Lyapunov spectrum for rational maps
Abstract.
We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere.
Key words and phrases:
Lyapunov exponents, multifractal spectra, iteration of rational map, Hausdorff dimension, nonuniformly hyperbolic systems2000 Mathematics Subject Classification:
Primary: 37D25, 37C45, 28D99 37F10Contents
1. Introduction
Let be a rational function of degree on the Riemann sphere and let be its Julia set. Our goal is to study the spectrum of Lyapunov exponents of . Given we denote by and the lower and upper Lyapunov exponent at , respectively, where
If both values coincide then we call the common value the Lyapunov exponent at and denote it by . For given numbers we consider also the following level sets
We denote by the set of Lyapunov regular points with exponent . If then is contained in the set of socalled irregular points
Recall that it follows from the Birkhoff ergodic theorem that for any invariant probability measure .
While the first results on the multifractal formalism go already back to Besicovitch [4], its systematic study has been initiated by work of Collet, Lebowitz and Porzio [5]. The case of spectra of Lyapunov exponents for conformal uniformly expanding repellers has been covered for the first time in [2] building also on work by Weiss [26] (see [13] for more details and references). To our best knowledge, the first results on irregular parts of a spectrum were obtained in [4]. Its first complete description (for digit expansions) was given by Barreira, Saussol, and Schmeling [3].
In this work we will formulate our results on the spectrum of Lyapunov exponents in terms of the topological pressure . For any let us first define
and
Let be the interval on which (a more formal and equivalent definition is given in (4) below).
Before stating our main results, we recall what is already known in the case that is a uniformly expanding repeller with respect to , that is, is a compact forward invariant (i.e., ) isolated set such that is uniformly expanding. Recall that is said to be uniformly expanding or uniformly hyperbolic on a set if there exist and such that for every and every we have . Recall that a set is said to be isolated if there exists an open neighborhood of such that for every implies . In our setting the Julia set is a uniformly expanding repeller if it does not contain any critical point nor parabolic point. Here a point is said to be critical if and to be parabolic if is periodic and its multiplier is a root of . If is a uniformly expanding repeller then for every we have
(see [2, 26]) and if and only if [23]. This gives the full description of the regular part of the Lyapunov spectrum. Moreover, in this setting of a uniformly expanding repeller , the interval coincides with the closure of the range of the function and the spectrum can be written as
where is the unique number satisfying
and where is the unique equilibrium state corresponding to the potential . If is not cohomologous to a constant then we have , and and are real analytic strictly convex functions that form a Legendre pair.
We now state our first main result.
Theorem 1.
Let be a rational function of degree with no critical points in its Julia set . For any , , we have
In particular, for any we have
If there exists a parabolic point in (and hence ) then
Moreover,
We denote by the set of all critical points of . Following Makarov and Smirnov [10, Section 1.3], we will say that is exceptional if there exists a finite, nonempty set such that
This set need not be unique. We will further denote by the largest of such sets (notice that it has no more than 4 points).
Theorem 2.
Let be a rational function of degree . Assume that is nonexceptional or that is exceptional but . For any we have
In particular, for any we have
and
Moreover,
and
If is exceptional and (this happens, for example, for Chebyshev polynomials and some Lattès maps) then the situation can be much different from the abovementioned cases. For example, the map possesses countably many points with Lyapunov exponent , two points with Lyapunov exponent , a set of dimension 1 of points with Lyapunov exponent , and no other Lyapunov regular points. Hence, for this map the Lyapunov spectrum is not complete in the interval .
The present paper does not provide a complete description of the irregular part of the Lyapunov spectrum even in the case . We do not know how big the set is except in the case when has only one critical point in (in which case consists only of the backward orbit of this critical point). Moreover, we do not know whether the set contains any points other than the backward orbits of critical points contained in and we only have some estimation for the Hausdorff dimension of the set even for values , .
