal parameter nest grow at the same rate as in the dynamical nest. This also yields the rigidity of the corresponding quadratics. The Fibonacci case has been worked out by LeRoy Wenstrom (in preparation).

• Geometry of real quadratics ([L3] and [L4], §5). We give a criterion when the geometry of a real quadratic polynomial is “essentially bounded”: It happens if and only if its ”essential period” is bounded. On each level with sufficiently high essential period the renormalized map has a big modulus.

• We prove local connectivity of the Mandelbrot set at all real infinitely renormalizable points with sufficiently high essential period on all levels (following §5 of [L4]). Rigidity for all real quadratics follows (compare Swiatek [Sw]).

• Extension of Sullivan’s complex a priori bounds onto infinitely renormalizable quadratics of essentially bounded type (joint work with Michael Yampolsky [LY]). Together with [L4] this yields complex bounds (and hence local connectivity of the Julia sets) for all real quadratics.

• Teichmüller metric on the space of quasi-quadratic maps following [L5], [L6].

• Applications to complex and real measurable dynamics.

Appendix: Conformal maps and geometry of curves.

A.1. Poincaré metric and distortion. A domain D ⊂ C is called hyperbolic if its universal covering space is conformally equivalent to the unit disk. This happens if and only if C\D consists of at least two point. Hyperbolic domains possess the hyperbolic (or Poincaré) metric ρ_{D} of constant negative curveture. This metric is obtained by pushing down the Poincaré metric dρ_{D} = |dz/(1 − z^{2})| from the unit disk D .

In the case of a simply connected hyperbolic domain D (“conformal disk”), dρ_{D} = p_{D}(z)|dz| is the pull-back of the ρ_{D} by the Riemann mapping D → D . In this case its density p_{D}(z) is comparable with 1/dist (z, ∂D) :

(1/4)dist (z, ∂D)^{−1} ≤ p(z) ≤ dist (z, ∂D)^{−1}  (A-1).

In the simply connected case, a set K ⊂ D has a bounded hyperbolic diameter diam_{D}K if and only if there is an annulus A ⊂ D of definite modulus surrounding K . More precisely, let μ_{min}(R) and μ_{max}(R) denote the minimal and maximal possible modulus of an annulus A ⊂ D surrounding K , where K runs over all subsets of hyperbolic diameter R. Then 0 < μ_{min} < μ_{max} < ∞ (all estimates are clearly independent. of D). This can be readily seen by passing to the disk model and moving one point of K to the origin. The extremal moduli correspond to the cases of a pair of points and hyperbolic disk of radius R. Moreover, both minimal an maximal moduli behave as log(1/R) + 0(1) (see [A], Ch. III).

Given a univalent holomorphic function f : D → C , the distortion of f on K is defined as

sup_{z,ζ∈K} log |\frac{f′(z)}{f′(ζ)}| .

Koebe Distortion Theorem. Let D be a conformal disk, K ⊂ D, r = diam_{D}K be the Poincaré diameter of K in D. Then the distortion of any univalent function f on K is bounded by a constant C_{D}(r) independent of a particular choice of K . Moreover C_{D}(r) = O(r) as r → 0 .

A.2. Moduli defect and capacity. Let D be a topological disk, Γ = ∂D, a ∈ D, and ψ : (D, a) → (D_{r}, 0) be the Riemann map onto a round disk of radius r with ψ′(a) = 1 . Then r ≡ r_{a}(Γ) is called the conformal radius of Γ about a. The capacity of Γ rel a is defined as

cap_{a}(Γ) = log r_{a}(Γ).

Lemma A.1. Let D_{0} ⊃ D_{1} ⊃ K , where D_{i} are topological disks and K is a connected compact. Assume that the hyperbolic diameter of K in D_{0} and the hyperbolic dist (K, ∂D_{1}) are both bounded by a L. Then there is an α(L) > 0 such that

mod(D_{1}\K) ≤ mod(D_{0}\K) − α(L).

Proof. Let us take a point z ∈ ∂D_{1} whose hyperbolic distance to K is at most L. Then there is an annulus of a definite modulus contained in D_{0} and enclosing both K and z.

Let us uniformize D_{0}\K by a round annulus A_{r} = {ζ : r < |ζ| < 1}, and let z̃ correspond to z under this uniformization. Then z̃ stays a definite Euclidian distance d from the unit circle.

If R ⊂ A_{r} is any annulus enclosing the inner boundary of A_{r} but not enclosing z̃ then by the normality argument mod (R) < mod(A_{r}) − α_{r}(d) with an α_{r}(d) > 0 . (Actually, the extremal annulus is just A_{r} slit along the radius from z̃ to the unit circle).

