Real bounds and Lyapunov exponents

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfy the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.


Introduction
This paper studies critical circle maps (as well as infinitely renormalizable unimodal maps) from the differentiable ergodic theory viewpoint. The ergodic aspects of one-dimensional dynamical systems have been the object of intense research for quite some time. In particular, the study of characteristic or Lyapunov exponents of invariant measures, or physical measures, was initiated in this context by Ledrappier, Bowen, Ruelle, and developed by Keller, Blokh and Lyubich among others. See [28, Chapter V] for a full account, and the references therein.
In this article we show that the Lyapunov exponent of a C 3 critical circle map (or of an infinitely renormalizable unimodal interval map) is always zero. The general approach leading to zero Lyapunov exponents is by arguing by contradiction and using Pesin's theory: non-zero Lyapunov exponent implies the existence of periodic orbits (see for instance [21,Supplement 4 and 5] and [34,Chapter 11]), and that would be a contradiction for critical circle maps with irrational rotation number. Our goal in this paper, however, is to prove that the exponent is zero directly from the real a-priori bounds (see Theorem 2.2), without using Pesin's theory. In fact, the only non-trivial result from ergodic theory we shall use here is Birkhoff's Ergodic Theorem.
By a critical circle map we mean an orientation preserving C 3 circle homeomorphism f with finitely many non-flat critical points c 1 , c 2 , . . . , c N (N ≥ 1). A critical point c is called non-flat if in a neighbourhood of c the map f can be written as f (t) = φ(t) |φ(t)| d−1 + f (c), where φ is a C 3 local diffeomorphism with φ(c) = 0, and where d > 1 is a real number known as the criticality (or order, or type, or exponent ) of such critical point. Critical circle maps have been studied by several authors in the last three decades. From a strictly mathematical viewpoint, these studies started with basic topological aspects [18], [42], then evolved -in the special case of maps with a single critical point -to geometric bounds [20], [37], and further to geometric rigidity and renormalization aspects, see [1], [9], [10], [11], [12], [14], [15], [16], [23], [25], [37], [38], [39], [40] and [41]. The geometric rigidity and renormalization aspects of the theory remain open for maps with more than one critical point, see Question 7.3. Such brief account bypasses important numerical studies by several physicists, as well as computer-assisted and conceptual work by Feigenbaum, Kadanoff, Lanford, Rand, Epstein and others; see [10] and references therein.
As we said before, this paper studies a critical circle map f from the differentiable ergodic theory viewpoint. We will focus on the case when the rotation number of f is irrational, in which case f is uniquely ergodic. Moreover, by a theorem of Yoccoz [42], f is minimal and therefore topologically conjugate to the corresponding rigid rotation. This implies that the support of its unique invariant Borel probability measure is the whole circle (see Section 3 for more details on the invariant measure). Our main result is the following.
Theorem A. Let f : S 1 → S 1 be a C 3 critical circle map with irrational rotation number, and let µ be its unique invariant Borel probability measure. Then log Df belongs to L 1 (µ) and it has zero mean:

Moreover, no critical point of f satisfies the Collet-Eckmann condition.
Recall that f satisfies the Collet-Eckmann condition at a critical point c ∈ {c 1 , c 2 , . . . , c N } if there exist C > 0 and λ > 1 such that Df n f (c) ≥ Cλ n for all n ∈ N (see for instance [28, Chapter V]), or equivalently lim inf n→∞ 1 n log Df n (f (c)) ≥ log λ > 0 . (1.1) The integrability of log Df was obtained by Przytycki in [31,Theorem B], where he also proved that log Df dµ ≥ 0 (see [35,Appendix A] for an easier proof). We will obtain the integrability again (see Proposition 3.1) on the way to proving that log Df dµ = 0. It is expected that log Df will not be integrable if we allow the presence of flat critical points, as in [18].
Theorem A applies to some classical examples of holomorphic dynamics in the Riemann sphere, see Theorem C in §6.
Remark 1.1. The analogue of Theorem A for diffeomorphisms is straightforward: if f is an orientation-preserving C 1 circle diffeomorphism, with irrational rotation number, the function ψ : S 1 → R defined by ψ = log Df is a continuous function and therefore, by the unique ergodicity of f , the sequence of continuous functions: 1 n n−1 j=0 ψ • f j converges uniformly to a constant, and this constant must be S 1 log Df dµ. By the chain rule n−1 j=0 ψ • f j = log Df n and, therefore, the sequence of continuous functions log Df n /n converges to the constant S 1 log Df dµ uniformly in S 1 . Since f n is a diffeomorphism for all n ∈ N, this constant must be zero. In our case, however, log Df is not a continuous function (it is defined only in S 1 \ {c 1 , c 2 , . . . , c N }, and it is unbounded, see Figure 1). Remark 1.2. The C 3 -smoothness hypothesis of Theorem A could be relaxed to C 2 (or even C 1+Zygmund , see [28, Section IV.2.1]) -indeed whatever smoothness is necessary for the real bounds to hold.
1.1. How the paper is organized. In §2 we briefly recall some classical combinatorial facts about circle maps, and also the well-known a-priori bounds on the critical orbits of a critical circle map. We deduce from these facts two useful lemmas concerning dynamical partitions. In §3, we establish the integrability of log Df with respect to the unique f -invariant probability measure, for any critical circle map without periodic points for which the real bounds of §2 hold true. In §4, we use the results of §2 and §3 to prove our main result, namely Theorem A. In §5, we prove Theorem B, an analogous result to Theorem A for infinitely renormalizable unimodal maps with non-flat critical point. In §6, we discuss an application of Theorem A to the ergodic theory of certain Blaschke products as well as quadratic polynomials. Finally, in §7, we conclude by stating a few open questions concerning both critical circle maps and rational maps of the Riemann sphere.

