Rich phase transitions in step skew-products
We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated to the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behavior of different types. However, in some sense the expanding behavior is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.
Key words and phrases:phase transitions, Lyapunov exponents, thermodynamic formalism, partially hyperbolic dynamics
2010 Mathematics Subject Classification:Primary: 37D35, 37D25, 37E05, 37D30, 37C29
Given a compact metric space and a continuous map , the arising dynamical system can be studied from various points of view. On the one hand, one can investigate the topological dynamics determined by . On the other hand, one can investigate the set of -invariant Borel probability measures and study measure theoretic aspects of the dynamics. Both sides are closely linked with each other. This link is characterized by the thermodynamic formalism and specified by the topological pressure functional (see  for full details). Given a continuous function , its topological pressure is defined in purely topological terms. It can be expressed in measure-theoretic terms via the variational principle
where denotes the entropy of the measure .
The set equipped with the weak topology forms a compact convex space that can have an extremely complicated structure. It is natural to aim for a closer understanding of this structure. One way is to characterize measures that are “relevant” and “designated” in a certain sense, for example, to focus on equilibrium measures. An invariant measure is said to be an equilibrium measure or equilibrium state of with respect to if it attains the supremum in the variational principle (1.1) (the measure maximizes what is sometimes also called the free energy of the potential ). Note that an equilibrium state for the zero potential is a measure of maximal entropy.
To show existence and uniqueness and to establish further specific properties of equilibrium measures are among the main problems in the thermodynamic formalism. On the other hand, particularly interesting are examples where existence or uniqueness fails. In many cases the coexistence of equilibrium states for some given potential is closely related to so-called phase transitions. Following nowadays standard notation, we say that the pressure function , , exhibits a phase transition at a characteristic parameter if it fails to be real analytic at . We say that it has a first order phase transition at if it fails to be differentiable at .
One major line of research in the thermodynamic formalism considers abstract dynamical systems such as Markov shifts and potentials with a certain regularity. See the classical texts by Ruelle  and Bowen  as well as the collection by Sarig . However, in the present paper we focus on smooth dynamical systems.
We will follow a classical approach to analyze smooth systems by studying their “basic pieces”. In the realm of uniformly hyperbolic dynamics such pieces are formed by the basic sets (sets that are compact, invariant, uniformly hyperbolic, topologically transitive, and locally maximal). Beyond uniformly hyperbolic dynamics a natural line of generalization is the investigation of homoclinic classes (such sets are topologically transitive and contain a dense subset of hyperbolic periodic points, see Definition 3.1).
Loosely speaking, when a system dynamically splits into basic pieces then this should also be reflected by the structure of its “dual” . Certainly, if a system has several transitive components (though they could be intermingled) then this will be reflected dynamically. Dobbs , for example, explains the mechanism that gives rise to phase transitions related to non-transitive behavior in renormalizable unimodal maps. If, however, a dynamical system is topologically transitive but there exist pieces that are “exposed” in a sense that dynamically and topologically they form extreme points then we still can observe the phenomenon of phase transitions and coexistence of equilibrium states. To further support this point of view we will discuss some examples and explain what we mean by “exposed”.
One well-understood case is when the transitive dynamics “splits” into a hyperbolic piece and a nonhyperbolic piece. Let us recall those examples of interval maps and the classically considered potential . In the example of Manneville and Pomeau  the interval splits into the single parabolic fixed point and the remaining set . The pressure exhibits a phase transition at that is related to a coexistence of the Dirac measure and an acip (absolutely continuous invariant probability measure) as equilibrium states. In the case of the Chebyshev polynomial the interval is transitive and exposes the post-critical fixed point (though the potential fails to be continuous due to the singularity) and exhibits a phase transition at . An example of Bruin  discusses quadratic maps with (several) ergodic measures supported on minimal Cantor sets with zero Lyapunov exponents that are coexisting with an acip. Note that in all these examples the nonhyperbolic part has zero entropy. See [4, Section 7], , and  for further discussion.
Of similar spirit are the examples of the Julia set of a polynomial or rational exceptional map on the complex plane discussed by Makarov and Smirnov in [14, 15]. Here, the Julia set possesses periodic points that are “dynamically exposed” in the sense that they are immediately post-critical (there is no branch of preimages dense in and disjoint with critical points, see also ). Chebyshev polynomials of degree are particular examples. In each of these cases the post-critical set is finite and thus carries only measures with zero entropy. Those measures are equilibrium states associated to a phase transition of the pressure function at a characteristic parameter.