We have to check that α_{r}(d) is not vanishing as r → 0 . Let us fix an outer boundary Γ of B (the unit circle + the slit in the extremal case). We may certainly assume that the inner boundary coincides with the r -circle. Then the defect mod (R) − log(1/r) monotonically increases to the cap_{0}(Γ) . By normality this capacity is bounded above by an −α(d) < 0 , and we are done. ⊔ ⊓

Let A be a standard cylinder of finite modulus, K ⊂ A. Define the width (K) ≡ width (K|A) ) as the modulus of the smallest concentric sub-cylinder A′ ⊂ A containing K .

Definite Grötzsch Inequality. Let A_{1} and A_{2} be homotopically non-trivial disjoint topological annuli in A. Let K be the set of points in their complement which are separated by A_{1} ∪ A_{2} from the boundary of A. Then there is a function β(x) > 0 ( x > 0 ) such that

mod(A) ≥ mod(A_{1}) + mod(A_{2}) + β(width(K)).

Proof. For a given cylinder this follows from the usual Grötzsch Inequality and the normality argument. Let us fix a K , and let mod (A) → ∞. We can assume that A_{i} are lower and upper components of A\K correspondingly. Then the modulus defect

mod(A) − mod(A_{1}) − mod(A_{2})

decreases by the usual Grötzsch inequality. At the limit the cylinder becomes the punctured plane, and the modulus defect converges to -(cap_{0}(K)+ cap_{∞}(K) ).

It follows from the area inequality that this sum of capacities is negative, unless K is a circle centered at the origin. Moreover the estimate depends only on width (K) . Indeed, let D_{0} and D_{∞} be the components of C \ K containing 0 and ∞ correspondingly. Let ϕ_{0} : B(0, R_{0}) → D_{0} and ϕ_{∞} : C\B(0, R_{∞}) → D_{∞} be the Riemann mappings normalized by: ϕ_{0}(z) ∼ z as z → 0 , and ϕ_{∞}(z) ∼ z as z → ∞. Then cap_{0}K = log R_{0} and cap_{∞}K = log(1/R_{∞}) .

As scaling does not change cap_{0}(K) + cap_{∞}(K) , we can assume that R_{∞}(K) = 1 . Let

ϕ_{∞}(z) = z − ∑_{k=1}^{∞} \frac{a_{k}}{z^{k}} .

Then:

area (C\D_{∞}) = \frac{i}{2} ∫_{|z|=1} ϕ_{∞}dϕ̄_{∞} = π(1 − ∑_{k=1}^{∞} k|a_{k}|^{2}) ≤ π,

with equality only in the case when ϕ_{∞} = id . Hence area (D_{0}) ≤ π with equality only in the case when K is the unit circle S^{1} .

As |ϕ_{0}|^{2} is a subharmonic function,

1 = |ϕ_{0}(0)|^{2} ≤ \frac{area (D_{0})}{πR_{0}^{2}} ≤ \frac{1}{R_{0}^{2}} ,

with equality only on the case when K = S^{1} . Hence cap_{0}(K) < 0 unless K = S^{1} . Moreover, by a normality argument cap_{0}(K) ≤ c(width (K)) < 0 . (Indeed, otherwise there would be a sequence of domains D_{0}^{m} as above converging to a domain Ω different from the unit disk, with cap_{0}(Ω) = 0 .) The lemma is proved. ⊔ ⊓

A.3. Eccentricity and pinching.

Let Γ be a Jordan curve surrounding a point a. Let d_{a}(Γ) and ρ_{a}(Γ) be the Euclidian radii of the inscribed and circumscribed circles about Γ centered at a. Then let us define the eccentricity of Γ about a as

e_{a}(Γ) = log \frac{ρ_{a}(Γ)}{d_{a}(Γ)}.

Lemma A.2. Let A ⊂ C\{0} be an annulus homotopically non-trivially embedded in the punctured plane, Γ ⊂ A be a homotopically non-trivial Jordan curve, and A_{i} be the components of A\Γ . Assume that mod (A_{i}) ≥ μ > 0 . If e_{0}(K) ≥ e then width (Γ|A) ≥ w(e) , where w(e) → ∞ as e → ∞.

Proof. Assume that there is a sequence of annuli A^{n} and curves Γ^{m} ⊂ A^{m} satisfying the assumptions of the lemma, such that width (Γ^{m}|A^{m}) ≤ w, while e_{0}(Γ^{m}) → ∞. Let us consider the uniformization ϕ_{m} : Ā^{m} → A^{m} of the A^{m} by round annuli centered at 0. Let Γ̄^{m} = ϕ^{−1}Γ^{m} . Then Γ̄^{m} is contained in a round annulus R̄^{m} of modulus ≤ w cocentric with Ā^{m} . Let R^{m} = ϕR̄^{m} .

Let us normalize the annuli A^{m} and Ā^{m} (by scaling and rotation) so that the inner radii of R^{m} and R̄^{m} are equal to 1, and ϕ_{m}(1) = 1 . Passing to a subsequence (without change of notations) we can find cocentric annuli R̄ ⊂ Ā such that the inner radius of R̄ is equal to 1, mod R̄ ≤ w, the both components of Ā\R̄ have moduli at least α(μ, w) , and R̄^{m} ⊂ R̄, Ā^{m} ⊃ Ā.

