On non-uniform specification and uniqueness of the equilibrium state in expansive systems
In , Bowen showed that for an expansive system with specification and a potential with the Bowen property, the equilibrium state is unique and fully supported. We generalize that result by showing that the same conclusion holds for non-uniform versions of Bowen’s hypotheses in which constant parameters are replaced by any increasing unbounded functions and with sublogarithmic growth (in ).
We prove results for two weakenings of specification; the first is non-uniform specification, based on a definition of Marcus in (), and the second is a significantly weaker property which we call non-uniform transitivity. We prove uniqueness of the equilibrium state in the former case under the assumption that , and in the latter case when . In the former case, we also prove that the unique equilibrium state has the K-property.
It is known that when or is bounded from below, equilibrium states may not be unique, and so this work shows that logarithmic growth is in fact the optimal transition point below which uniqueness is guaranteed. Finally, we present some examples for which our results yield the first known proof of uniqueness of equilibrium state.
Key words and phrases:Expansive, non-uniform specification, uniqueness of equilibrium state
2010 Mathematics Subject Classification:Primary: 37D35; Secondary: 37B10, 37B40
A central question in the theory of topological pressure is knowing when a dynamical system and potential admit a unique equilibrium state. Often, such proofs use as hypotheses a mixing/shadowing property for the system and a regularity condition on the potential .
One of the first and most important results of this type was proved by Bowen in , using the hypotheses of expansiveness and specification on and the so-called Bowen property on . Informally, is expansive if there exists a fixed distance so that any unequal points of will be separated by distance at least under some iterate of . Specification is the ability, given arbitrarily many orbit segments, to find a periodic point of whose orbit “shadows” (meaning it stays within some small distance of) those orbit segments, with gaps dependent only on the desired shadowing distance. The Bowen property is simply boundedness (w.r.t. ) of the differences of the partial sums over pairs whose first iterates under stay within some predetermined distance. (See Section 2 for formal definitions.) Bowen’s theorem can then be stated as follows.
() If is an expansive system with specification and is a Bowen potential, then has a unique equilibrium state for , which is fully supported.
The assumptions of specification for and the Bowen property for each have associated constant bounds independent of a parameter ; for specification there is the bound on the gap size required between shadowing orbit segments of length , and for the Bowen property there is the bound on the associated variation of the th partial sum.
The main results of this work show that the same conclusions hold even for unbounded and , as long as they grow sublogarithmically with . We have results using two different versions of specification; the first is called non-uniform specification, and the second, much weaker, property is called non-uniform transitivity.
If is an expansive dynamical system (with expansivity constant ) with non-uniform specification with gap bounds (at scale ), is a potential with partial sum variation bounds (at scale ), and , then has a unique equilibrium state for , which is fully supported.
If is an expansive dynamical system (with expansivity constant ) with non-uniform transitivity with gap bounds (at scale ), is a potential with partial sum variation bounds (at scale ), and if , then has a unique equilibrium state for , which is fully supported.
For non-invertible surjective , there is a canonical way to create an invertible system called the natural extension. It’s well-known that the natural extension has the same simplex of invariant measures as that of the original system, and so Theorems 1.2 and 1.3 can also be applied to any whose natural extension satisfies their hypotheses. In particular, we note that whenever is positively expansive (see Theorem 2.2.32(3) of ), its natural extension is expansive.
We also prove results about preservation of these properties under expansive factors and products, which are unavoidably a bit technical, and so we postpone formal statements to Section 3.
For any and satisfying the hypotheses of Theorem 1.2 and associated unique equilibrium state , is a K-system.
Since K-systems have positive entropy, this also answers a question of Climenhaga from  about so-called hyperbolic potentials. Following , a potential is said to be hyperbolic for if every equilibrium state has positive entropy. Climenhaga’s question was the following:
Question 1.7 (, Question 3.20).
Is there an axiomatic condition on a subshift , weaker than specification (perhaps some form of non-uniform specification), guaranteeing that every Hölder potential on is hyperbolic? Is there such a condition that is preserved under passing to (subshift) factors?