The paper is organized as follows. In Section 2 we introduce the tools we are going to use in this paper. In particular, we construct a family of uniformly expanding Cantor repellers with pressures pointwise converging to the pressure on (Proposition 1). Section 3 discusses general properties of the spectrum of exponents. In Section 4 we obtain upper bounds for the Hausdorff dimension. Here we use conformal measures to deal with conical points (Proposition 2) and we prove that the set of nonconical points with positive upper Lyapunov exponent is very small using the pullback construction (Proposition 3). In Section 5 we derive lower bounds for the dimension. To do so, we first consider the interior of the spectrum and we will use the sequence of Cantor repellers from Section 2 to obtain for any a big uniformly expanding subset of points with Lyapunov exponent from which we derive our estimates. We finally study the boundary of the spectrum and the irregular part of the spectrum using a construction, that generalizes the wmeasure construction from [8].
2. Tools for nonuniformly hyperbolic dynamical systems
2.1. Topological pressure
Given a compact invariant set , we denote by the family of invariant Borel probability measures supported on . We denote by the subset of ergodic measures. Given , we denote by
the Lyapunov exponent of . Notice that we have for any [17].
Given , we define the function by
(1) 
Given a compact invariant uniformly expanding set , the topological pressure of (with respect to ) is defined by
(2) 
where denotes the entropy of with respect to . We simply write and if we consider the full Julia set and if there is no confusion about the system. A measure is called equilibrium state for the potential (with respect to ) if
For every we have the following equivalent characterizations of the pressure function (see [21], where further equivalences are shown). We have
(3) 
Here in the first equality the supremum is taken over the set of all ergodic invariant Borel probability measures on that have a positive Lyapunov exponent and are supported on some invariant uniformly expanding subset of . In the second equality the supremum is taken over all uniformly expanding repellers . In fact, in the second equality it suffices to take the supremum over all uniformly expanding Cantor repellers, that is, uniformly expanding repellers that are limit sets of finite graph directed systems satisfying the strong separation condition with respect to , see Section 2.2.
Let us introduce some further notation. Let
(4) 
where the given characterizations follow easily from the variational principle.
Recall that, given , we define
(5) 
and
(6) 
Note that
2.2. Building bridges between unstable islands
We describe a construction of connecting two given hyperbolic subsets of the Julia set by “building bridges” between the sets.^{1}^{1}1This is a precise realization of an idea of Prado [15].
We call a point nonimmediately postcritical if there exists some preimage branch , , that is dense in and disjoint from . If is nonexceptional or if it is exceptional but then for every hyperbolic set all except possibly finitely many points (in particular, all periodic points) are nonimmediately postcritical.
We will now consider a set that is an uniformly expanding Cantor repeller (ECR for short), that is, a uniformly expanding repeller and a limit set of a finite graph directed system (GDS) satisfying the strong separation condition (SSC) with respect to . Recall that by a GDS satisfying the SSC with respect to we mean a family of domains and maps satisfying the following conditions (compare [12, pp. 3, 58]):

There exists a finite family of open connected (not necessarily simply connected) domains in the Riemann sphere with pairwise disjoint closures.

There exists a family of branches of mapping into with bounded distortion (not all pairs must appear here).
Note that a general definition of GDS allows many maps from each to each . Here however there can be at most one, since we assume that critical points are far away from and that the maps are branches of . 
We have
We assume that we have and hence that for each there exists and for each there exists such that .
We can view , , as vertices and as edges from to of a directed graph .
This definition easily implies that is uniformly expanding on the limit set of such a GDS, and that is a repeller for . Clearly is a Cantor set. In fact a sort of converse is true (though we shall not use this fact in this paper, but it clears up the definitions). Namely we observe the following fact.
Lemma 1.
If is an invariant compact uniformly expanding set that is a Cantor set, then is contained in the limit set of a GDS satisfying the SSC (this limit set can be chosen to be contained in an arbitrarily small neighborhood of ). Hence, is contained in an ECR set.
Proof.
