Abundant rich phase transitions
in step skew products
We study phase transitions for the topological pressure of geometric potentials of transitive sets. The sets considered are partially hyperbolic having a step skew product dynamics over a horseshoe with one-dimensional fibers corresponding to the central direction. The sets are genuinely non-hyperbolic containing intermingled horseshoes of different hyperbolic behavior (contracting and expanding center).
We prove that for every there is a diffeomorphism with a transitive set as above such that the pressure map of the potential ( the central direction) defined on has rich phase transitions. This means that there are parameters , , where is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum of on .
Key words and phrases:thermodynamic formalism, Lyapunov exponents, phase transitions, partially hyperbolic dynamics, skew product, transitivity
2000 Mathematics Subject Classification:37D25, 37D35, 28D20, 28D99, 37D30, 37C29
Thermodynamical formalism is a branch of ergodic theory, that studies quantifiers of invariant measures such as entropy and Lyapunov exponents. Those are interrelated by the pressure of the so-called geometric potentials associated to the differential of the map. Recall that, given a continuous map of a compact metric space and a continuous function , its topological pressure with respect to is defined in purely topological terms and can be expressed via the variational principle
where denotes the entropy of the measure and the supremum can be taken over all ergodic -invariant probability measures (see ). A measure is called an equilibrium measure for with respect to if it attains the supremum in the above principle.
This theory is quite well understood in the uniformly hyperbolic setting, see [4, 19]. In this setting, for a given Hölder continuous function (sometimes called potential) the pressure function , , is analytic and its derivatives are related to stochastic properties of the dynamics and of equilibrium states. Little is known beyond this hyperbolic context and naturally, a crucial point is the differentiability of the pressure function.
Going beyond the hyperbolic scenery, in the non-hyperbolic context there are essentially two types of dynamics called critical and non-critical (see Preface in ). The former one is present for instance in the quadratic family and for diffeomorphisms with homoclinic tangencies, while the latter one is mainly associated to partially hyperbolic dynamics. In this paper we consider partially hyperbolic transitive sets of a diffeomorphism with one-dimensional center direction and study the differentiability of the pressure for the geometric potential . Let us observe that this map is always convex and thus non-differentiable in at most a countable number of points (see [21, Chapter 9] for more information). Moreover, in this setting equilibrium states for always exist for every (see [7, Proposition 6] and also ).
Before stating the result of the paper, let us recall that the pressure function , , is said to exhibit a phase transition at a parameter if it fails to be real analytic at . This transition is of first order if it fails to be differentiable at . In some cases, the existence of a phase transition is associated to the existence of (at least) two (ergodic) equilibrium states with different Birkhoff averages. Note that to each ergodic measure there is associated a sub-gradient of the pressure function whose slope is given by the Birkhoff average of with respect to the measure. The transition is said to be rich if there are at least two equilibrium states for with positive entropy.
Given any and any manifold of dimension at least , there is a diffeomorphism of having a compact invariant set that is topologically transitive and partially hyperbolic with splitting
with non-trivial bundles and one-dimensional center bundle having the following properties:
Consider the continuous potential , the topological pressure has rich phase transitions. More precisely, there are parameters such that is real analytic in each interval and not differentiable at . Further, for every , , there exist at least two equilibrium states for with respect to both with positive entropy.
The dynamics of the transitive set is a step skew product with one-dimensional fibers, semi-conjugate to a shift map over three symbols, and its central direction has mixed contracting and expanding behaviors. For instance, the sets of those hyperbolic periodic points which are contracting in the central direction and those which are expanding in the central direction are both dense in . This sort of sets have also a rich fiber structure and contain uncountably many curves (called spines) tangent to the central direction. Topological properties and the rich fibre structure of such sets, called porcupine horseshoes, were studied in [12, 9, 11], compare also so-called bony attractors with a somehow similar structure in [14, 16]. The occurrence of phase transitions for the geometric potential were studied in [17, 9] (existence of one transition) and  (existence of one rich transition).
