ON MEASURE AND HAUSDORFF DIMENSION OF JULIA SETS FOR HOLOMORPHIC COLLET–ECKMANN MAPS

FELIKS PRZYTYCKI

Abstract. Let f : \overline{ℂ} → \overline{ℂ} be a rational map on the Riemann sphere , such that for every f-critical point c ∈ J which forward trajectory does not contain any other critical point, |(f^{n})′(f(c))| grows exponentially fast (Collet–Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0.

For f satisfying additionally Masato Tsujii’s condition that the average distance of f^{n}(c) from the set of critical points is not too small, we prove that Hausdorff dimension of Julia set is less than 2. This is the case for f(z) = z^{2} + c with c real, 0 ∈ J, for a positive line measure set of parameters c.

Introduction

It is well-known that if f : \overline{ℂ} → \overline{ℂ} a rational map of the Riemann sphere is hyperbolic, i.e. expanding on its Julia set J = J(f) namely |(f^{n})′| > 1 for an integer n > 0, then Hausdorff dimension HD(J) < 2.

The same holds for a more general class of subexpanding maps, namely such maps that all critical points in J(f) are non-reccurrent, supposed J(f) ≠ \overline{ℂ}, see [U].

On the other hand there is an abundance of rational maps with J ≠ \overline{ℂ} and HD(J) = 2, [Shi].

Recently Chris Bishop and Peter Jones proved that for every finitely generated not geometrically finite Kleinian groups for the Poincaré limit set Λ one has HD(Λ) = 2. As geometrically finite exhibits some analogy to subexpanding in the Kleinian Groups – Rational Maps dictionary, the question arised, expressed by Ch. Bishop and M. Lyubich at MSRI Berkeley conference in January 1995, isn’t it true for every nonsubexpanding rational map with connected Julia set, that HD(J) = 2 ?

Here we give a negative answer. For a large class of “non-uniformly” hyperbolic so called Collet–Eckmann maps, studied in [P1], satisfying an additional Tsujii condition, HD(J) < 2.

Date: (version of July 26, 1995).

Key words and phrases. Math. Rev. 58F23.

The author acknowledges support by Polish KBN Grant 2 P301 01307 “Iteracje i Fraktale, II”. He expresses also his gratitude to the MSRI at Berkeley (partial support by NSF grant DMS-9022140)

and ICTP at Trieste, where parts of this paper were written.

Notation. For a rational map f : \overline{ℂ} → \overline{ℂ} denote by Crit(f) the set of all critical points of f, i.e.points where f′ = 0. Let ν := sup{multiplicity off^{n}atc : c ∈ Crit(f) ∩ J}. Finally denote by Crit′(f) the set of all critical points of f in J(f) which forward trajectories do not contain other critical points.

We prove in this paper the following results:

Theorem A. Let f be a rational map on the Riemann sphere f : \overline{ℂ} → \overline{ℂ}, and there exist λ > 1, C > 0 such that for every f-critical point c ∈ Crit′(f)

|(f^{n})′(f(c))| ≥ Cλ^{n},  (0.1)

there are no parabolic periodic points, and J(f) ≠ \overline{ℂ}. Then Vol(J(f) = 0, where Vol denotes Riemann measure on \overline{ℂ}.

Theorem B. In the conditions of Theorem A assume additionally that

lim_{t→∞} lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{n} max(0, − log(dist(f^{j}(c), Crit(f))) − t) = 0.  (0.2)

Then HD(J(f)) < 2.

For f(z) = z^{2}+c, c ∈ [−2, 0] real, it is proved in [T] that (0.1) and (0.2) are satisfied for a positive measure set of parameters c for which there is no sink in the interval [c, c^{2}+ c]. Tsujii’s condition in [T] called there weak regularity is in fact apparently stronger than (0.2). The set of subexpanding maps satisfying (0.1) and weak regularity has measure 0, [T]. Thus Theorem B answers Bishop–Lyubich’s question.

Remark. In [DPU] it is proved that for every rational map f\overline{ℂ} → \overline{ℂ}, c ∈ Crit′

lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{n} − log dist(f^{j}(c), Crit(f)) ≤ C_{f}

where C_{f} depends only on f. Here in the condition (0.2) it is sufficient, for Theorem B to hold, to have a positive constant instead of 0 on the right hand side, unfortunately apparently much smaller than C_{f}.