In the present paper we present examples of local diffeomorphisms with a locally maximal nonhyperbolic set that is at the same time a homoclinic class. We do not aim for generality but instead try to provide the simplest example possible. For that we choose a map that is a skew-product of interval diffeomorphisms over a Smale horseshoe with three legs. Although the dynamics on is topologically transitive, the spectrum of the Lyapunov exponents associated to the one-dimensional central direction contains positive and negative values and has a gap. In this example the potential is continuous and equilibrium states for exist for every (see ). The spectral gap is immediately related to a phase transition of the pressure function at some characteristic parameter . Moreover, for there exist two equilibrium states both of positive entropy (we call this a rich phase transition). To our knowledge, this is the first example of a phase transition in a topologically transitive local diffeomorphism associated to several equilibrium states with positive entropy. This work is an extension of [9, 13] where topological properties and phase transitions related to homoclinic classes are studied. In these examples the exposed sets are single fixed points.
Inside the locally maximal set coexist intermingled sets of different type of hyperbolicity which can be seen, for example, from the coexistence of periodic points with unstable manifolds of different dimensions. However, there exists a measure of maximal entropy with nonpositive central Lyapunov exponent. Hence, we may conclude that the dynamics on is not predominantly expanding. Moreover, in some parameter range this measure is unique and its central exponent is negative.
Besides these dynamical features, our examples possess also topologically a rich structure of the fibers of the skew-product map. Following the approach in , one can show that there exists uncountably many fibers that contain a single point only and uncountably many fibers that contain a continuum.
Let us briefly explain to what corresponds an “exposed piece of dynamics” in our example. We consider genuinely nonhyperbolic homoclinic classes containing infinitely many hyperbolic periodic points of different type. Although this class is transitive, it properly contains a “lateral” horseshoe whose saddles are not homoclinically related to periodic hyperbolic points outside . This lateral horseshoe is a kind of extreme of . One of the equilibrium states involved in the phase transition is supported on the horseshoe that has positive central Lyapunov exponents and positive entropy. The other coexisting state lives on and also has positive entropy.
Let us conclude with a heuristic remark. That a homoclinic classes is properly contained in a bigger one seems to be the underlying mechanism for the gap in the spectrum and hence for the phase transition. However, this configuration is somehow atypical. Indeed, for typical diffeomorphisms homoclinic classes either are disjoint or coincide  and the spectrum of Lyapunov exponents has no gaps .
The paper is organized as follows. We provide the details of our example in Section 2. In Section 3 we explain its topological properties and establish topological transitivity, we postpone the proof to Section 7. In Section 4 we prove the existence of a gap in the spectrum of central Lyapunov exponents, see Proposition 4.2 and Corollary 5.1. In Section 5.1 we prove the existence of a rich phase transition (see Theorem 1). Moreover, in Section 5.2 we discuss periodic fibers that give rise to a measure of maximal entropy with nonpositive or even negative central exponent (see Proposition 5.8 and Corollary 5.12). In Section 6 we briefly discuss more general potentials and provide a sufficient condition for the existence of a phase transition in our setting.
2. A class of skew-products
In this section we construct the class of maps that will be studied in this paper. Consider the cube and a diffeomorphism defined on having a horseshoe in conjugate to the full shift of three symbols. Denote by the conjugation map . We consider the following naturally associated sub-cubes , , of
Let and . We consider the map defined by
where , , , , are assumed to be injective interval maps satisfying properties that we are going to specify now.
To produce a simple example, we will assume that is affine. We also assume that the rate of expansion (contraction) of the horseshoe is stronger than any expansion (contraction) of , , and . In this way the -invariant splitting given by
is dominated and and are uniformly hyperbolic. We denote by and the corresponding strong stable and strong unstable manifolds associated to and .
The following conditions (F0),(F1), and (F2) will imply that the system of the fiber maps is of cycle type and mixes expansion and contraction behavior – compare also Figure 1.
The map is increasing and has exactly two hyperbolic fixed points, the point (repelling) and the point (attracting). Let and . Moreover, for all .
The map is an affine contraction with negative derivative
where . We denote by the attracting fixed point of . Note that (cycle condition).
The map is increasing and has two hyperbolic fixed points, the point (repelling) and the point (attracting). We have .
The next condition (F01) guarantees that suitable compositions of exhibit some expanding behavior (see Step 7.3). We formulate this condition in the simplest case where the derivative is decreasing in .
The derivative is decreasing in and satisfies
Note that given , this condition is clearly satisfied if is sufficiently close to .
To establish the existence of phase transitions, we will require one further property (F012) giving constraints to the variation of and and to . It will be specified in Section 4.