The real bounds
Let f be a C 3 critical circle map as defined in the introduction, that is, f is an orientation preserving C 3 circle homeomorphism with finitely many non-flat critical points of odd type. As we have pointed out already, our standing assumption is that the rotation number ρ(f ) = θ ∈ [0, 1) is irrational. Therefore it has an infinite continued-fraction expansion, say θ = [a 0 , a 1 , ..., a n , a n+1 , ...] = lim n→∞ 1 a 0 + 1 We define recursively a sequence of return times of θ by: q 0 = 1, q 1 = a 0 and q n+1 = a n q n + q n−1 for n ≥ 1. In particular the sequence {q n } n≥1 grows at least exponentially fast when n goes to infinity (we will use this fact in the proof of Proposition 3.1 and in the proof of Proposition 4.3 below). We recall that the numbers q n are also obtained as the denominators of the truncated expansion of order n of θ: p n q n = [a 0 , a 1 , a 2 , ..., a n−1 ] = 1 a 0 + 1 We recall also the following well-known estimates.
A crucial combinatorial fact is that P n (c) is a partition (modulo boundary points) of the circle for every n ∈ N. We call it the n-th dynamical partition of f associated with the point c. Note that the partition P n (c) is determined by the finite piece of orbit f j (c) : 0 ≤ j ≤ q n + q n+1 − 1 . As we are working with critical circle maps, our partitions in this article are always determined by a critical orbit. Our proof of Theorem A is based on the following result.
Theorem 2.2 (The real bounds). There exists a constant K 0 > 1 with the following property. Given a C 3 critical circle map f with irrational rotation number there exists n 0 = n 0 (f ) such that, for each critical point c of f , for all n ≥ n 0 , and for every pair I, J of adjacent atoms of P n (c) we have: where |I| denotes the Euclidean length of an interval I in the real line.
Of course for a particular f we can choose K 0 > 1 such that (2.1) holds for all n ∈ N. Theorem 2.2 was proved byŚwiatek and Herman (see [20] and [37]) in the case when f has a single critical point. The original proof is based on the so-called cross-ratio inequality ofŚwiatek. As it turns out, this inequality is valid also in the case when the map f has several critical points (all of non-flat type), see [30]. This fact combined with the method of proof presented in [11, Section 3] yields the above general result. A detailed proof will appear in [8].
Note that for a rigid rotation we have |I n | = a n+1 |I n+1 | + |I n+2 |. If a n+1 is big, then I n is much larger than I n+1 . Thus, even for rigid rotations, real bounds do not hold in general.
In the case of maps with a single critical point, one also has the following corollary, which suitably bounds the distortion of first return maps. Corollary 2.3. Given a C 3 critical circle map f with irrational rotation number and a unique critical point c ∈ S 1 , there exists a constant K 1 > 1 such that the following facts hold true for each n ≥ n 0 : (i) For all x, y ∈ f (I n+1 (c)), we have (ii) For all x, y ∈ f (I n (c)), we have The control on the distortion of the return maps in the above corollary follows from Koebe's distortion principle (see [11,Section 3]). When f has two or more critical points, the estimates given in the Corollary may fail, because the intervals f (I n (c)) and f (I n+1 (c)) could in principle contain other critical points of f qn+1−1 and f qn−1 , respectively.
Remark 2.4. We shall henceforth use the constant K = max{K 0 , K 1 } > 1 whenever we invoke the real bounds.
For our purposes, an important consequence of Corollary 2.3 is the following auxiliary result.
Lemma 2.5. Let f be as in Corollary 2.3. There exists C > 1 such that for all n ∈ N and for all x ∈ I n (c) \ I n+2 (c): Proof of Lemma 2.5. For each k ∈ N, let us write I k instead of I k (c) in this proof. Fix n ∈ N and x ∈ I n \I n+2 . By Corollary 2.3 the map f qn+1−1 : f (I n ) → f qn+1 (I n ) has bounded distortion. In particular, there exists C 0 > 1 such that: Since I n+1 ⊂ f qn+1 (I n ) ⊂ I n ∪ I n+1 we obtain from the real bounds that (1/K)|I n | ≤ f qn+1 (I n ) ≤ (1 + K)|I n |. Therefore: Since c is a non-flat critical point of f of odd type d there exist 0 < C 1 < C 2 such that C 1 |I n | d ≤ |f (I n )| ≤ C 2 |I n | d for all n ∈ N, and then: Again using that c is non-flat there exist 0 < A < B such that A|x − c| d−1 ≤ Df (x) ≤ B|x−c| d−1 for all x in a small but fixed neighbourhood around the critical point. In particular, for all x ∈ I n \ I n+2 and for all n ∈ N, since |x − c| ≥ |I n+2 | ≥ |I n |/K 2 , again by the real bounds. With this at hand we deduce that for all x ∈ I n \ I n+2 and for all n ∈ N. This finishes the proof of the lemma, provided we take The following consequence of the real bounds was inspired by [11, Lemma A.5, page 379]. It holds under the general assumptions of Theorem 2.2, for maps with an arbitrary number of critical points. For each n ≥ 1 let: where d(c, I) denotes the Euclidean distance between an interval I ⊂ S 1 and the critical point c. Proof of Lemma 2.6. Given a critical point c, let us write in this proof, for simplicity of notation, P k , I k instead of P k (c), I k (c) respectively, for each k ∈ N. Note that the transition from P n to P n+1 can be described in the following way: the interval I n = [c, f qn (c)] is subdivided by the points f jqn+1+qn (c) with 1 ≤ j ≤ a n+1 into a n+1 + 1 subintervals. This sub-partition is spread by the iterates of f to all f j (I n ) = f j ([c, f qn (c)]) with 0 ≤ j < q n+1 . The other elements of the partition P n , namely the intervals f j (I n+1 ) with 0 ≤ j < q n , remain unchanged. On one hand, for any I ∈ P n \ {I n , I n+1 } we have: On the other hand: This gives us: By the real bounds |In\In+2| |In+2| ≤ K 2 for all n ≥ 1 and we are done.