We can multiply the standard sphere Riemann metric by a positive smooth function such that with respect to this new metric we have on . It is easy to show that one can find an arbitrarily small number such that the neighborhood consists of a finite number of connected open domains with pairwise disjoint closures. We account for our GDS the branches of on the sets such that each intersects . Then maps each into some because it is a contraction (by the factor ). Hence the family of maps satisfies the assumptions of a GDS with the SSC. ∎
In the proof of the following lemma we will “build bridges” between two ECR’s.
Lemma 2.
For any two disjoint ECR sets , that both contain nonimmediately postcritical points there exists an ECR set containing the set . If is topologically transitive on each , , , then is topologically transitive on .
Proof.
Let , be two sets satisfying the assumptions of the lemma and let and be two nonimmediately postcritical points. Consider a family of open connected domains and a family of branches of mapping to that define the DGS’s satisfying the SSC that have as their limit sets, for , , respectively. Let
We can assume that each is an arbitrarily small neighborhood of , by replacing by , where by we denote the family of all compositions
For each , let us choose a backward trajectory of the point (the “bridge”) such that
for all , , and , . Let us denote by the branch of that maps to , that is, let
Let be an open disc centered at that is contained in and satisfies for all , , (note that this is possible provided we choose small enough) and
Let us consider an integer such that the component of containing is contained in , that is, that
for , . Now let us replace by , let us replace each map by the family of its restrictions to contained in and and let us denote by the union of those families. This defines a GDS with graph , for , . Now we restrict each bridge to the element of that contains . As the next step we consider
where for we choose an arbitrary prolongation of the bridge by maps . Finally, we consecutively thicken slightly along the bridges such that . For each let us denote by the branch of from to for , , . Let us denote by the family of all these branches. By construction the family of domains
and the family of maps form our desired GDS with a graph satisfying the SSC and hence defines an ECR set that containes .
Finally notice that this system has topologically transitive limit set since its transition graph is transitive. This follows from the assumption that due to topological transitivity of the graphs are transitive and from the construction of the bridges. ∎
2.3. Hyperbolic subsystems and approximation of pressure
Our approach is to “exhaust” the Julia set by some family of subsets and to show that the corresponding pressure functions converge towards the pressure of . In particular, in order to be able to conclude convergence of associated spectral quantities, it is crucial that each such is an invariant uniformly expanding and topologically transitive set.
We start by stating a classical result from PesinKatok theory. In follows for example from [22, Theorems 10.6.1 and 11.2.3]. Recall that an iterated function system (IFS) is a GDS that is given by a complete graph.
Lemma 3.
For every ergodic invariant measure that is supported on and has a positive Lyapunov exponent, for every continuous function and for every , there exist an integer and an ECR set that is topologically transitive and a limit set of an IFS, such that
(7) 
where we use the notation , and in particular
(8) 
Our aim is to apply Lemma 3 to potentials and to use the resulting ECR sets to construct a sequence of ECR sets on which the pressure function converges pointwise to . If the ECR sets generated by Lemma 3 are pairwise disjoint, we can simply build bridges between such sets applying Lemma 2. In the general situation we start with the following lemma.
Lemma 4.
Let be a topologically transitive ECR set. Let be a finite number of continuous functions. Then for any open disc intersecting and for any one can find a set and a natural number such that is an ECR set and for every satisfies
Proof.
Consider in a clopen set contained in . Let . Let us choose large enough such that in the case that and the pullback satisfies . Note that it is enough to take
where is the expanding constant on in the metric defined as in the proof of Lemma 1. Note that the topological transitivity of implies that every pullback of can be continued by a bounded number of consecutive pullbacks until it hits , say this number is bounded by a constant . This way we obtain an IFS for , where , with its limit set contained in .
Recall the equivalent definition of tree pressure established in [21, Theorems A, A.4]. Due to topological transitivity in the definition of pressure we can consider separated sets that are contained in the set of preimages . Therefore, for any the pressures with respect to and with respect to differ by at most from each other. As can be chosen arbitrarily big, this proves the lemma. ∎
The following approximation results are fundamental for our approach.
Proposition 1.