The class of subshifts with non-uniform specification with gap bounds satisfying is closed under (subshift) factors, and for any such subshift, every Hölder potential is hyperbolic.
We also collect results from the literature which demonstrate that when one of is and the other quantity grows logarithmically, uniqueness of the equilibrium state is still not guaranteed; this shows that our hypotheses cannot be weakened by too much.
Full shifts correspond to the case , and for those there is the following example, based on a classical example of Hofbauer from .
Theorem 1.9 ().
For every , there exists a potential on the full shift on with partial sum variation bounds where has multiple equilibrium states for , whose supports are disjoint.
The example in , the so-called Double Hofbauer model, was actually for one-sided full shifts, rather than the two-sided ones treated here. Briefly, they define the potential in terms of the largest nonnegative integer where . It is not hard to adapt this to a two-sided version, which satisfies Theorem 1.9, by instead choosing maximal for which . In fact, this is essentially the idea behind our later Example 5.9.
For any positive increasing with , there exists a subshift with non-uniform specification with gap bounds and multiple ergodic measures of maximal entropy whose supports are disjoint.
The reader may notice that there is a gap in the results we’ve presented for full shifts; Theorem 1.9 shows that non-uniqueness can happen for with arbitrarily close to , and Theorems 1.2 and 1.3 guarantee uniqueness only when approaches in general or along a subsequence. In fact, a careful reading of our proofs shows that these theorems hold as long as the limit or liminf of is smaller than .
We chose not to use this as our hypothesis both due to an aesthetic preference for and to be on equal footing, and because we wanted hypotheses invariant under products in order to use Ledrappier’s arguments to prove the K-property for the unique equilibrium state. However, it seems that the actual “transition point” for is likely of the form for , and it would be an interesting technical problem to prove this and find the in question; we will not, however, treat that question in this work though.
The techniques used to prove Theorems 1.2 and 1.3 are somewhat similar to the proofs of previous weaker results from , which applied only to measures of maximal entropy on subshifts and showed only that two such measures , could not have disjoint supports. Roughly speaking, the proof in  involved combining “large” collections of words based on and to create more words in than there should be by definition of . The assumption of disjoint supports of and implied that the words from the two collections could not have overlap above a certain length, which ensured that all words created were distinct.
For the results in this work, several changes must be made. First of all, obviously ‘words’ must be replaced with ‘orbit segments shadowed by a very small distance’ for general expansive systems, and ‘number of words’ must be replaced by ‘partition function for an -separated set’ for general potentials. These changes require some technical results about changes of scale for expansive systems (see Section 3), but the ideas are all essentially present in previous work of Bowen and others.
The main advance in this work is dealing with whose supports may not be disjoint. The best we can do then is to assume ergodicity of , which implies their mutual singularity, and therefore the existence of disjoint compact sets with arbitrarily small (and positive distance ). The new idea here is the use of the maximal ergodic theorem, which allows us to define “large” collections of orbit segments based on and where all initial segments of -orbit segments have a large proportion of visits to , and all terminal segments of -orbit segments have a large proportion of visits to . This means that initial segments of -orbit segments and terminal segments of -orbit segments are separated by at some point, mimicking the lack of overlap from the proof in . This is enough to create an -separated collection by combining segments from the two collections whose partition function is larger than the pressure should allow, achieving the desired contradiction.
In various works (including , , and ), Climenhaga and Thompson have defined different weakenings of the specification property, which allowed them to both generalize Bowen’s results in a different direction, even treating some non-expansive systems and continuous flows. Without going into full detail here, their definitions involve decomposing all orbit segments of points in the system into prefixes, cores, and suffixes, where the sets of possible prefixes/suffixes are “small” in some sense, and where for any , the collection of segments whose prefix and suffix are shorter than has specification (in the sense that one can always find a point which shadows arbitrarily many such segments, of any lengths, with constant gaps). Existence of such a Climenhaga-Thompson decomposition is less restrictive than non-uniform specification in that specification properties must hold only for a subset of orbit segments, but is more restrictive in that the property required for that subset (weak specification) is significantly stronger. To our knowledge, neither of non-uniform specification or a Climenhaga-Thompson decomposition implies the other.