The integrability of log Df
As before let f be a C 3 critical circle map with finitely many non-flat critical points and with rotation number ρ(f ). Since we assume that ρ(f ) is irrational, f admits a unique invariant Borel probability measure µ. Moreover, by a theorem of Yoccoz [42], f has no wandering intervals and therefore there exists a circle homeomorphism h : S 1 → S 1 which is a topological conjugacy between f and the rigid rotation by angle ρ(f ), that we denote by R ρ(f ) . More precisely, the following diagram commutes: where Leb denotes the normalized Lebesgue measure in the unit circle (the Haar measure for the multiplicative group of complex numbers of modulus 1). Therefore µ is just the push-forward of Lebesgue measure under h −1 , that is, µ(A) = h −1 * Leb (A) = Leb h(A) for any Borel set A in the unit circle (recall that the conjugacy h is unique up to post-composition with rotations, so the measure µ is well-defined).
In this section we prove that log Df belongs to L 1 (µ). As before, let us denote by c 1 , c 2 , . . . , c N the critical points of f .
Let ϕ : S 1 → R be given by ϕ = | log Df |. For each 1 ≤ j ≤ N and each n ≥ 1, let J n (c j ) = I n (c j ) ∪ I n+1 (c j ). We define E n = N j=1 J n (c j ) and consider ϕ n : S 1 → R given by: ϕ n = χ S 1 \En · ϕ , that is, ϕ n = 0 on each J n (c j ) and ϕ n = ϕ on the complement of their union. We will use the following four facts: As explained above, the measure µ is the pullback of the Lebesgue measure under any topological conjugacy between f and the corresponding rigid rotation. In particular, for each 1 ≤ j ≤ N and for all k ≥ 1, we have µ(I k (c j )) = |q k θ − p k | and by Theorem 2.1: for all k ≥ 1 and each 1 ≤ j ≤ N .
Fact 3. By combinatorics, we have µ(I k (c j ) \ I k+2 (c j )) = a k+1 µ(I k+1 (c j )), for all k ≥ 0 and for each 1 ≤ j ≤ N . Fact 4. Since each c j is a non-flat critical point, there exist C 0 > 0 and a neighbourhood V j of c j such that for all x ∈ V j we have: We may assume, of course, that the V j 's are pairwise disjoint. With all these facts at hand we are ready to prove the desired integrability result.
Proof of Proposition 3.1. Note that the sequence {ϕ n } converges monotonically to ϕ = | log Df |. Let n 0 be the smallest positive integer such that J n0 (c j ) ⊆ V j for all 1 ≤ j ≤ N . We only look at values of n greater than n 0 . Then, since ϕ n is identically zero on E n and agrees with ϕ everywhere else, we can write The first integral on the right-hand side is a fixed number independent of n. Hence it suffices to bound the last double sum. Using (3.1) and the fact that in I k (c j ) \ I k+2 (c j ) the closest point to c j is f q k+2 (c j ), we see that (see Figure 2) Applying facts 1, 2 and 3 to this last sum, we see that However we know from Theorem 2.1 that Since the q k 's grow exponentially fast (at least as fast as the Fibonacci numbers), we have Hence the left-hand side of (3.6) is uniformly bounded. Taking this information back to (3.3) and then to (3.2), we deduce that there exists a constant C 2 > 0 such that But then, by the Monotone Convergence Theorem, ϕ is µ-integrable, as desired.
Remark 3.2. The proof of Proposition 3.1 yields, mutatis mutandis, a slightly stronger result, namely that log Df ∈ L p (µ) for every finite p ≥ 1. However, this more general fact will not be needed in this paper.

Proof of Theorem A
In this section we present two different proofs of Theorem A. The first proof works only when the map f has a single critical point, whereas the second works in the general multicritical case.

4.1.
First proof: the unicritical case. Here we assume that f has a single critical point c. In particular, we are free to use Lemma 2.5. Once again we write P k , I k , instead of P k (c), I k (c), etc., for simplicity of notation.
Consider the Borel set A ⊂ S 1 defined in the following way: x ∈ A iff there exists an increasing sequence {n k } k∈N ⊂ N such that for each k ∈ N there exists (a necessarily unique) j k ∈ {0, 1, ..., q n k +1 − 1} such that f j k (x) ∈ I n k \ I n k +2 .
Proof of Lemma 4.1. The first assertion follows immediately from the definition of A, hence we focus on proving that A has full µ-measure. For each n ≥ 1 consider the disjoint union: We claim that µ(A n ) > 1/3 for all n ≥ 1. Indeed, µ(A n ) = q n+1 µ(I n \ I n+2 ) = a n+1 q n+1 µ(I n+1 ), since µ(I n \I n+2 ) = a n+1 µ(I n+1 ). As explained at the beginning of Section 3, the measure µ is the pullback of the Lebesgue measure under any topological conjugacy between f and the corresponding rigid rotation. In particular: where θ ∈ R \ Q is the rotation number of f . By Theorem 2.1: and then: Since q n+2 = a n+1 q n+1 + q n < (a n+1 + 1)q n+1 we deduce that Since a n+1 ≥ 1 for all n ≥ 0 we obtain the claim, that is, µ(A n ) > 1/3 for all n ≥ 1. Moreover, since: Proof of Proposition 4.3. Recall that, since c is a non-flat critical point of f , there exists L > 0 such that for any x, y ∈ S 1 \ {c} we have: Let x ∈ B and let {n k } k∈N be its corresponding increasing sequence of natural numbers. Recall that for each k ∈ N there exists (a necessarily unique) j k ∈ {0, 1, ..., q n k +1 − 1} such that f j k (x) ∈ I n k \ I n k +2 . Then we have: where the second inequality is given by Lemma 2.5. By combinatorics we have the following facts: (1) The points f i (x) and f i+qn k +1 (x) do not belong to I n k ∪ I n k +1 for any . By the real bounds: Therefore, with the notation of Lemma 2.6, we have that: and then: At this point, recall that the sequence {q n } n≥1 grows at least exponentially fast with n, whereas S n grows at most linearly with n, by Lemma 2.6. Hence both terms in the right-hand side of this last inequality go to zero as k → ∞, and we are done.
