Lyapunov spectrum

Lyapunov spectrum
for exceptional rational maps

K. Gelfert Instituto de Matemática, UFRJ, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil gelfert@im.ufrj.br F. Przytycki Instytut Matematyczny Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-956 Warszawa, Poland feliksp@impan.gov.pl M. Rams Instytut Matematyczny Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-956 Warszawa, Poland rams@impan.gov.pl  and  J. Rivera-Letelier Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile riveraletelier@mat.puc.cl
Abstract.

We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain -finite conformal measures. This allows to extend previous results to exceptional rational maps.

Key words and phrases:
2000 Mathematics Subject Classification:
Primary: 37D25, 37C45, 28D99, 37F10

1. Introduction and main results

We are going to study the Lyapunov exponents of a rational function acting on the Riemann sphere, of degree at least . In particular, continuing the investigations in [5], we are interested in the case that the map  is exceptional. Slightly modifying [8, Section 1.3], we call  exceptional if there exists a finite, nonempty, and forward invariant set such that

(1)

Here is the Julia set of  and is the set of critical points of . Every such set  has at most  points (see Lemma 1), hence there is a maximal set with this property, which we denote by . If  is non-exceptional we put . When  is clear from the context we denote  simply by .

1.1. Main results

Given , denote by and the lower and upper Lyapunov exponent at , respectively. If both values coincide then we call the common value the Lyapunov exponent at and denote it by . Similarly, for a -invariant probability measure  we denote by  its Lyapunov exponent. Let be the set of all -invariant Borel probability measures supported on and be the one of all non-atomic ones. Let and be the sets of ergodic measures contained in and , respectively. Let

(see Corollary 1 for equivalent definitions of ).

For given numbers we consider the level sets

We denote by the set of Lyapunov regular points with exponent . We will describe the complexity of such level sets in terms of their Hausdorff dimension . To do so, given a parameter let us consider the potential and the pressure function

(2)

Notice that if is nonempty the potential is unbounded and  does not coincide with the the classical topological pressure for (see [17]). We define the hidden variational pressure

(3)

(following the terminology in [15]). After Makarov and Smirnov [8, Theorem B], the pressure function fails to be real analytic on the interval if and only if is exceptional and

Moreover, by [8, Theorem A] the function is real analytic on the interval and

(4)

For any let

(5)

Our main result is the following theorem.

Theorem 1.

Let be a rational function of degree at least . For any numbers , with we have

In particular, for any we have

For we have

Moreover,

and

The result of the above theorem has been shown in [5] in the particular case that is non-exceptional.

To prove our main result, in this paper we will create new technical tools in order to deal with exceptional rational maps and then show how these tools can be applied to adapt the original proofs in [5]. The paper is organized as follows. In Section 2 we collect some known results about exceptional maps that will be used in the rest of the paper. In Section 3 we will introduce the concept of hidden pressure using backward branches of , analogously to the tree pressure from [15]. In the case of exceptional rational maps we not always have at hand a finite conformal measure with dense support, see Proposition 1. For that reason, in Section 4 we introduce -finite conformal measures that are associated to the hidden pressure. Finally, in Section 5 we apply these tools to prove Theorem 1. In Section 5.1 we provide a lower bound for dimension using the fact that for any rational map we can find an increasing family of uniformly expanding Cantor repellers contained in using a construction of bridges that has been established in [5] and applies to the setting of this paper without changes. In Section 5.2 we provide an upper bound for dimension applying Frostman’s Lemma to an appropriate -conformal measure at a conical point. Finally, in Section 5.3, we show the existence of periodic orbits in with exponent as large as possible.

We give an alternative proof of this result in Appendix A via a variant of Bowen’s periodic specification property, [1].

2. Exceptional maps and phase transitions

For a critical point  we will denote by  the local degree of  at . The following result has been proved by the same computation first in [4, Lemma 2].

Lemma 1.

If  is a finite subset of  such that , then . If  is a polynomial then .

Proof.

Using that  has critical points counted with multiplicity, by (1) we have

so . If is a polynomial, then it has at most  finite critical points counted with multiplicity, so in this case . ∎

The following is an example of a one parameter family of exceptional rational maps such that for some parameters the exceptional set contains a critical point: for  put

The point  is critical of multiplicity , the point is fixed of multiplier , and the point is critical of maximal multiplicity and the only preimage of . Thus, when  belongs to the Julia set we have . By choosing suitable , the fixed point  could be repelling, Cremer, etc.

If  is exceptional, then the set contains at least one periodic point. Observe that it hence must consist of a finite number of periodic points plus possibly some of their preimages. We write , where denotes the subset of all neutral periodic points in plus its pre-images and where denotes the subset of all expanding periodic points in plus its pre-images. We refer to [7] for further details on exceptional maps and numerous examples.