Crucial in proving Theorems A and B is the following intermediate result:

Theorem 0.1 (on the existence of pacim), see [P1]. Let f : \overline{ℂ} → \overline{ℂ} satisfies the assumptions of Theorem A. Let μ be an α-conformal measure on the Julia set J = J(f) for an arbitrary α > 0. Assume there exists 0 < λ′ < λ such that for every n ≥ 1 and every c ∈ Crit′(f)

∫ \frac{dμ}{dist(x, f^{n}(c))^{(1−1/ν)α}} < C^{−1}(λ′)^{αn/ν}.  (0.3)

Then there exists an f-invariant probability measure m on J absolutely continuous with respect to μ (pacim).

Recall that a probability measure μ on J is called α-conformal if for every Borel B ⊂ J on which f is injective μ(f(B) = ∫_{B} |f′|^{α}dμ. In particular |f′|^{α} is Jacobian for f and μ. The number α is called the exponent of the conformal measure.

If Vol(J) > 0 then the restriction of Vol to J, normalized, is 2-conformal and obviously satisfies (0.3). If HD(J) = 2 then by [P1] we know there exists a 2conformal measure μ on J but we do not know whether it is not too singular, namely whether it satisfies (0.3). Under the additional assumption (0.2) we shall prove that it is so for every α-conformal measure.

Notation. Const will denote various positive constants which may change from one formula to another, even in one string of estimates.

1. More on pacim. Proof of Theorem A.

Proposition 1.1. In the situation of Theorem 0.1 there exists K > 0 such that μ-a.e. \frac{dm}{dμ} ≥ K.

Proof. In Proof of Theorem 0.1 [P1] one obtains m as a weak* limit of a subsequence of the sequence of measures \frac{1}{n} ∑_{j=0}^{n−1} f_{∗}^{j}(μ).

It is sufficient to prove that there exists K > 0 and n_{0} > 0 such that for μ-a.e. y ∈ J(f)

\frac{df_{∗}^{n}(μ)}{dμ}(y) = L^{n}(1) ≥ K.  (1.1)

Here L denotes the transfer operator, which can be defined for example by L(φ)(y) = ∑_{f(z)=y} |f′(z)|^{−α}φ(z). 1 is the constant function of value 1. We can assume y ∉ ⋃_{n>0} f^{n}(Crit(f)) because

μ(⋃_{n>0} f^{n}(Crit(f))) = 0  (1.2)

If a critical value for f^{n} were an atom then a critical point would have μ measure equal to ∞.

It is sufficient to prove (1.1) for y ∈ B(x, δ) ∩ J(f) for an a priori chosen x and an arbitrarily small δ and next to use the fact that there exists m ≥ 0 such that f^{m}(B(x, δ)) ⊃ J(f) (called topological exactness). Indeed

L^{n}(1)(w) = ∑_{f^{m}(y)=w} L^{n−m}(1)(y)|(f^{m})′|^{−α} ≥ (sup |(f^{m})′|)^{−α}L^{n−m}(y_{0})

where y_{0} ∈ f^{−m}({w}) ∩ B(x, δ).

Recall the estimate from [P1]. For an arbitrary γ > 1 there exists C > 0 such that for every x ∈ J(f)

L^{n}(1)(x) ≤ C + C ∑_{c∈Crit(f)∩J} ∑_{j=0}^{∞} \frac{γ^{j}λ^{−jα/ν}}{dist(x, f^{j}(f(c)))^{(1−1/ν)α}} .  (1.3)

By the assumptions (0.1) and (0.3) the above function is μ-integrable if γ is small enough.

Pay attention to the assumption (0.3). It concerns only c ∈ Crit′. Fortunately there is only a finite number of summands in (1.3) for which f^{j_{0}}(c) ∈ Crit, j_{0} ≥ j . Each summand is integrable because up to a constant it is bounded by L^{j}(1).

So

∑_{c∈Crit(f)∩J} ∑_{j=s}^{∞} \frac{γ^{j}λ^{−jα/ν}}{dist(x, f^{j}(f(c)))^{(1−1/ν)α}} → 0 μ − a.e. ass → ∞.  (1.4)

Fix from now on an arbitrary x ∈ J(f) for which (1.4) holds, (dm/dμ)(x) ≥ 1 and x ∉ ⋃_{n>0} φ^{n}(Crit(f)) (possible by (1.2) and by ∫ (dm/dμ)dμ = 1).

We need now to repeat from [P1] a part of Proof of Theorem 0.1:

For every y ∈ B(x, δ) and n > 0

L^{n}(1)(y) = ∑_{y′∈f^{−n}(y),regular} |(f^{n})′(y′)|^{−α} + ∑_{(y′,s)singular} L^{n−s}(1)(y′)|(f^{s})′(y′)|^{−α}

= ∑_{reg,y} + ∑_{sing,y} .