Let us briefly describe the diffeomorphism in a neighborhood of . 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 . Let . We consider a map defined by
where are injective interval maps specified in Section 3.1. We also assume that the rate of expansion (contraction) of the horseshoe is stronger than any expansion (contraction) of . In this way the -invariant splitting given by
is dominated and and are uniformly hyperbolic. We consider the maximal invariant set of in the cube
Let us emphasize that transitivity is a key property in our setting, as even in the hyperbolic case naturally there can occur phase transitions related to the existence of several basic pieces. Note also that there are phase transitions in the critical case associated to failure of transitivity related to renormalizable dynamics, see .
We note that the phase transitions are obtained from corresponding gaps in the spectrum of the central Lyapunov exponents. The intervals between the spectral gaps correspond to “lateral horseshoes” which are associated to sub-shifts, exhibiting appropriate Birkhoff averages and entropy. This will imply that these transitions are rich. Observe that the support of the relevant equilibrium states for jumps between invariant subsets as moves through the parameter of the phase transition.
The set that we consider is a special type of transitive set, called a homoclinic class. It was shown in  that -generically such classes have a convex Lyapunov spectrum. Thus, to obtain these gaps, we need to consider non-generic situations where the homoclinic class contains saddles of different types related by heterodimensional cycles (see [3, Chapter 6]). Our study is also motivated by [2, Question 6.4] about the existence of multiple spectral gaps for homoclinic classes.
Two transitive hyperbolic sets of different index (dimension of the unstable manifold) are related by a heterodimensional cycle if their invariant manifolds meet cyclically (there are heteroclinic orbits going from one of the sets to the other and vice versa). Let us emphasize that the existence of such cycles is a typical feature in the partially hyperbolic setting. The understanding of the interplay of the sets in the cycle and the “dominating” hyperbolicity is a key step in the understanding of the global dynamics.
The heterodimensional cycle studied here are provided by a cycle condition in the fiber dynamics (see condition (F1)) which plays a somewhat similar role as the post-critical point in the quadratic family – typical orbits only slowly approach to cycle sets which correspond to the critical point and the post-critical and give rise to some transient behavior and hence to spectral gaps.
A key feature of our construction is the existence of a cycle configuration involving horseshoes (expanding in the central direction) and a saddle (contracting in the central direction) depicted in Figure 1. Each horseshoe is, in some sense, an “exposed” parts of the dynamics. Each one has different averages and thus is responsible for an interval of exponents in the central Lyapunov spectrum and these intervals are separated by gaps. These horseshoes play the role of exposed piece of the dynamics similar to the critical/post-critical point in the quadratic family.
Let us now contrast our non-critical example (the central dynamics does not have critical points) with the critical case of the quadratic family. Observe that the dynamics of a complex rational map on its (transitive) Julia set can have at most two phase transitions ([18, Main Theorem]). Further, [5, 6] present examples of complex quadratic polynomials having a first order phase transition with a unique associated equilibrium state and with a phase transition with infinitely differentiable pressure function and no equilibrium state associated to the transition parameter, respectively.
In principle the number of phase transitions in a system can be arbitrarily large. The example  of a topologically mixing countable Markov shift with a one-parameter family of piecewise constant potentials possesses a positive Lebesgue measure Cantor set of parameters at which the pressure is not analytic. However, in this example there do not exist equilibrium states associated to the parametrized potentials. The phase transition coincides with the change of the system from being recurrent to being transient (see also  for further discussion and references).
This paper is organized as follows. In Section 2 we select certain sub-shifts in the base dynamics that give rise to the subsets . In Section 3 we define the fibre maps and complete the definition of the diffeomorphism . In this section we present some topological properties of the underlying iterated function system and explain how these properties imply the transitivity of . Finally, in Section 4 we prove Theorem 1.
2. Base dynamics
We now go to the details of the construction of . The first step is the choice of appropriate sub-shifts in the base (horseshoe). The main result of this section is Proposition 2.1 stating differentiability properties of the pressure that is the symbolic counterpart of the one in Theorem 1.