3. Dynamical properties
We introduce some notations and state some dynamical properties of . The skew product structure of allows us to reduce the study of its dynamics to the study of the IFS associated by the maps (see also Section 4).
Consider the sequence space endowed with the metric for , , where if and otherwise. Every sequence is given by , where and . We denote by the periodic sequence of period such that for all and always refer to the least period of a sequence.
3.1. Fixed points and invariant sets
Denote by , , , , the fixed points of the horseshoe map . The structure of the horseshoe and the choice of imply that possesses five fixed points given by
Observe that the u-index (dimension of the unstable manifold) of is while the one of is , for all .
Let us consider the lateral two-legged horseshoe of
Notice that is invariant with respect to since for every and , we have with . This lateral horseshoe is topologically transitive, uniformly hyperbolic, and contains the saddles and .
3.2. Homoclinic classes
We will focus on the maximal invariant set of in the cube , , that will be a special type of transitive set called a homoclinic class. Inside the set coexist intermingled hyperbolic sets of different “types” (-indices). This will give rise to the existence of heterodimensional cycles associated to periodic points in . This is the underlying mechanism to produce a rich dynamics mixing hyperbolicity of different types.
Definition 3.1 (Homoclinic class).
Given a saddle point of its homoclinic class is the closure of the transverse intersections of the stable and unstable manifolds of the orbit of . Two saddles and are homoclinically related if the invariant manifolds of their orbits meet cyclically and transversely. A homoclinic class is non-trivial if it contains at least two different orbits.
In the following we restrict our attention to the dynamics inside the cube . We call the closure of the set of points that are in the transverse intersections of the stable and unstable manifolds of the orbit of and whose orbit is entirely contained in the homoclinic class of relative to . We denote this set by .
Observe first that homoclinically related saddles all have the same -index. Note also that the (relative) homoclinic class coincides with the closure of all saddle points that are homoclinically related to relative to the cube (i.e., the orbits of the transverse intersections are contained in ). However, this closure may contain periodic points that are not homoclinically related to . Indeed, this paper illustrates such a situation and, in fact, is an essential ingredient of our example. Finally, a homoclinic class is always transitive (existence of a dense orbit) and uncountable if non-trivial.
Finally, we see that is nonhyperbolic and transitive.
There is a saddle of -index two such that the set is the homoclinic class of relative . In particular, the set is topologically transitive.
This proposition is a version of the results in  considering skew product dynamics over the shift of two symbols whose central dynamics is given by the maps and . The only difference is that the structure of here is slightly more complicated due to the existence of the additional “leg” of the horseshoe and its associated map . This extra “leg” is also responsible for the existence of the lateral horseshoe and gives rise to more combinatorics in the orbits. We postpone the proof to Section 7.
From similar observations we conclude also the following relations showing the special nature of the lateral horseshoe
4. Lyapunov exponents of the IFS. Spectral gap
In this section we study dynamical properties of the underlying iterated function system (IFS) generated by the maps , establishing the existence of a gap in the spectrum of central Lyapunov exponents (Proposition 4.2). This gap will correspond to a gap in the spectrum of the central exponents of diffeomorphism due to the skew product structure (see Corollary 5.1).
We use the following notation for concatenated maps of the IFS. Given a finite sequence , , let
Given a finite sequence let
Similarly, given a finite sequence , let
An one-sided infinite sequence is said to be admissible for a point if the map is well-defined at for all . By writing we always mean that is admissible for .
Given and a sequence that is admissible for , the (forward) Lyapunov exponent of with respect to is defined by
whenever this limit exists. Otherwise we denote by and the lower and the upper Lyapunov exponent defined by taking the lower and the upper limit, respectively. Note that, in fact, depends only on the positive part of only. Hence, when considering exponents of a pair in the following, we will disregard the hypothesis that is admissible.
Let us consider the symbolic description of the lateral horseshoe together with its stable manifold. This is given by the set of “exceptional points” of the IFS:
As codes all points in the lateral horseshoe together with its stable manifold, its central Lyapunov spectrum is an interval.
To show the following lemma, it is enough to observe that, by construction, and are the smallest and largest central exponents of .
To prove the main result of this section, we need an additional assumption to be satisfied.
We have for all . There exists an interval such that and satisfy
and assume that
Finally, let us also assume that with satisfying
Clearly, (F012) can be guaranteed if is close enough to and if is non-linear close to and does not contract the length of the unit interval too much. Then we can choose very small and can guarantee that is close to and that is close to in order to guarantee (4.2).
We obtain the following gap of the full range of possible exponents.
Under the hypothesis (F012) we have .