With Proposition 3.1, Proposition 4.2 and Proposition 4.3 at hand our main result -in the unicritical case -follows in a straightforward manner.
Proof of Theorem A. By Proposition 3.1 we already know that log Df belongs to L 1 (µ). Hence by Birkhoff's Ergodic Theorem we have for µ-almost every x ∈ S 1 . Combining this fact with Proposition 4.2 and Proposition 4.3 we obtain: Finally we have to prove that f does not satisfy the Collet-Eckmann condition. Indeed, if there were constants C > 0 and λ > 1 such that Df n f (c) ≥ Cλ n for all n ∈ N we would have:  For this purpose it would be enough to prove that the limit exists (since f (c) belongs to B), for instance by proving that the critical value of f is a µ-typical point for the Birkhoff's averages of log Df . Note, however, that this fact does not follow directly from the unique ergodicity of f since log Df is not a continuous function (it is defined only in S 1 \ {c}, and it is unbounded, see Figure 1 in the introduction).

4.2.
Second proof: the general multicritical case. Let us now give a proof of Theorem A that works in general. Our proof relies on Proposition 4.8 below, which can be regarded as a suitable replacement for Lemma 2.5. As before, let {q n } n∈N be the sequence of return times given by the irrational rotation number of f (see Section 2). Let us denote by c 1 , c 2 , ..., c N the critical points of f (N ≥ 1) and let d i > 1 denote the criticality of each c i . Conjugating f by a suitable C 3 -diffeomorphism (which does not affect its Lyapunov exponent) we may assume that each c i has an open neighbourhood V (c i ) where f is a power-law of the form: We also assume, of course, that V (c i ) ∩ V (c j ) = Ø whenever i = j.
Recall from the real bounds (Theorem 2.2) that, for each c ∈ {c 1 , c 2 , ..., c N }, the dynamical partitions P n (c) n∈N have the comparability property: any two consecutive atoms of P n (c) have comparable lengths. We will also need the following three further consequences of the real bounds.
Lemma 4.5. There exists B 0 = B 0 (f ) > 1 such that for each c ∈ {c 1 , c 2 , ..., c N }, for each n ∈ N and for each atom ∆ ∈ P n (c) we have: Lemma 4.6. There exists B 1 = B 1 (f ) > 1 with the following property: let ∆ ∈ P n (c) and denote by ∆ * the union of ∆ with its two immediate neighbours in P n (c).
If 0 ≤ j < k ≤ q n are such that the intervals f j (∆ * ), f j+1 (∆ * ),..., f k−1 (∆ * ) do not contain any critical point of f , then the map f k−j : f j (∆) → f k (∆) has distortion bounded by B 1 , that is: Proof of Lemma 4.6. The real bounds imply that f j (∆) has space inside f j (∆ * ).
Lemma 4.7. There exists B 2 = B 2 (f ) > 1 with the following property: if c = c ′ are critical points of f and ∆ ∈ P n (c), ∆ ′ ∈ P n (c ′ ) for some n ∈ N are such that Lemma 4.7. This follows from the combinatorial fact that ∆ is contained in the union of two adjacent atoms of P n (c ′ ), one of which is ∆ ′ , and likewise for ∆ ′ .
For each k ≥ 0 and each critical point c we will use the notation J k (c) = The key to prove Theorem A is the following fact. (1) For each x ∈ S 1 and all n ≥ 0 we have log Df qn (x) ≤ C.
In what follows we denote by C 0 , C 1 , C 2 , C 3 ,... positive constants (greater than 1, in fact) depending only on f . Moreover, for any two positive numbers a and b we use the notation a ≍ b to mean that C −1 a ≤ b ≤ Ca for some constant C > 1 depending only on f .
Proof of Proposition 4.8. Let us fix once and for all a critical point c ∈ {c 1 , c 2 , ..., c N }. We assume that n ≥ 0 is large enough so that each atom of P n (c) contains at most one critical point of f . Let x ∈ S 1 and let ∆ ∈ P n (c) be such that x ∈ ∆. Let ∆ * ⊇ ∆ be as in Lemma 4.6. Just by taking n larger still, we may assume that, for 0 ≤ k < q n , each f k (∆ * ) contains at most one critical point of f . We say that 0 ≤ k < q n is a critical time for x if f k (∆ * ) contains a critical point of f . Let us write 0 ≤ k 1 < k 2 < ... < k m < q n for the sequence of all critical times for x. Note that m ≤ 3N since the family f k (∆ * ) 0≤k<qn has intersection multiplicity equal to 3. Using these critical times and the chain rule we can write: We proceed to estimate each term in the product (4.3) above. From Lemma 4.6 (with j = 0 and k = k 1 ) we have: Again from Lemma 4.6 (with k j + 1 and k j+1 replacing j and k respectively) we have for all j ∈ {1, ..., m − 1}: For each j ∈ {1, ..., m} let β j ∈ {c 1 , c 2 , ..., c N } be the (unique) critical point of f in f kj (∆ * ), and let d j be its criticality. Since we are assuming that n is sufficiently large, we may suppose that f kj (∆ * ) ⊆ V (β j ) for all j ∈ {1, ..., m}. Then, from the power-law expression (4.1) we have: and recall that d j − 1 > 1 for all j ∈ {1, ..., m}. Still using the power-law expression we see that: Using Lemma 4.6 yet again, we also see that: . (4.8) Let us now prove item (1) of the conclusion of Proposition 4.8. Note that (4.6) yields: where C 0 = C 0 (f ) > 0. Combining all these facts, namely (4.4)-(4.9), we deduce the following (upper) telescoping estimate: where in the last line we have used (4.7) and finally Lemma 4.5. This proves item (1). In order to prove item (2) note first that all estimates provided above are two-sided, except (4.9). In order to get a lower bound for the left side of (4.9) we use the hypothesis in (2). Since f kj (x) / ∈ J 2n (β j ) we have: From the real bounds we know that there exists λ ∈ (0, 1) depending only on f such that C −1 4 λ n I n (β j ) ≤ I 2n (β j ) ≤ C 4 λ n I n (β j ) . Moreover, we claim that I n (β j ) is comparable to f kj (∆) . Indeed, this follows from Lemma 4.7 because I n (β j ) ∈ P n (β j ) intersects an atom of P n (c) in f kj (∆ * ), and this atom has length comparable to f kj (∆) (such atom is either f kj (∆) itself, or one of its neighbours). Using these facts in (4.11) we deduce that: (4.12) Using this lower estimate in place of the upper estimate (4.9) and proceeding as in (4.10) we arrive at the estimate Df qn (x) ≥ C 6 λ nα , and then: log Df qn (x) ≥ −nα log 1 λ + log C 6 ≥ −C 7 n .