(1.5)

We shall recall the definitions of regular and singular: Take an arbitrary subexponentially decreasing sequence of positive numbers b_{j}, j = 1, 2 … with ∑ b_{j} = 1/100. Denote by B_{[k} the disc B(x, (∏_{j=1}^{k}(1 − b_{j}))2δ). We call s the essentially critical time for a sequence of compatible components W_{j} = Compf^{−j}(B_{[j}), compatible means f(W_{j}) ⊂ W_{j−1}, if there exists a critical point c ∈ W_{s} such that f^{s}(c) ∈ B_{[s}.

We call y′ regular in (1.5) if for the sequence of compatible components W_{s}, s = 0, 1, …, n, W_{n} ∋ y′ no s < n is essentially critical.

We call a pair (y′, s) singular if f^{s}(y′) = y and for the sequence of compatible components W_{j}, for j = 0, 1, …, s, with W_{s} ∋ y′, the integer s is the first (i.e., the only) essentially critical time.

If δ is small enough then all s in ∑_{sing,x} are sufficiently large that ∑_{sing,x} ≤ 1/2. This follows from the estimates in [P1]; here is the idea of the proof: Transforming ∑_{sing,x} in (1.5) using the induction hypothesis (1.3) we obtain the summands

C \frac{γ^{j}λ^{−jα/ν}}{dist(x, f^{s+j−1}(f(c)))^{(1−1/ν)α}} , j = 0, …, n − s

multiplied by

Const|(f^{s−1})′(x′)|^{−α/ν}a_{s} < γ^{s−1}λ^{−(s−1)α/ν}.

The numbers a_{s} are constants arising from distortion estimates, related to b_{s}. The numbers γ^{s} swallow them and other constants.

(There is a minor inaccuracy here: (s, x′) is a singular pair where the summand appears, provided x′ is not in the forward trajectory of another critical point, otherwise one moves back to it, see [P1] for details.)

Now ∑_{sing,x} ≤ 1/2 follows from (1.4).

The result is that ∑_{reg,x} ≥ 1/2. So by the uniformly bounded distortion along regular branches of f^{−n} on B(x, δ) we obtain

L^{n}(1)(y) ≥ ∑_{reg,y} ≥ Const ∑_{reg,x} ≥ Const > 0

The name regular concerned formally y′ ∈ f^{−n}(y) but in fact it concerns the branch of f^{−n} mapping y to y′ not depending on y ∈ B(x, δ).

By distortion of any branch g of f^{−n} on a set U we mean

sup_{z∈B} |g′(z)|/ inf_{z∈B} |g′(z)|.

This proves Proposition 1.1. □

Corollary 1.2 In the situation of Theorem 0.1 for measure-theoretic entropy h_{m}(f) > 0.

Proof. Denote dm/dμ by u.

Consider an open set U ⊂ \overline{ℂ} intersecting J(f) such that there exist two branches g_{1} and g_{2} of f^{−1} on it. Then by the f-invariance of m we have Jac_{m}(g_{1}) + Jac_{m}(g_{2}) ≤ 1 (= 1 if we considered all branches of f^{−1}). Jac_{m}(g_{i}) means Jacobian with respect to m for g_{i}.

We have m(U) > 0 because μ does not vanish on open sets in J (by the topological exactness of f on J) and by Proposition 1.2. At m-a.e. x ∈ U

Jac_{m}(g_{i})(x) = u(g_{i}(x))|g′_{i}(x)|u(x)^{−1} > 0,

(we used here also (1.4)).

Hence Jac_{m}(g_{i}) < 1, so Jac_{m}(f) > 1 on the set g_{i}(U), i = 1, 2 of positive measure m. Now we use Rokchlin’s formula and obtain

h_{m}(f) = ∫ log Jac_{m}(f)dm > 0.  □

Let χ_{m} = ∫ log |f′|dm denote characteristic Lyapunov exponent.

Corollary 1.3 In the situation of Theorem 0.1, χ_{m} > 0.

Proof. This Corollary follows from Ruelle’s inequality h_{m}(f) < 2χ_{m}, see [R].