Arguing by contradiction, let us assume that for every there exist a point together with a sequence such that and that .
The Lyapunov exponent of a trajectory is the average of along the trajectory. Hence, if the upper Lyapunov exponent of a trajectory is greater than , the trajectory must return to infinitely many times. The only way to enter is by coming from after applying , and the only way to get into is by applying . To reach a contradiction, we only need to prove that the average of along the piece of trajectory between two such consecutive visits to is not greater than .
Note that it is enough to consider the case that for infinitely many since otherwise we would have . Further, we can freely assume that as otherwise we could replace by some iterate. Note that
Thus, we can further assume that the orbit hits the interval infinitely many times. Indeed, otherwise this orbit would be contained in the interval in which the derivatives , , and are bounded from above by and thus the upper Lyapunov exponent would be bounded from above by . Hence, without loss of generality, possibly replacing by some positive iterate, we can assume that and .
For every we write Note that by our choices in (4.1) the only way of entering in is by coming from after applying and the only way of entering and staying in is by applying .
We define three increasing sequences , , and of positive integers as follows: ,
where is the smallest number with this property. By the above observation . Condition (4.1) implies that for every index whenever . By (4.1) we also have for every index . In particular, writing , the latter implies
Moreover, we have for all , which implies
Let us denote by the number of iterates of the point staying in before entering :
We have .
By (4.7) we have
Since and since is affine, we have . Hence, as and , we have
Finally, we have
which proves the claim. ∎
By Claim 4.3, is bounded from above by , where is defined by
Let us now estimate the finite-time Lyapunov exponent associated to the sequence .
We have .
First observe that if , with
by our hypothesis (4.4) the claim is automatically satisfied.
To prove the claim in the other case , it is enough to assume that the number of iterations in the interval is the maximum possible (clearly this is the case which bounds the derivative from above), that is, let us suppose that . Then
From and (4.9) we conclude
We have proved the claim. ∎
5. Thermodynamical formalism
In the first part of this section we establish the existence of rich phase transitions using the gap in the Lyapunov spectrum (Theorem 1). In the second part we construct a maximal entropy measure with nonpositive central exponent.
5.1. Co-existence of equilibrium states with positive entropy
Due to of the skew product structure and our hypotheses, the splitting in (2.2) is dominated and for every Lyapunov regular point coincides with the Oseledec splitting provided by the multiplicative ergodic theorem. In particular, the Lyapunov exponent associated to the central direction at such a point is well-defined and, in fact, is the Birkhoff average of the continuous function
Observe that given a Lyapunov regular point and a sequence given by , we have
Finally note that the remaining exponents are associated to the stable and the unstable directions and , respectively, and are uniformly bounded away from zero.
Let us first recall some general facts. We denote by the set of -invariant Borel probability measures supported on a set and by the subset of ergodic measures. For let
Considering the spectrum of ergodic measures, based on Proposition 4.2, the following result about the set of all possible central exponents is an immediate consequence.
Note that there are two possibilities for an ergodic measure. If its support is contained in then its central Lyapunov exponent is contained in . Otherwise, any generic point is outside and then by Proposition 4.2 has central Lyapunov exponent in .
Given a continuous potential , an -invariant Borel probability measure is called an equilibrium state of with respect to if
where denotes the measure theoretic entropy of . Since the central direction is -dimensional such maximizing measure indeed exists by [8, Theorem A]. Note that we have the following variational principle
where is the topological pressure of with respect to (see  for the definition and further properties that are used in the following). Denote by the topological entropy of . Note that an equilibrium state for the zero potential is a measure of maximal entropy . It is immediate that the two-legged horseshoe defined in (3.2) satisfies
As the central direction does not contribute to the entropy, we also have
Let us investigate the following one-parameter family of continuous potentials defined by
and will denote . Note that is convex (and hence continuous and differentiable on a residual set). One says that exhibits a phase transition at a characteristic parameter if it fails to be real analytic at . We say that it has a first order phase transition at if it fails to be differentiable at .
Let us recall some basic facts. A number is said to be a sub-gradient at if for all . Note that any equilibrium state of the potential with respect to provides a sub-gradient of at given by . By definition
In particular, the entropy of is the intersection of the tangent line with the -axis. Moreover, if is differentiable at then
In particular, in this case, all equilibrium states of have the same exponent. In our case, non-differentiability is equivalent to the existence of a parameter and (at least) two equilibrium states for with different central exponents.
A first order phase transition of is said to be rich if there are two associated equilibrium states with different central exponents and with positive entropy.
The next result establishes the existence of a rich phase transition.