With Proposition 4.8 at hand we are ready to prove Theorem A in the general multicritical case.
Proof of Theorem A. The fact that no critical point of f satisfies the Collet-Eckmann condition follows at once from item (1) of Proposition 4.8. By Proposition 3.1 we know that log Df ∈ L 1 (µ), and then we know from Birkhoff's ergodic theorem that: for µ-almost every x ∈ S 1 . For each n ≥ 0 let: and consider We claim that A has full µ-measure. Indeed, since: we deduce that q n µ J 2n (c j ) → 0 (exponentially fast in n, in fact) and since µ(A n ) ≥ 1 − N q n µ J 2n (c j ) we see that µ(A n ) → 1 as n → +∞. This implies the claim that µ(A) = 1. Now for each x ∈ A we have from Proposition 4.8 that there exists a sequence n k → +∞ such that: and letting k → +∞ we get that: Therefore: lim n→+∞ log Df n (x) n = 0 for µ-almost every x ∈ A, and then we are done since A has full µ-measure.
Remark 4.9. Note that for maps with two or more critical points the analogue of Question 4.4 has, in general, a negative answer. Indeed, it may well happen that one of the critical points, say c 1 , lies in the pre-orbit of one of the other critical points. In that case, the Lyapunov exponent of f (c 1 ) will be equal to −∞.

Analogous results for unimodal maps
The main result of this paper, Theorem A, has an analogue in the context of infinitely renormalizable unimodal maps. We wish to state the result (see Theorem B below) and this will involve a slight digression into the renormalization theory of unimodal maps. We refer the reader to [28, Chapter VI] for general background on the subject.
Let I 0 = [−1, 1] ⊂ R, and consider C 3 maps f : I 0 → I 0 which are unimodal with f ′ (0) = 0, f (0) = 1, i.e., with a unique critical point at 0 and critical value at 1. Without loss of generality assume that f is even, in the sense that f (−x) = f (x) for all x ∈ I 0 . We assume throughout that the critical point is non-flat , as defined in the introduction. Such an f is said to be renormalizable if there exist p = p(f ) > 1 and λ = λ(f ) = f p (0) such that f p |[−|λ|, |λ|] is unimodal and maps [−|λ|, |λ|] into itself. With p smallest possible, the first renormalization of f is the map Rf : I 0 → I 0 given by The intervals ∆ j = f j ([−|λ|, |λ|]), for 0 ≤ j ≤ p − 1, have pairwise disjoint interiors, and their relative order inside I 0 determines a unimodal permutation θ of {0, 1, . . . , p − 1}. Thus, renormalization consists of a first return map to a small neighbourhood of the critical point rescaled to unit size via a linear rescale (see Figure 3).
x Since Rf is again a normalized unimodal map, one can ask whether Rf is also renormalizable, and if the answer is yes then one can define R 2 f = R(Rf ), and so on. Thus, a unimodal map f is said to be infinitely renormalizable if the entire sequence f, Rf, R 2 f, . . . , R n f, . . . is well-defined.
We assume henceforth that f is infinitely renormalizable. Let us denote by I f ⊆ I 0 the closure of the orbit of the critical point of f . The set I f is a Cantor set with zero Lebesgue measure, and is the global attractor of f both from the topological and metric points of view: the set B(I f ) = x ∈ I 0 : ω(x) = I f is a residual set in I 0 , and it has full Lebesgue measure (see [28,Section V.1] and the references therein). Let us point out, however, that B(I f ) has empty interior.
For each n ≥ 0, we can write where q 0 = 1, λ 0 = 1, q n = n−1 i=0 p(R i f ) and λ n = n−1 i=0 λ(R i f ) = f qn (0). The positive integers a i = p(R i f ) ≥ 2 are called the renormalization periods of f , and the q n 's are the closest return times of the orbit of the critical point (in perfect analogy with the case of critical circle maps). Note that q n+1 = a n q n = i=n i=0 a i ≥ 2 n+1 , in particular the sequence q n goes to infinity at least exponentially fast.
Next, consider the renormalization intervals ∆ 0,n = [−|λ n |, |λ n |] ⊂ I 0 , and ∆ i,n = f i (∆ 0,n ) for i = 0, 1, . . . , q n − 1. The collection C n = {∆ 0,n , . . . , ∆ qn−1,n } consists of pairwise disjoint intervals. Moreover, {∆ : ∆ ∈ C n+1 } ⊆ {∆ : ∆ ∈ C n } for all n ≥ 0 and we have It is also well-known that f | I f is a homeomorphism, and that the dynamics of f restricted to I f is conjugate to that of an adding machine (see [28,Section III.4,Prop. 4.5]). More precisely, consider the following inverse system of cyclic groups (each endowed with the discrete topology): Here, the morphisms obviously correspond to multiplication by the successive periods a n . The inverse limit A of this system together with the translation τ induced by x → x + 1 on each factor (with carryover to the left) is a compact abelian group, called the adding machine with periods (a 0 , a 1 , . . .) (cf. [34, p. 212]). The system (A, τ ) is a minimal and uniquely ergodic dynamical system (and so is f | I f ). The assertion is that there exists a homeomorphism H : commutes. In particular, if ν denotes the unique invariant probability measure under τ , then the pushforward µ = H * ν is the unique invariant probability measure for f | I f . It is not difficult to check that this measure satisfies µ(∆ i,n ∩ I f ) = 1 q n = 1 a 0 a 1 · · · a n−1 for each n ≥ 0 and each 0 ≤ i ≤ q n − 1. This occurs simply because, at each level n, the intervals ∆ i,n with 0 ≤ i ≤ q n − 1 are permuted by f .