Proof of Theorem A. Suppose Vol(J(f)) > 0. After normalization we obtain a 2-conformal measure μ on J(f) and by Theorem 0.1 and Corollary 1.3 a pacim m with χ_{m} > 0. By Pesin’s Theory [Pesin] in the iteration in dimension 1 case [Le] ([Le] is on the real case, but the complex one is similar), for m-a.e. x, there exists a sequence of integers n_{j} → ∞ and r > 0 such that for every j there exists a univalent branch g_{j} of f^{−n_{j}} on B_{j} := B(f^{n_{j}}(x), r) mapping f^{n_{j}}(x) to x and g_{j} has distortion bounded by a constant. By χ_{m} > 0 diamg_{j}(B(f^{n_{j}}(x), r) → 0. (This follows also automatically from the previous assertions by the definition of Julia set [GPS].) Now we can forget about the invariant measure m and go back to Vol. Because J(f) is nowhere dense in \overline{ℂ}, there exists ε > 0 such that for every z ∈ J(f)

\frac{Vol(B(z, r) \ J(f))}{Vol(B(z, r))} > ε.

Bounded distortion for g_{j} on B(z, r), z = f^{n_{j}}(x) allows to deduce that the same part of each small disc≈ g_{j}(B_{j}) around x is outside J(f) , up to multiplication by a constant. This is so because we can write for every X ⊂ B(z, r), y ∈ B(z, r)

Volg_{j}(X) ≈ |g′_{j}(y)|^{2}Vol(X)  (1.6)

where ≈ means up to the multiplication by a uniformly bounded factor. So x is not a density point of J(f). On the other hand a.e. point is a density point. So VolJ(f) = 0 and we arrived at a contradiction. □

2. Proof of Theorem B.

As mentioned in the Introduction, the following result is crucial:

Lemma 2.1. Under the conditions of Theorem B, i.e. in the situation of Theorem A and assuming (0.2), the condition (0.3) holds for every α-conformal measure, α > 0.

Proof. Step 1. Denote the expression from (0.2)

max(0, − log inf_{c∈Crit′(f)} dist(f^{n}(c), Crit(f)) − t)

by φ_{t}(n). Consider the following union of open-closed intervals

A′_{t} := ⋃_{n}(n, n + φ_{t}(n) · K_{f}] and write A_{t} := ℤ_{+} \ A′_{t},

for an arbitrary constant K_{f} > ν/ log λ.

By (0.2) for every a > 0 there exist t > 0 and n(a, t) such that for every n ≥ n(a, t)

A_{t} ∩ [n, n(1 + a)] ≠ ∅  (2.1)

Moreover, fixed an arbitrary integer M > 0, we can guarantee for every n′ ≥ n(1 + a), n ≥ n(a, t)

♯(A_{t} ∩ \{j ∈ [n, n′] : jdivisable byM\}) ≥ \frac{1}{2M} (n′ − n).  (2.2)

Observe that for every n_{0}, n, (2.1) transforms into

[n_{0} + n, n_{0} + n + a(n_{0} + n)] = [n_{0} + n, n_{0} + n + a(\frac{n_{0}}{n} + 1)n].

The result is that if n ≥ bn_{0} for an arbitrary b > 0 then

A_{t} ∩ [n_{0} + n, n_{0} + n + a(b^{−1} + 1)n] ≠ ∅.  (2.3)

Denote in the sequel a(b^{−1} + 1) by a′.

Step 2. Observe now that if n ∈ A_{t} then for every c ∈ Crit′(f) there exist branches g_{s}, s = 1, 2, …, n − 1 of f^{−s} on B_{n} := B(f^{n}(c), δ) with uniformly bounded distortions, where δ = ε exp −tν for a constant ε small enough. Sometimes to exhibit the dependence on n we shall write g_{s,n}.

Indeed, define g_{s} on B_{[s} = B(f^{n}(c), ∏_{j=1}^{s}(1−b_{j})2δ) for s = 1, 2, …, n−1 according to the procedure described in Proof of Theorem A. If there is an obstruction, namely s an essential critical time, then for every z ∈ B_{[s}

|g′_{s−1}(z)| ≤ λ^{−s}ϑ^{s} ≤ exp(−sν/K_{f})  (2.4)

for ϑ > 1 arbitrarily close to 1 (in particular such that K_{f} > \frac{ν}{log λ−log ϑ}) and for s large enough. The constant ϑ takes care of distortion. (2.4) holds for z = f^{s}(q), where q is the critical point making s critical time, without ϑ by (0.1) (with the constant C instead). The small number ε takes care of s small, which cannot be then essential critical.