Assume that is nonexceptional or that is exceptional but . Then there exists a sequence of positive integers and a sequence of invariant uniformly expanding topologically transitive sets such that for every , we have
(9) 
For every we have
(10) 
and
(11) 
where and are defined as in (5) and (4) but with instead of .
Proof.
To prove (9) it is enough to construct an ECR set such that
(12) 
for all . Recall that we have (3). As is uniformly Lipschitz continuous, we only need to check (12) for a finite number of potentials . Given , we apply Lemma 3 to potentials , obtaining a family of ECR sets on which the pressure . We then apply Lemma 4 to construct a family of pairwise disjoint ECR sets that satisfy . Those sets are also disjoint ECR sets for and by our assumption contain nonimmediately postcritical points. Hence we can consecutively apply Lemma 2 to them. We obtain an ECR set satisfying (12) for all . This proves (9).
2.4. Conformal measures
The dynamical properties of any measure with respect to are captured through its Jacobian. The Jacobian of with respect to is the (essentially) unique function determined through
(13) 
for every Borel subset of such that is injective. In particular, its existence yields the absolute continuity .
2.5. Hyperbolic times and conical limit points
When is not uniformly expanding, we can still observe a slightly weaker form of nonuniform hyperbolicity. We recall two concepts that have been introduced.
A number is called a hyperbolic time for a point with exponent if
It is an immediate consequence of the Pliss lemma (see, for example, [14]) that for a given point , for any there exist infinitely many hyperbolic times for with exponent .
We denote by
the maximal distortion of a map on a set . After [6], we will call a point conical if there exist a number , a sequence of numbers and a sequence of neighborhoods of such that and that is bounded uniformly in .
3. On the completeness of the spectrum
In the following two lemmas we will investigate which numbers can occur at all as upper/lower Lyapunov exponents.
Lemma 5.
We have
If does not contain any critical point of then we have
Proof.
Consider an arbitrary and a sequence such that
and
in the weak topology. The limit measure is invariant, [25, Theorem 6.9]. Define
Notice that is a monotonically decreasing sequence of continuous functions that converge pointwise to . Hence we obtain
where the equality follows from the Lebesgue monotone convergence theorem. This proves the first statement.
The second statement follows simply from the fact that is continuous on if has no critical points in . ∎
Remark 2.
We remark that the same method of proof gives a slightly stronger result than the fact that for every . Namely we have
see [19, Proposition 2.3. item 2].
Lemma 6.
If has a finite Lyapunov exponent then , that is, we have
If there are no critical points in then is empty. If there is only one critical point in then consists only of this critical point and its preimages.
Proof.
Let be a Lyapunov regular point with exponent and assume that . It is enough for us to prove that there exists a periodic point with Lyapunov exponent arbitrarily close to , the contradiction will then follow from the definition of .
Note first that if exists and is finite then must be a Cauchy sequence and hence satisfies
(14) 
By [17, Corollary to Lemma 6], we then can conclude that . Choose now a small number . Let be a hyperbolic time for with exponent (recall that has infinitely many hyperbolic times with that exponent, and hence that can be chosen arbitrarily big). Because of (14), there exists an integer such that for all we have
(15) 
and we can assume that .
We start with a construction that is standard in Pesin theory. For each , , we define
where is the greatest degree of a critical point of . This way we define a sequence of balls that are centered at points of the backward trajectory of and that have diameters shrinking slower than the derivative of along this branch and at the same time have diameters much smaller than their distance from any critical point. As we have , (15) implies that for big enough for any the set does not contain any critical point and that
(16) 
where is some constant that depends only on .
Claim:
If is sufficiently big then for any , ,
the map is univalent and has bounded distortion on the set
and satisfies .
To prove the above claim, note that the ball shrinks as increases. Hence, it is enough to prove the statement for . The statement for the initial finitely many steps , , is then automatically provided is big enough. Let us assume that is sufficiently big such that also
By construction of the family and by (15), for each we have
Hence, if for we have for every , , then (16) implies
On the other hand, recall that is a hyperbolic time for with exponent . Hence, if
then . The above claim now follows by induction over .
Let us consider the set
Notice that