Since B(I f ) has full Lebesgue measure in I 0 and f | I f is uniquely ergodic, it is easy to check that the measure µ is the unique physical measure for f in I 0 , see [28, Section V.1, Theorem 1.6]. Now, the exact analogue of Theorem A can be stated as follows.
Theorem B. Let f : I 0 → I 0 be a C 3 infinitely renormalizable unimodal map with non-flat critical point, and let µ be the unique f -invariant Borel probability measure supported in the closure of its post-critical set. Then log |Df | belongs to L 1 (µ) and it has zero mean:

Moreover, f does not satisfy the Collet-Eckmann condition.
Note that Theorem B is stated for C 3 infinitely renormalizable unimodal maps of any combinatorial type, with non-flat critical point of any criticality and without any assumption about the Schwarzian derivative.
If f has negative Schwarzian derivative and non-degenerate critical point (that is, D 2 f (c) = 0), Theorem B goes back to Keller [22,Theorem 3,page 722]. It is also well-known that an infinitely renormalizable unimodal map does not satisfy the Collet-Eckmann condition (otherwise it would admit an absolutely continuous invariant probability measure, see [3] and the references therein, which is impossible since its non-wandering set has zero Lebesgue measure [28, Section VI.2, Theorem 2.1], see also [27]). The lack of the Collet-Eckmann condition will be re-obtained here, just as in Theorem A, by showing that there exists a subsequence of log Df n f (c) /n converging to zero.
Remark 5.1. Just as in the case of critical circle maps, Theorem B suggests the question whether lim log Df n f (c) /n = 0, see Question 4.4. Again, for this purpose it would be enough to prove that the limit exists, for instance by proving that the critical value of f is a µ-typical point for the Birkhoff's averages of log |Df |. We remark that Nowicki and Sands have proven in [29] that lim inf Combined with what we said above and will prove below, this fact implies that, if the limit exists, it must be equal to zero.

Proof of Theorem B.
Sullivan has shown in [36] that, at every level of renormalization, each renormalization interval ∆ j,n has a definite space around itself inside I 0 . We can state this particular result by Sullivan as follows (cf. [28, Section VI.2, Lemma 2.1]). Given a closed interval ∆ and δ > 0, we define the δscaled neighborhood of ∆ to be the open interval V ⊃ ∆ such that each component of V \ ∆ has length equal to δ|∆|.
This fact combined with Koebe distortion principle implies that f k−j : ∆ j,n → ∆ k,n is a diffeomorphism with bounded distortion for all 1 ≤ j ≤ k ≤ q n . This is the contents of the following lemma, whose proof can be found in [28, Section VI.2, Theorem 2.1, item 1].
for all x ∈ ∆ j,n and all n ≥ 0.
The proposition below follows from the fact that the successive renormalizations of f form a bounded sequence in the C 1 -topology, which in turn is a consequence of Lemma 5.2 and Lemma 5.3. Nevertheless, we provide a proof as a courtesy to the reader. Proof of Proposition 5.4. Since 0 ∈ I 0 is a non-flat critical point with exponent d > 1 there exist 0 < a < b such that a|x| d−1 ≤ Df (x) ≤ b|x| d−1 for all x ∈ ∆ 0,n and all n ∈ N. In particular for all x ∈ ∆ 0,n and all n ∈ N. Now, given n ∈ N, j ∈ {0, 1, ..., q n − 1} and x ∈ ∆ j,n note that f qn−j (x) ∈ ∆ 0,n and then f qn−j+1 (x) ∈ ∆ 1,n . From (5.2) we deduce that: and also that 1 From the chain rule and (5.4), (5.5) and (5.6) we get Since Df (y) ≥ a|y| d−1 for all y ∈ ∆ 0,n and since d > 1 we get: Df qn (x) ≤ b K 2 1 d 2 d a =: C 0 for all x ∈ ∆ j,n .
Proposition 5.4 gives us large iterates where |Df | is far away from infinity. With this at hand, we just need to find large iterates where |Df | is far away from zero. For this purpose we give the following definition: Definition 5.5. For each x ∈ I f and each n ≥ 0 we define the n-th entrance time of x, denoted by v n (x), to be: Note that v n (x) ∈ {0, 1, ..., q n − 1} and that v n+1 (x) ≥ v n (x) for all n ≥ 0 and all x ∈ I f . Note also that v n f (c) = q n − 1 for all n ≥ 0.
Lemma 5.6. For µ-almost every x ∈ I f we have v n (x) → +∞ as n → +∞. 1 Another way to obtain a positive lower bound for the sequence of ratios |∆ 1,n |/|∆ 0,n | d is by noting that |∆ 1,n | = φ(∆ 0,n ) d = Dφ(xn) d |∆ 0,n | d , where the C 3 local diffeomorphism φ, which fixes the origin, is given by the definition of non-flat critical point (see the introduction) and xn is some point in ∆ 0,n .
Proof of Lemma 5.6. For each n ≥ 0 we define: We also define: The set A is f -invariant, and x ∈ A if and only if v n (x) → +∞ as n → +∞. We claim that µ(A n ) = 1 − 1/a n ≥ 1/2 for all n ≥ 0.
We shall use the fact, due to Guckenheimer in the late seventies [17], that at each renormalization level the interval containing the critical point is the largest (up to multiplication by a constant). More precisely: Lemma 5.7. There exists a constant ε = ε(f ) > 0 such that |∆ 0,n | ≥ ε|∆ j,n | for all j ∈ {0, 1, ..., q n − 1} and all n ≥ 0.