By the Koebe Theorem, the family of functions ϕ_{m} is normal in A. Hence these functions are uniformly bounded on R̄ contradicting the assumption that the eccentricities of Γ^{m} about 0 go to ∞. ⊔ ⊓

“Pinching” of a Jordan curve means creating of a narrow region which in limit makes the curve non-simple. Below we will quantify this process.

Let us take a number 0 < k < 1 called the “pinching parameter”. Let us define the k -pinching of a Jordan curve Γ as ξ_{k}(Γ) = inf dist (z_{1}, z_{2}) , where the infimum is taken over all pairs of points z_{i} ∈ Γ such that both components Γ_{1} and Γ_{2} of Γ\{z_{1}, z_{2}} have diameter at least k diam Γ .

We say that a curve Γ is (k, ϵ) -pinched if ξ_{k}(Γ) < ϵ. Note that if the curve is symmetric about 0 and e_{0}(Γ) ≥ e, then it is (0.5 − e^{−1}, e^{−1}) -pinched.

The following lemma shows that a sufficiently pinched curve has a definite width:

Lemma A.3. Let Γ , A and A_{i} be the same objects as in Lemma A.2. Let also mod (A_{i}) ≥ μ > 0 . If Γ is (k, ϵ) -pinched, then there exists a w = w(μ, k) > 0 such that width (Γ|A) ≥ w > 0 for all sufficiently small ϵ > 0 .

Proof. Otherwise we can find a sequence A^{m} of annuli as above with width (Γ^{m}|A^{m}) → 0 as m → ∞, and Γ^{m} is (k, 1/m) -pinched. As in the previous lemma, let ϕ_{m} : Ā^{m} → A^{m} be the uniformizations by round annuli normalized in such a way that Γ and Γ̄ = ϕ^{−1}Γ pass through 1.

Then the curves Γ̄^{m} should converge to the unit circle in the Hausdorff metric. Moreover, the family ϕ_{m} is well-defined and normal on a cocentric annulus Ā of modulus, say, \sqrt{μ}. Hence any Hausdorff limit of the famlily of curves Γ^{m} is an analytic Jordan curve. On the other hand, these curves should be (k, 0) -pinched (that is, they are non-simple). Contradiction. ⊔ ⊓

Lemma A.4. Let Γ , A and A_{i} be the same objects as in Lemma A.2, and mod (A_{i}) ≥ μ > 0 . Let D be a topological disk bounded by Γ , and b ∈ D. Then there is a function δ(L) → 0 as L → ∞ such that

dist (b, ∂Γ) ≤ δ(L) diam Γ,

provided ρ_{D}(0, b) ≥ L.

(Thus, if b is hyperbolically far away from 0 then it is Euclideanly close to the ∂D, in the scale of D.)

Proof. Otherwise there is a sequence of the curves Γ^{m} = ∂D^{m} as above, and points b^{m} ∈ D^{m} such that diam D^{m} = 1 ,

ρ_{D^{m}}(0, b_{m}) → ∞,  (A-2),

and

dist (b^{m}, Γ^{m}) ≥ δ.  (A-3)

Passing to a Caratheodory limit along some subsequence (without change of notations), we have: (D^{m}, 0, b^{m}) → (D, 0, b) , where D is a topological disk (note that b ∈ D due to (A-3)). But then ρ_{D^{m}}(0, b_{m}) → ρ_{D}(0, b) < ∞, contradicting (A-2). ⊔ ⊓

Lemma A.5. Let Γ be a Jordan curve which does not pass through 0. If it is (k, ϵ) - pinched then its pull-back under the quadratic map Φ : z ↦ z^{2} is (C^{−1}k, C\sqrt{ϵ}) -pinched, where C > 1 is an absolute constant.

Proof. Let z_{1} and z_{2} be two points on Γ such that dist (z_{1}, z_{2}) < ϵ, while diam Γ_{i} > k, where Γ_{i} are complementary components of Γ\{z_{1}, z_{2}}. Let us mark the Φ -preimages of the corresponding objects with twilde (select the closest preimages of the points z_{i} ).

We can assume that diam Γ = 1/4 . If dist (Γ, 0) > 1/4 then the distortion of the quadratic map on Γ is bounded by an absolute constant, and the conclusion follows. Otherwise Γ is contained in the unit disk. Hence 1/2 ≥ diam Γ̃ ≥ 1/8 and diam Γ̃_{i} ≥ (1/2)diam Γ_{i} ≥ k/8 . Moreover, dist (z̃_{1}, z̃_{2}) ≤ \sqrt{ϵ}, and we are done. ⊔ ⊓

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Mikhail Lyubich MSRI, 1000 Centennial Drive, Berkeley, CA 94720-5070 E-mail: misha@msri.org Permanent address: Mathematics Department and IMS, SUNY Stony Brook