2.1. Construction of sub-shifts
We now choose a sub-shift of that is a union of a finite number of disjoint topologically mixing pieces , each of which having a different frequency of the symbol . We define the sub-shifts recursively.
We first give a general construction. Given a triple of integers , we say that a finite word is -admissible if
A finite word , , is said to be -admissible if every sub-word , , of length is admissible.
Fix some positive integer . We define a finite sequence of triples of positive integers as follows. We start with
Consider a strictly increasing sequence . For we choose recursively with
By construction, if the numbers are large enough, we have
For a fixed sequence of triples, we define for each the sets
We consider also
Given , consider the continuous potential given by
The following is our main technical result about the topological pressure of on the above defined sub-shifts (the proof is in Section 2.4). We will study the topological pressure of with respect to . By definition,
There exist numbers such that for every and every we have
Moreover, is real analytic in each interval . Further, the left/right derivatives satisfy
and there exist equilibrium states for with respect to that both have positive entropy and Birkhoff average .
In the following subsections we specify certain averaging properties of the sub-shift guaranteed by an appropriate choice of the sequence . However, we need to start from topological properties of .
2.2. Topological properties
Let be -admissible. Then there exist a word of length not greater than such that is -admissible.
Without weakening of assumptions we can assume that are of length . Note that if and are -admissible words of length that differ at most in one place then the word is also -admissible. So, when we take a sequence of -admissible words , , such that and differ in only one place then the word is -admissible. As and differ in at most places, the assertion follows. ∎
is topologically mixing for every .
2.2.1. Birkhoff averages
2.2.2. Entropy constants
Associated to the sequence of triples there are also defined entropy constants that we will estimate in the following.
There exist constants such that
We can assume that . Note that by a slight modification of [21, Theorem 7.13 (i)] the entropy of the sub-shift satisfies
where denotes the number of all cylinders , , in . Note that in particular the limit exists and it is enough to consider some subsequence.
Observe that the number of cylinders of length in is given by
hence (up to a multiplicative constant) we can write
(we remind that is bounded by ).
By Lemma 2.2 for every pair of -admissible finite words there is a finite word of length such that is -admissible.
Hence, for every the number of cylinders of length at most is bounded from below by
Substituting (2.6) we get the lower bound.
Similarly, we can estimate the number of periodic orbits of period from above by , which gives the upper bound. ∎
The above enables us to further specify the choice of in such a way that for each we have
Associated to the triples there are cones defined by
Observe their geometric meaning: the graph of the pressure function lies inside , see Section 2.4.
By the above choices, we have and . By construction, two consecutive cones and intersect each other in a rectangle that projects to the first coordinate to the interval (compare Figure 2).
2.2.4. Choice of the triples
We now specify the choice of triples .
Given any , there are triples satisfying conditions (2.1) in such a way that for every
and by definition (2.5) we have
Hence, for big enough we have
Hence, can be made arbitrarily large.
With the above we can proceed recursively. Observe that for we have if is large enough. For suppose that we have already constructed for every satisfying Properties 1–3. By definition, . If is large enough then . ∎
2.3. Properties of the sub-shifts
By construction, the sets all are pairwise disjoint and -invariant. Observe that implies that . It turns out that .
If then for all and all .
If then in the word the symbol 0 appears either at least times (if ) or precisely or times (if ). We know that for some .
If then it means that in the word the symbol 0 appears at least times, which is impossible because this word is a sub-word of . If then it means that in the word the symbol 0 appears no more than , which is impossible because this word contains all the symbols except the first one from and the latter word has at least 0’s. Hence, and we proceed by induction. ∎
2.4. Proof of Proposition 2.1
Given , consider the lower and upper Birkhoff averages of at with respect to given by
The definition of implies that for every and every we have
Consider the set defined in (2.3). We will study the topological pressures of the potential with respect to the shift maps and . Observe that for every we have
To locate the graph of the pressure function note that
and that any sub-gradient of the pressure function is in . Thus, using the cones defined above, we get that
Note that for every the potential is Hölder continuous. By Corollary 2.3, for each the sub-shift is uniformly expanding and topologically mixing, thus the function is real analytic . In particular, by the above, derivatives are in the interval .