The argument used by Guckenheimer relies on the Minimum Principle for maps with negative Schwarzian derivative (see [28, Section II.6, Lemma 6.1] for its statement) so at this point we need the fact that f is of class C 3 .
With Proposition 5.4, Lemma 5.6 and Lemma 5.7 at hand we are ready to prove Theorem B: Proof of Theorem B. The integrability of log |Df | was obtained by Przytycki [31, Theorem B] so let us prove that I0 log |Df | dµ is zero. For this purpose define A ⊂ I f to be the set of points x ∈ I f such that v n (x) → +∞ as n → +∞, where v n (x) is the sequence of entrance times of x ∈ I f , see Definition 5.5. We also define B ⊂ I f to be the set of points x ∈ I f such that: Now let x ∈ A. For each n ∈ N, f vn(x) maps ∆ qn−vn(x),n ∋ x into ∆ 0,n . By Lemma 5.3 there exists K 1 = K 1 (f ) > 1 such that: for all n ≥ 1.
Applying Lemma 5.7 we deduce in particular that for all x ∈ A and for all n ≥ 1: Therefore: Let us finish Section 5 with the following remark about the µ-integrability of log |Df |.
Lemma 5.8. There exists C = C(f ) > 1 such that: Proof of Lemma 5.8. We just need to imitate the procedure of Section 3, during the proof of Proposition 3.1. Indeed, let ψ = log |Df | and for each n ∈ N write ψ n = ψ · χ I0\∆0,n . Using the power-law at the critical point c = 0 we see that: for all x ∈ ∆ 0,n \{0} and n ∈ N big enough. Now observe that: Since |∆ 0,n+1 | ≤ |x| ≤ |∆ 0,n | for all x ∈ ∆ 0,n \ ∆ 0,n+1 we see from (5.11) and (5.12) that: for all j ∈ N we obtain from (5.13) and a simple telescoping trick that: and since ψ n ր ψ as n goes to infinity we are done.
Note in particular that if f is infinitely renormalizable of bounded type, then the series on the right-hand side of (5.10) converges, since by the real bounds we know that |∆ 0,n | ≍ |∆ 0,n+1 | (the so-called bounded geometry, see [28, Section VI.2, Theorem 2.1, item 2]), and the q n 's grow exponentially fast. Note also that the integrability of log |Df | proved by Przytycki in [31, Theorem B] already mentioned shows that, in the general case, the series on the left-hand side of (5.10) always converges.

Neutral measures on Julia sets
In this section we show some applications of Theorems A and B to holomorphic dynamics. Let f : C → C be a rational map of the Riemann sphere with degree greater than or equal to two, and let M f be the set of all f -invariant ergodic Borel probability measures in the Riemann sphere, whose support is contained in the Julia set of f . By [31, Theorem A] we know that log |Df | is µ-integrable for any µ ∈ M f (here Df denotes the derivative considered with respect to the spherical metric). The Lyapunov exponent of f with respect to µ is the real number χ(µ) = C log |Df | dµ. By Birkhoff's Ergodic Theorem, χ(µ) = lim n→+∞ log |Df n (z)| n for µ-almost every z ∈ C. If f is hyperbolic, i.e. if each critical point is either periodic or contained in the basin of an attracting periodic orbit, then there exists χ > 0 such that χ(µ) > χ for any µ ∈ M f . If f is not hyperbolic, then either f has a parabolic periodic orbit or one of its critical points lies in its Julia set (both phenomena can occur simultaneously). In the first case, the average of the Dirac measures supported along some parabolic periodic orbit gives an element µ ∈ M f such that χ(µ) = 0 (take for example f (z) = z 2 + 1/4 and µ = δ 1/2 ∈ M f ). Being purely atomic, these measures are not so interesting. In the second case (when the Julia set contains a critical point) one may have a non-atomic measure µ ∈ M f such that χ(µ) = 0. Following [31] we call such a measure neutral.
As an example, consider the one-parameter family f ω : C → C of Blaschke products in the Riemann sphere given by: for ω ∈ [0, 1). (6.1) Just as any Blaschke product, every map in this family commutes with the geometric involution around the unit circle Φ(z) = 1/z (note that Φ is the identity in the unit circle). In particular every map in this family leaves invariant the unit circle (Blaschke products are the rational maps leaving invariant the unit circle), and its restriction to S 1 is a real-analytic critical circle map with a unique critical point at 1, which is of cubic type, and with critical value e 2πiω (the fact that f ω has topological degree one, when restricted to the unit circle, follows from the Argument Principle, since it has two zeros and one pole in the unit disk). By monotonicity of the rotation number (see for instance [19] and [28]) we know that for each irrational number θ in [0, 1) there exists a unique ω in [0, 1) such that the rotation number of f ω | S 1 is θ. As an application of Theorem A we have the following result.
Theorem C. Let ω in [0, 1) such that the rotation number of f ω | S 1 is irrational. Then the rational function f ω admits a non-atomic invariant ergodic Borel probability measure µ whose support is contained in the Julia set J ω of f ω and such that χ(µ) = 0. Moreover, f ω does not satisfy the Collet-Eckmann condition.
Proof of Theorem C. The measure µ is precisely the unique invariant measure supported on the unit circle. By Theorem A, we only need to prove that the unit circle is contained in the Julia set. Since S 1 is f ω -invariant and f ω | S 1 is minimal (by Yoccoz's result [42]), either we have S 1 ⊂ J ω , either S 1 ∩ J ω = Ø. Suppose, by contradiction, that S 1 ∩ J ω = Ø, and let U be the Fatou component of f ω containing S 1 . By the invariance of the unit circle, U is mapped into itself by f ω , and therefore it must be a Siegel disk or a Herman ring (precisely because it has an invariant simple closed curve on its interior). But in that case f ω : U → U would be a biholomorphism, which is impossible since it has a critical point in the unit circle. Therefore S 1 ⊂ J ω , and Theorem C follows directly from Theorem A.