We will say that  has a phase transition in the negative spectrum if the function  fails to be real analytic on . In this case we put

We have  and, since the function is convex, for each  we have .

In the following proposition we gather several results in [8, 15]. A measurable subset  of  is said to be special if is injective. Given a function , a Borel probability measure on is said to be -conformal outside if for every special set we have

If we simply say that  is -conformal.

Proposition 1.

Let  be a rational map of degree at least  and let . Then we have the following properties.

  1. Suppose that  does not have a phase transition in the negative spectrum, or that  has a phase transition in the negative spectrum and . Then  and there is a finite -conformal measure whose support is equal to .

  2. Suppose that  has a phase transition in the negative spectrum and that . Then  is exceptional, there is an expanding periodic point such that and for every neighborhood  of  and every measure  that is -conformal measure outside  we have

Proof.

The equality  in part  follows from the definition of . The existence of the conformal measure in part  follows from [8, Lemma ] if  and from [15, Theorem A] if .

The fact that  is exceptional and that there is an expanding periodic point  such that  in part  is given by [8, Theorem B]. To complete the proof of part , let  be a -conformal measure . Since  is topologically exact on , it follows that the support of  is equal to . Let  be the period of  and let  be sufficiently small so that  and so that the inverse branch  of  fixing  is defined on  and satisfies . Then there is a distortion constant  such that, if we put , then for each integer  we have by the conformality of 

Thus

proving the proposition. ∎

3. Hidden tree pressure

The goal of this section is to prove equivalence of three pressure functions: the hidden variational pressure defined in (3) as well as the hidden hyperbolic pressure and the hidden tree pressure defined in (6) and (8) below.

Given , the hidden hyperbolic pressure is defined as

(6)

where the supremum is taken over all compact -invariant (i.e. ) isolated expanding subsets of . We call such a set uniformly expanding repeller. Here isolated means that there exists a neighborhood of such that for all implies .

Proposition 2.

for every .

Proof.

The inequality follows from the variational principle. On the other hand [16, Theorem 11.6.1] implies that for any we have and hence . ∎

Before defining the hidden tree pressure, let us recall some concepts from [13][15], and [16, Chapter 12.5]. Given and , we consider the tree pressure of at defined by

A point is said to be safe if

where denotes the spherical distance. A point is said to be expanding if there exist numbers and such that for all sufficiently large the map is univalent on and satisfies . Here, for a subset  of  and we denote by the connected component of  containing .

We point out that every point in  outside a set of Hausdorff dimension zero is safe and that for each safe point  we have , see [13, 15] and compare with [13, Theorem 3.4].

Lemma 2.

There exists an expanding safe point in .

Proof.

Notice that

Since is finite and for any and , this inclusion implies that the set of points that fail to be safe has zero Hausdorff dimension. Thus, the existence of an expanding safe point outside follows from the existence of uniformly expanding Cantor repellers outside , for example as derived in [5, Lemma 4]. Note that such repellers always have a positive Hausdorff dimension by Bowen’s formula (see, for example, [16, Chapter 9.1]). ∎

Let us now define the hidden tree pressure that is an analogue of the tree pressure, obtained by considering a restricted tree of preimages. Given a subset  of  and  which is not in the forward orbit of a critical point, we define

(7)

and we consider the hidden tree pressure of at defined by

(8)

Usually the point  will be expanding safe and the set  will be a neighborhood of . Note that after Lemma 2 there are such  and .

Lemma 3.

If , is a sufficiently small neighborhood of , and is expanding safe, then the pressure does not depend on .

To prove the above lemma we need the following technical lemma.

Lemma 4.

For an arbitrary neighborhood  of  and an arbitrary number there exists a number and positive integers such that for every point there exist numbers , and a point such that the set is -dense in and satisfies

Proof.

By the locally eventually onto property of  on  there is an integer  such that for each  the set  is -dense in . We put

For each integer let be defined by

We will show that for each  there is an integer such . Since for each the function  is continuous and for each  the sequence is non-decreasing, it follows that there is a number so that is strictly positive on . This will imply the desired assertion with

We distinguish three cases:
1) If is not in the forward orbit of a critical point then .
2) If is in the forward orbit of a citical point that is not pre-periodic then there exists a number such that is disjoint from . Hence, we obtain that .
3) If is in the forward orbit of a pre-periodic critical point then, there is  and an infinite backward trajectory starting at  that is disjoint from and in particular this backward trajectory is longer than . Hence, we can choose numbers , and a point such that is not in the forward orbit of a critical point and such that for each we have . In particular, we have . Thus, if we put

then . ∎

Proof of Lemma 3.

