Real bounds and Lyapunov exponents
Abstract.
We prove that a 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 satisfies the ColletEckmann condition. This result is proved directly from the wellknown real apriori 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.
Key words and phrases:
Lyapunov exponents, real bounds, critical circle maps, infinitely renormalizable unimodal maps, neutral measures on Julia sets2010 Mathematics Subject Classification:
Primary 37E10; Secondary 37A05, 37E20, 37F10.1. Introduction
This paper studies critical circle maps (as well as infinitely renormalizable unimodal maps) from the differentiable ergodic theory viewpoint. The ergodic aspects of onedimensional 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 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: nonzero 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 apriori bounds (see Theorem 2.2), without using Pesin’s theory. In fact, the only nontrivial result from ergodic theory we shall use here is Birkhoff’s Ergodic Theorem.
By a critical circle map we mean an orientation preserving circle homeomorphism with finitely many nonflat critical points (). A critical point is called nonflat if in a neighbourhood of the map can be written as , where is a local diffeomorphism with , and where 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 computerassisted 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 from the differentiable ergodic theory viewpoint. We will focus on the case when the rotation number of is irrational, in which case is uniquely ergodic. Moreover, by a theorem of Yoccoz [42], 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 be a critical circle map with irrational rotation number, and let be its unique invariant Borel probability measure. Then belongs to and it has zero mean:
Moreover, no critical point of satisfies the ColletEckmann condition.
Recall that satisfies the ColletEckmann condition at a critical point if there exist and such that for all (see for instance [28, Chapter V]), or equivalently
(1.1) 
The integrability of was obtained by Przytycki in [31, Theorem B], where he also proved that (see [35, Appendix A] for an easier proof). We will obtain the integrability again (see Proposition 3.1) on the way to proving that . It is expected that 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 is an orientationpreserving circle diffeomorphism, with irrational rotation number, the function defined by is a continuous function and therefore, by the unique ergodicity of , the sequence of continuous functions:
converges uniformly to a constant, and this constant must be . By the chain rule and, therefore, the sequence of continuous functions converges to the constant uniformly in . Since is a diffeomorphism for all , this constant must be zero. In our case, however, is not a continuous function (it is defined only in , and it is unbounded, see Figure 1).
Remark 1.2.
1.1. How the paper is organized
In §2 we briefly recall some classical combinatorial facts about circle maps, and also the wellknown apriori 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 with respect to the unique 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 nonflat 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.
2. The real bounds
Let be a critical circle map as defined in the introduction, that is, is an orientation preserving circle homeomorphism with finitely many nonflat critical points of odd type. As we have pointed out already, our standing assumption is that the rotation number is irrational. Therefore it has an infinite continuedfraction expansion, say
We define recursively a sequence of return times of by:
and for .
In particular the sequence grows at least exponentially fast when 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 are also obtained as the denominators of the truncated expansion of order of :
We recall also the following wellknown estimates.
Theorem 2.1.
For all we have:
2.1. Dynamical partitions
Denote by the interval , where denotes a critical point of , and define as:
A crucial combinatorial fact is that is a partition (modulo boundary points) of the circle for every . We call it the nth dynamical partition of associated with the point . Note that the partition is determined by the finite piece of orbit
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 with the following property. Given a critical circle map with irrational rotation number there exists such that, for each critical point of , for all , and for every pair of adjacent atoms of we have:
(2.1) 
where denotes the Euclidean length of an interval in the real line.
Of course for a particular we can choose such that (2.1) holds for all . Theorem 2.2 was proved by Świa̧tek and Herman (see [20] and [37]) in the case when has a single critical point. The original proof is based on the socalled crossratio inequality of Świa̧tek. As it turns out, this inequality is valid also in the case when the map has several critical points (all of nonflat 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 . If is big, then is much larger than . 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 critical circle map with irrational rotation number and a unique critical point , there exists a constant such that the following facts hold true for each :

For all , we have

For all , 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 has two or more critical points, the estimates given in the Corollary may fail, because the intervals and could in principle contain other critical points of and , respectively.
Remark 2.4.
We shall henceforth use the constant 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 be as in Corollary 2.3. There exists such that for all and for all :
Proof of Lemma 2.5.
For each , let us write instead of in this proof. Fix and . By Corollary 2.3 the map has bounded distortion. In particular, there exists such that:
Since we obtain from the real bounds that . Therefore:
Since is a nonflat critical point of of odd type there exist such that for all , and then:
Again using that is nonflat there exist such that for all in a small but fixed neighbourhood around the critical point. In particular,
for all and for all , since , again by the real bounds. With this at hand we deduce that
for all and for all . 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 let:
where denotes the Euclidean distance between an interval and the critical point .
Lemma 2.6.
For each critical point of , the sequence is bounded.
Proof of Lemma 2.6.
Given a critical point , let us write in this proof, for simplicity of notation, , instead of , respectively, for each . Note that the transition from to can be described in the following way: the interval is subdivided by the points with into subintervals. This subpartition is spread by the iterates of to all with . The other elements of the partition , namely the intervals with , remain unchanged. On one hand, for any we have:
On the other hand:
This gives us:
By the real bounds for all and we are done. ∎
3. The integrability of
As before let be a critical circle map with finitely many nonflat critical points and with rotation number . Since we assume that is irrational, admits a unique invariant Borel probability measure . Moreover, by a theorem of Yoccoz [42], has no wandering intervals and therefore there exists a circle homeomorphism which is a topological conjugacy between and the rigid rotation by angle , that we denote by . More precisely, the following diagram commutes:
where denotes the normalized Lebesgue measure in the unit circle (the Haar measure for the multiplicative group of complex numbers of modulus ). Therefore is just the pushforward of Lebesgue measure under , that is, for any Borel set in the unit circle (recall that the conjugacy is unique up to postcomposition with rotations, so the measure is welldefined).
In this section we prove that belongs to . As before, let us denote by the critical points of .
Let be given by . For each and each , let . We define and consider given by:
that is, on each and on the complement of their union. We will use the following four facts:

From the real bounds (Theorem 2.2) there exists such that for all and each .