The inequality (2.4) and rooting (1/ν to pass from s − 1 to s) imply φ_{t}(f^{n−s}(c)) ≥ s/K_{f}, so n ∉ A_{t}, a contradiction.

Step 3. By uniformly bounded distortion for the maps g_{j,n}, n ∈ A_{t} we obtain (compare (1.6)) for every n_{0} > 0 large enough, c ∈ Crit′

μB(f^{n_{0}}(c), r_{j}) ≈ r_{j}^{α}  (2.5)

for a sequence r_{j}, j = 1, 2, … such that

r_{1} > exp −Lbn_{0}  (2.6)
r_{j+1} > r_{j}^{1+σ}  (2.7)
and r_{j+1} < r_{j}/2.  (2.8)

Here L := 2 sup |f′| and b, σ are arbitrarily close to 0.

Indeed, we can find r_{j} satisfying the conditions above by taking

r_{j} := diamg_{n_{j},n_{0}+n_{j}}(B(f^{n_{0}+n_{j}}(c), δ))

where n_{j} ∈ A_{t} are taken consecutively so that

n_{j+1} ∈ [n_{j} + (1 + ϑ)n_{j}, n_{j} + (1 + ϑ)(1 + a′)n_{j} for j ≥ 2 and
n_{1} ∈ [n_{0} + bn_{0}, n_{0} + bn_{0} + a′bn_{0}],

where ϑ > 0 is an arbitrary constant close enough to 0.

This gives

r_{j+1}/r_{j} ≥ exp(−2(log L)a′n_{j})  (2.9)

To conclude we need to know that r_{j} shrink exponentially fast with n_{j} → ∞. For that we need the following fact (see for example [GPS], find the analogous fact in Proof of Theorem A):

(*) For every r > 0 small enough and ξ, C > 0 there exists m_{0} such that for every m ≥ m_{0}, x ∈ J(f) and branch g of f^{−m} on B(x, r) having distortion less than C we have diamg(B(x, r)) < ξr.

Apply now (2.2) to n = n_{0}, n′ = n_{j} +n_{0}. We obtain a “telescope”: For all consecutive τ_{1}, τ_{2}, …, τ_{k(j)} ∈ A_{t} ∩ [n_{0}, n_{j} + n_{0}] divisible by M

g_{τ_{i+1}−τ_{i},τ_{i+1}}(B(f^{τ_{i+1}}(c), δ)) ⊂ B(f^{τ_{i}}(c), δ/2)

for M ≥ m_{0} from (*).

Hence using (2.2)

r_{j} ≤ 2^{−n_{j}/2M}.  (2.10)

Denote 2a′ log L by γ and (log 2)/2M by γ′. (2.9) and (2.10) give

r_{j+1} ≥ r_{j} exp −γn_{j} ≥ r_{j}(exp −γ′n_{j})^{γ/γ′} ≥ r_{j}^{1+γ/γ′}.

As γ′ is a constant and γ can be done arbitrarily small if a is small enough, we obtain (2.7).

The condition (2.8) follows from the fact that for n_{0} large enough all n_{j+1} − n_{j} are large enough to apply (*).

Conclusion. We obtain

∫ \frac{dμ}{dist(x, f^{n_{0}}(c))^{(1−1/ν)α}} ≤

μ(\overline{ℂ} \ B(f^{n_{0}}(c), r_{1}))\frac{1}{r_{1}^{(1−1/ν)α}} + ∑_{j≥2} μ(B(f^{n_{0}}(c), r_{j−1}) \ B(f^{n_{0}}(c), r_{j}))\frac{1}{r_{j}^{(1−1/ν)α}} ≤

exp(Lbn_{0}(1 − 1/ν)α) + Const ∑_{j≥2} \frac{r_{j−1}^{α}}{r_{j}^{(1−1/ν)α}} ≤

(exp(Lb(1 − 1/ν)α))^{n_{0}} + Const ∑_{j≥2} r_{j−1}^{α}r_{j−1}^{−(1−1/ν)α(1+σ)}.

The latter series has summands decreasing exponentially fast for σ small enough so it sums up to a constant, hence the first summand dominates. We obtain the bound by (λ′)^{n_{0}} with λ′ > 1 arbitrarily close to 1. Thus (0.3) has been proved. □

Remark 2.2. The only result in our disposal on the abundance of non-subexpanding maps satisfying (0.1) and (0.2) is Tsujii’s one concerning z^{2} + c, c real (see Introduction). For this class however the exponential convergence of

diamCompf^{−n_{j}}(B(f^{n_{j}+n_{0}}(0), δ)

to 0 follows from [N] (the component containing f^{n_{0}}(c)). So restricting our interests to this class we could skip (2.2) and considerations leading to (2.10) above.