Let , be two neighborhoods of . Without loss of generality we can assume that . By Lemma 4, every backward branch of starting at and ending at some point can be modified to end at some . The modification involves only removing at most last steps, that decreases at most by a constant factor because , and replacing them by at most steps, which stay in a uniformly bounded from below distance from critical points. Hence we conclude that and differ at most by . This proves the lemma. ∎

We denote by the maximal distortion of a map on a set . We establish one preliminary approximation result.

Proposition 3.

Given , a sufficiently small neighborhood of and an expanding safe point , for every there exists a uniformly expanding repeller such that

Proof.

We start by recalling the construction used in [15, Proposition 2.1] to prove an analogous statement for  and then we modify it using Lemma 4 to prove the proposition.

As  is expanding safe, there exist , and so that for all  the map is univalent on and . Hence, in particular, the distortion is bounded from above uniformly in  by some number . Given , let be the smallest integer satisfying . Hence, with the above, we have and , where and . Let be such that for any .

Let us choose positive constants , and large enough so that and that for every , , for every point on the component the map is univalent and satisfies

(9)

Note that with this choice we have for large ,

As and covers , we can conclude that for every preimage there exists a component  of contained in . The map , and hence , is univalent on . Thus, the map

(10)
{overpic} [scale=.4]P_proof.eps
Figure 1. Construction of the uniformly expanding repeller

has no critical points, and is a uniformly expanding repeller with respect to .

Let us now slightly modify the construction of by (10) and ignore all those backward branches that correspond to a point . Given , let us consider the positive integers and the number provided by Lemma 4. Then, by Lemma 4, for each point there exist numbers , , a point in . Any such branch stays -far from . Note that the distortion of  on

is bounded by a constant  independent of  and . Given an integer , put

Note that for distinct  and  in  the sets  and  are disjoint. Setting

the sets

are uniformly expanding repellers for  and , respectively. Both of these sets are disjoint from  by construction. On the other hand there, letting

we have

Since , there is  such that

Hence, if we put , then

Since , , are independent of  and , we obtain the desired assertion by taking a sufficiently large . ∎

We are now ready to prove one further equivalence.

Proposition 4.

Given a sufficiently small neighborhood  of and an expanding safe point , for every we have

Proof.

By Proposition 3, we have .

In view of Lemma 3, to prove the inequality it is enough to show that for each expanding repeller  that does not intersect , there is a neighborhood  of  such that . Notice that for every and every neighborhood  of  disjoint from  we have,

This follows easily considering the contribution of the backward branches of  contained in  in the sum in (7).

Let  be a neighborhood of  on which  is uniformly expanding. Thus, there is a constant  and for every  there is  such that for every integer  and every  there is  shadowing  and so that

It follows that for each neighborhood  disjoint from  we have .

By the eventually onto property of  on , we have for some . Fix and let  neighborhood of  disjoint from . Then we have

This shows and completes the proof of the inequality . ∎

4. -finite conformal measures

Recall that is a rational map  of degree at least . If  is exceptional, then  is the maximal finite and forward invariant subset of  satisfying . Otherwise .

In the following proposition we adapt the classical method by Patterson and Sullivan to construct a -conformal measure on  for each . For a map without a phase transition in the negative spectrum or for a map with a phase transition in the negative spectrum at some parameter , we obtain a finite conformal measure supported on , as in part  of Proposition 1. For a map having a phase transition in the negative spectrum at some parameter this construction gives us a conformal measure outside , which is finite outside each neighborhood of . Recall that by part  of Proposition 1, existence of phase transition implies that there does not exist a finite -conformal measure for .

Proposition 5.

Let  be a rational function of degree at least . For each there exists a Borel measure on  that is -conformal outside , finite outside any neighborhood of , gives zero measure to and whose support is equal to .

Proof.

As a first step we will apply the Patterson-Sullivan method while considering only those inverse branches outside a given neighborhood  of . We obtain in this way a measure that is -conformal outside the set . We will obtain a measure -conformal outside  by taking the limit of the measures obtained by repeating this construction with  replaced by smaller and smaller neighborhoods.

We start with the following lemma. Recall that denotes the set of neutral periodic points in  plus its preimages.

Lemma 5.

Given , for every  there exist positive numbers and such that for every and every integer we have

where the sum is taken over all satisfying for every .

Proof.

Let  be sufficiently small so that for each periodic point  of minimal period  we have

Hence, there is some constant  such for every integer  and every point  satisfying for every  we have

Reducing  if necessary, we may assume that for every  the map  is injective on  and the set  is disjoint from . So for each  and there is at most one point  such that . By induction we can conclude that for each , , and , there is at most one point