As explained above, the measure is the pullback of the Lebesgue measure under any topological conjugacy between and the corresponding rigid rotation. In particular, for each and for all , we have and by Theorem 2.1:

By combinatorics, we have , for all and for each .

Since each is a nonflat critical point, there exist and a neighbourhood of such that for all we have:
(3.1) We may assume, of course, that the ’s are pairwise disjoint.
With all these facts at hand we are ready to prove the desired integrability result.
Proposition 3.1.
The function is integrable, i.e., .
Proof of Proposition 3.1.
Note that the sequence converges monotonically to . Let be the smallest positive integer such that for all . We only look at values of greater than . Then, since is identically zero on and agrees with everywhere else, we can write
(3.2) 
The first integral on the righthand side is a fixed number independent of . Hence it suffices to bound the last double sum. Using (3.1) and the fact that in the closest point to is , we see that (see Figure 2)
(3.3) 
Applying facts 1, 2 and 3 to this last sum, we see that
(3.4) 
However we know from Theorem 2.1 that
(3.5) 
Putting (3.5) into (3.4) we get
(3.6) 
Since the ’s grow exponentially fast (at least as fast as the Fibonacci numbers), we have
Hence the lefthand 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 such that
But then, by the Monotone Convergence Theorem, is integrable, as desired. ∎
Remark 3.2.
The proof of Proposition 3.1 yields, mutatis mutandis, a slightly stronger result, namely that for every finite . However, this more general fact will not be needed in this paper.
4. Proof of Theorem A
In this section we present two different proofs of Theorem A. The first proof works only when the map 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 has a single critical point . In particular, we are free to use Lemma 2.5. Once again we write , instead of , etc., for simplicity of notation.
Consider the Borel set defined in the following way: iff there exists an increasing sequence such that for each there exists (a necessarily unique) such that .
Lemma 4.1.
The set is invariant and .
Proof of Lemma 4.1.
The first assertion follows immediately from the definition of , hence we focus on proving that has full measure. For each consider the disjoint union:
We claim that for all . Indeed, , since . As explained at the beginning of Section 3, the measure is the pullback of the Lebesgue measure under any topological conjugacy between and the corresponding rigid rotation. In particular:
where is the rotation number of . By Theorem 2.1:
and then:
Since we deduce that
Since for all we obtain the claim, that is, for all . Moreover, since:
we have . The ergodicity of under now implies that , since is invariant. ∎
Now we consider the Borel set given by , where denotes the preorbit of the critical point, that is:
Proposition 4.2.
The set has full measure, i.e., ; moreover, the critical value belongs to .
Proof of Proposition 4.2.
Since has no atoms, (recall that is invariant and has no periodic orbits). In particular . The critical point of belongs to by definition, and by invariance, so does its critical value. Since there are no periodic orbits, and then . ∎
The relation between and the integrability of is given by the following:
Proposition 4.3.
Let and let be its corresponding increasing sequence of natural numbers. Then:
Proof of Proposition 4.3.
Recall that, since is a nonflat critical point of , there exists such that for any we have:
Let and let be its corresponding increasing sequence of natural numbers. Recall that for each there exists (a necessarily unique) such that . Then we have:
where the second inequality is given by Lemma 2.5. By combinatorics we have the following facts:

The points and do not belong to for any .

For each there exist and , adjacent elements of the partition , such that the points and belong to (the possibility that is not excluded; if they are different, we may suppose that ). By the real bounds: .

If in then .
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 belongs to . Hence by Birkhoff’s Ergodic Theorem we have
for almost every . Combining this fact with Proposition 4.2 and Proposition 4.3 we obtain:
Finally we have to prove that does not satisfy the ColletEckmann condition. Indeed, if there were constants and such that for all we would have:
but this is impossible, since by Proposition 4.2 we know that belongs to , and this implies by Proposition 4.3 that there exists a subsequence of converging to zero. ∎
Question 4.4.
Theorem A suggests the question whether . For this purpose it would be enough to prove that the limit exists (since belongs to ), for instance by proving that the critical value of is a typical point for the Birkhoff’s averages of . Note, however, that this fact does not follow directly from the unique ergodicity of since is not a continuous function (it is defined only in , 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 be the sequence of return times given by the irrational rotation number of (see Section 2). Let us denote by the critical points of () and let denote the criticality of each . Conjugating by a suitable diffeomorphism (which does not affect its Lyapunov exponent) we may assume that each has an open neighbourhood where is a powerlaw of the form:
(4.1) 
We also assume, of course, that whenever .
Recall from the real bounds (Theorem 2.2) that, for each , the dynamical partitions have the comparability property: any two consecutive atoms of have comparable lengths. We will also need the following three further consequences of the real bounds.
Lemma 4.5.
There exists such that for each , for each and for each atom we have:
∎
Lemma 4.6.
There exists with the following property: let and denote by the union of with its two immediate neighbours in . If are such that the intervals , ,…, do not contain any critical point of , then the map has distortion bounded by , that is:
(4.2) 
Proof of Lemma 4.6.
The real bounds imply that has space inside . Moreover, the map