By [N] diam(Comp(f^{−n}(B(x, δ))) ∩ ℝ) < Cλ̃^{−n} for some constants C > 0, λ̃ > 1, δ small enough and every component Comp. Just the uniform convergence of the diameters to 0 as n → ∞ follows from [P1], but I do not know how fast is it.

Proof of Theorem B. Suppose that HD(J) = 2. Then there exist a 2-conformal measure μ on J. This follows from the existence of an α-conformal measure for α = HD_{ess}(J), where HD_{ess} is the essential Hausdorff dimension which can be defined for example as supremum of Hausdorff dimensions of expanding Cantor sets in J, see [DU] [P2] and [PUbook], and from HD_{ess}(J) = HD(J), see [P1]. The former holds for every rational map, the latter was proved in [P1] only for Collet–Eckmann maps.

By Lemma 2.1 the condition (0.3) holds, hence there exists a pacim m ≪ μ. Moreover χ_{m} > 0 by Corollary 1.3. As in Proof of Theorem A by Pesin Theory there exists X ⊂ J, m(X) = μ(X) = 1, such that for every x ∈ X there exists a sequence of integers n_{j}(x) → ∞, r > 0 and univalent branches g_{j} of f^{−n_{j}} on B(f^{n_{j}}(x), r) mapping f^{n_{j}} to x with uniformly bounded distortion. Write B_{x,j} := g_{j}(B(f^{n_{j}}(x), r)).

We obtain for every x ∈ X applying (1.6) to Vol and μ (similarly as in Proof of Theorem A)

μ(B_{x,j}) ≤ ConstVol(B_{x,j}) ≤ Const(VolB(x, diamB_{x,j})).

If VolX = 0 then there exists a covering of X by discs B(x_{t}, diamB_{x_{t},j_{t}}), t = 1, 2, … which union has Vol < ε for ε arbitrarily close to 0, of multiplicity less than a universal constant (Besicovitch’s theorem). Hence

ε ≥ Const ∑_{t} VolB(x_{t}, diamB_{x_{t},j_{t}}) ≥ Constμ ∑_{t} B_{x_{t},j_{t}} ≥ 1,

a contradiction. Hence VolJ ≥ VolX > 0.

This contradicts Theorem A that VolJ = 0 and the proof is over.

Remark that we could end the proof directly: As in Proof of Theorem A we show that no point of X is a point of density of the Vol measure. Hence VolX = 0. (I owe this remark to M. Urbański.) □

References

[BJ] Ch. Bishop, P. Jones: Hausdorff dimension and Kleinian groups. Preprint SUNY at Stony Brook, IMS 1994/5.

[DPU] M. Denker, F. Przytycki, M. Urbański: On the transfer operator for rational functions on the Riemann sphere. Preprint SFB 170 Göttingen, 4 (1994). To appear in Ergodic Th. and Dyn. Sys..

[DU] M. Denker, M. Urbański: On Sullivan’s conformal measures for rational maps of the Riemann sphere. Nonlinearity 4 (1991), 365-384.

[GPS] P. Grzegorczyk, F. Przytycki, W. Szlenk: On iterations of Misiurewicz’s rational maps on the Riemann sphere. Ann. Inst. H. Poincaré, Phys. Théor. 53 (1990), 431-444.

[Le] F. Ledrappier: Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 77-93.

[N] T. Nowicki: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies niform hyperbolicity. Ergodic Th. & Dynamic. Sys. 8 (1988), 425-435.

[Pesin] Ya. B. Pesin: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32 (1977), 45-114.

[P1] F. Przytycki: Iterations of holomorphic Collet–Eckmann maps: conformal and invariant measures. Preprint n^{o} 57, Lab. Top. Université de Bourgogne, Février 1995.

[P2] F. Przytycki: Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119(1) (1993), 309-317.

[PUbook] F. Przytycki, M. Urbański: To appear.

[R] D. Ruelle: An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 83-87.

[Shi] M. Shishikura: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia set. Preprint SUNY at Stony Brook, IMS 1991/7.

[T] M. Tsujii: Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math. 111 (1993), 113-137.

[U] M. Urbański: Rational functions with no recurrent critical points. Ergodic Th. and Dyn. Sys. 14.2 (1994), 391-414.

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00 950 Warszawa, Poland

  E-mail address: feliksp@impan.impan.gov.pl