By the main result of [32], and since there is only one critical point in the Julia set J ω , we deduce that f ω does not satisfy any of the standard definitions of non-uniform hyperbolicity for rational maps in the Riemann sphere (topological Collet-Eckmann condition, uniform hyperbolicity of periodic orbits in the Julia set, etc.).
We remark that Theorem B in §5 yields analogous examples to those given in Theorem C above, in the context of polynomials. For instance, one can take f (z) = z 2 + c ∞ , where c ∞ ∈ [−2, 1/4] denotes the infinitely renormalizable parameter for period doubling (the so-called Feigenbaum parameter). In that case, the neutral measure is the unique invariant measure supported in the closure of the post-critical set of f , which is non-atomic as explained in Section 5.

6.1.
Other examples. Let θ ∈ [0, 1] be an irrational number, and consider the quadratic polynomial P θ : C → C given by: The origin is a fixed point of P θ , and DP θ (0) = e 2πiθ has modulus one. Let us assume that θ is of bounded type, that is, there exists ε > 0 such that: for any integers p and q = 0. By a classical result of Siegel P θ is linearizable around the origin: there exists a simply-connected component Ω θ of the Fatou set of P θ , a Siegel disk, that contains the origin and where P θ is conformally conjugate to its linear part z → e 2πiθ z, an irrational rotation acting on the unit disk. A famous theorem of Douady [7] (see also the recent paper [43]) asserts that ∂ Ω θ is a quasi-circle that contains the (unique) critical point of P θ . Moreover, there exist ω = ω(θ) ∈ (0, 1) and a quasiconformal homeomorphism φ θ : C → C with φ θ (D) = Ω θ and such that φ θ •f ω = P θ •φ θ on C\D, where f ω is the Blaschke product given by (6.1). In particular φ θ : S 1 → ∂ Ω θ is a quasisymmetric homeomorphism which conjugates f ω with P θ . Combining this with Theorem C we get the following result.
Corollary 6.1. Let θ ∈ [0, 1] be an irrational number of bounded type, and let P θ be the quadratic polynomial given by (6.2). Then the harmonic measure on the boundary of the Siegel disk Ω θ , viewed from the origin, is a neutral measure for P θ .
Corollary 6.1 follows from Theorem C and the general fact that a quasiconformal conjugacy between rational maps preserves zero Lyapunov exponents (one way to see this is to combine Koebe's distortion lemma with the fact that a quasiconformal homeomorphism is bi-Hölder and the argument in [33,Lemma 8.3]).
More generally, the harmonic measure at the boundary of any Siegel disk or at the boundary of any Herman ring is a neutral measure, as stated in the introduction of [31]. In particular, by taking some polynomial or rational function f with a Siegel disk whose boundary does not contain critical points (for such an example see [2] and the references therein) we obtain µ ∈ M f such that χ(µ) = 0, µ has no atoms and supp(µ) contains no critical point of f . Note, however, that for any rational function f and any neutral measure µ we have that supp(µ) must be contained in the closure of the forward orbit of some critical point of f , otherwise f would be expanding on supp(µ) and then µ would not be neutral.
Finally, let us mention that Cortez and Rivera-Letelier have constructed in [5] and [6] real quadratic polynomials for which the ω-limit set of the critical point is a minimal Cantor set (necessarily contained in the Julia set) supporting any prescribed number of neutral measures (finite, countable infinite or uncountable). This is in sharp contrast with the examples discussed above, where f | supp(µ) is uniquely ergodic.

Further questions
We conclude with some questions (in addition to Question 4.4 posed at the end of Section 4).
Let f be a rational map of the Riemann sphere with degree greater than or equal to two, and let µ be a neutral measure for f , that is, a non-atomic f -invariant ergodic Borel probability measure, whose support is contained in the Julia set of f , and with Lyapunov exponent equal to zero (just as in the situation of Theorem C).
Question 7.1. Is it true that h top (f | supp(µ) ) = 0? Is f | supp(µ) a minimal dynamical system? Is it true, at least, that supp(µ) has no periodic orbits? Question 7.2. What are all the examples of rational maps having some non-atomic invariant ergodic Borel probability measure, supported inside its Julia set, and with zero Lyapunov exponent?
A difficult problem in the context of critical circle maps with finitely many critical points is the one of geometric rigidity. More precisely, let f and g be two orientation preserving C 3 circle homeomorphisms with the same irrational rotation number, and with N ≥ 1 non-flat critical points of odd type. Denote by S f = {c 1 , ..., c N } the ordered critical set of f , by S g = {c ′ 1 , ..., c ′ N } the ordered critical set of g, and suppose that the criticalities of c i and c ′ i are the same for all i ∈ {1, ..., N } (the cubic case is the generic one). Finally, denote by µ f and µ g the corresponding unique invariant measures of f and g.
By Yoccoz's result [42] we know that f and g are topologically conjugate to each other. By elementary reasons, the condition µ f [c i , c i+1 ] = µ g [c ′ i , c ′ i+1 ] for all i ∈ {1, ..., N − 1} is necessary (and sufficient) in order to have a topological conjugacy between f and g that sends the critical points of f to the critical points of g. Under this assumption, it turns out that this conjugacy is in fact a quasisymmetric homeomorphism. This follows from a recent general result of Clark and van Strien [4]. In this context it also follows from the real bounds (see [11,Corollary 4.6] for the case of a single critical point, and the forthcoming article [8] for the multicritical case). Question 7.3. Is this conjugacy a smooth diffeomorphism?
In the case of exactly one critical point this question has been answered in the affirmative, in the real-analytic category. This is due to the efforts of several mathematicians during the last twenty years (see [1], [9], [10], [11], [12], [23], [25], [38], [39], [40] and [41]). A positive answer was recently obtained also in the C 3 category (see [14], [15] and [16]). To the best of our knowledge, the case of more than one critical point remains completely open.