Note that is convex and thus differentiable for all but at most countably many , and the left and right derivative and are defined for all . Moreover, is differentiable at if, and only if, all equilibrium measures for have the same average (refer to [21, Section 9.5] for more details).
Write . By the remarks above, we have . Since the intervals are pairwise disjoint, the graphs of and intersect in exactly one point .
The choice of the cones implies that for and the choice of implies that . Hence, the pressure changes from the cone to . Since in each cone the slope of the pressure is in the interval given by the “opening” of the cone and these “openings” are disjoint this implies that at the pressure is not differentiable.
Fix and let us focus on the sub-dynamics of . First, let us obtain an equilibrium state with the claimed properties. Let be an weak accumulation point of as , where is the (unique) equilibrium state for with respect to . As above, we observe . In particular, we hence have
Upper semi-continuity of the entropy function implies that is an equilibrium state for with respect to . By a standard argument we can assume that is ergodic. Thus, typical Birkhoff averages are equal to , proving the first property. Observe that for every we have
or, in other words, is the intersection of the tangent line to the pressure at with the vertical axis. Assume, by contradiction, that . As is strictly convex, this would imply that for the entropy would be negative, which is a contradiction.
Analogous arguments apply to .
Finally, Lemma 2.6 item 3 implies that , ending the proof of the proposition.
3. Smooth realization
Fix , the triples , and the sets as in Section 2.1. Each cylinder associates a sub-cube of defined as the connected component of containing , where is the conjugation map defined in Section 1. To produce a simple example, we will assume that is affine in . To write the map in a compact way, write and define as in (1.1) by
where the fibre maps are given by
Consider the sub-cubes . To complete the definition of in we take some appropriate -continuation such that
Note that the skew product is constant on each such sub-cube . As each fibre map of the skew product is determined by at most symbols of the symbolic representation of , the map is an -step skew product.
Figure 4 illustrates roughly the construction of level-1 rectangles.
Recall the definition of in (2.3) and consider the lateral set
3.1. The underlying iterated function system
We start with the following conditions.
The map is increasing and has exactly two hyperbolic fixed points in , the point (repelling) and the point (attracting). Let and .
The map is an affine contraction where . We denote by the attracting fixed point of . Note that (cycle condition).
The map is increasing and has two hyperbolic fixed points in , the point (repelling) and the point (attracting). Let .
Before going into further details, let us explain some heuristic ingredients.
A key point is the existence of a fundamental domain , close to , such that , for some large , is close to and and the restriction of to is uniformly expanding. This is the essential point for defining expanding itineraries (see Section 3.1.1) and hence proving transitivity of . This is guaranteed by the following.
The derivative is decreasing in , in particular, for all , and satisfies
Note that given , condition (F01) is clearly satisfied if is sufficiently close to .
The maps are chosen as follows.
The maps are close to and satisfy , , and . Moreover, for simplicity, we take in .
Finally, a key point is the existence of a gap in the spectrum (see Section 3.1.2). For simplicity, let us assume that is close to but smaller than . Due to monotonicity of the maps and its derivatives, the only way that a point may have central Lyapunov exponent close to is when its orbit stays close to for a long time. But due to the cycle configuration, such an orbit previously stayed close to for a long time. The latter compensates the expansion and as a result the central Lyapunov exponent is smaller than some number (see definition (3.5)).
The choice of is such that they are close to in such a way that the expansion they introduce do not destroy the spectral gap . The precise technical condition is literally (mutatis mutandi) the same as (F012) in , however stating the condition not only for but for and , respectively111 (F012) We have for all . The interval , as above, is such that with the intersections are all empty. Moreover, we assume that Finally, let us also assume that with satisfying .
3.1.1. Expanding properties of the iterated function system
First, we provide more details on the IFS. The maps are chosen (see [10, Remark 7.2]) such that there are numbers , close to as above, and , and such that every interval has associated a natural number satisfying the following:
Let be the smallest number such that