Finally, we summarize the structure of the paper: Section 2 contains relevant definitions and background on topological dynamics, ergodic theory, and thermodynamic formalism. Section 3 contains some results about preservation of various hypotheses under changes of scale, expansive factors, and products. Section 4 contains the proofs of Theorems 1.2 and 1.3, including various auxiliary results. Finally, Section 5 contains some examples for which our results imply uniqueness of equilibrium state and for which we believe this to be previously not known.
The author would like to thank Jerome Buzzi and Sylvain Crovisier for pointing out that the proof of Theorem 1.2 could be easily adapted to use the weaker hypothesis of non-uniform transitivity (yielding Theorem 1.3), and would also like to thank François Ledrappier for discussions about the use of the techniques from  for proving the K-property for unique equilibrium states.
A dynamical system is given by a pair where is a compact metric space and is a homeomorphism.
A dynamical system is expansive if there exists (called an expansivity constant) so that for all unequal , there exists for which .
A particular class of expansive dynamical systems are given by subshifts, to which the next few definitions refer.
Given a finite set called the alphabet, a subshift is given by which is closed (in the product topology) and invariant under the left shift map defined by for .
Every subshift is expansive; simply choose any so that . Then, any must have for some , and then , so .
The language of a subshift , denoted by , is the set of finite strings of letters from (called words) which appear in some .
We now return to definitions for more general expansive systems.
Given an expansive dynamical system and any and , a set is called -separated if for all unequal , there exists so that .
Given a continuous function (called a potential), the partial sums of are the functions defined by .
For a dynamical system and potential , the th partition function of at scale are the functions
The topological pressure at scale of is
The topological pressure of for a potential is
Lemma 2.9 ().
If is expansive with expansivity constant , then for any , .
We also need some definitions from measure-theoretic dynamics. All measures considered in this paper will be -invariant Borel probability measures on for an expansive dynamical system, and we denote the space of such measures by .
A measure on is ergodic if any measurable set which is invariant, i.e. , has measure or .
Not all -invariant measures are ergodic, but a well-known result called the ergodic decomposition shows that any non-ergodic measure can be written as a “weighted average” (formally, an integral) of ergodic measures. Also, whenever ergodic measures and are unequal, in fact they must be mutually singular (written ), i.e. there must exist a set with . (See Chapter 6 of  for proofs and more information.)
When a measure is ergodic and , the ergodic averages converge -a.e. to the “correct” value ; this is essentially the content of Birkhoff’s ergodic theorem. We will need the following related result, which deals with the supremum of such averages rather than their limit.
Theorem 2.11 (, Theorem 2 (Maximal Ergodic Theorem)).
For , define . Then for any ,
The following corollary is immediate.
For nonnegative and as in Theorem 2.11, and any ,
We note that by considering instead, both of these results also hold when is replaced by .
We also need concepts from measure-theoretic entropy/pressure; for more information/proofs, see .
For any , and finite measurable partition of , the information of with respect to is
where terms with are omitted from the sum.
For any and finite measurable partition of , the entropy of with respect to is
Note that by subadditivity, it is always true that .
For any , the entropy of is
We say that has the K-property if for every partition consisting of nonempty sets, .
Note that any with the K-property trivially has .
A partition is a generating partition for if separates -a.e. points of .
If is a generating partition, then .
By expansivity, any partition of sets whose diameters are all less than is a generating partition for all , and so we have the following fact:
If is expansive with expansivity constant and consists of sets whose diameters are all less than , then for any measure .
The relationship between topological pressure and measure-theoretic entropy is given by the following Variational Principle:
Theorem 2.20 ().
For any dynamical system and continuous ,
For any , an equilibrium state for and is a measure on for which .
If is expansive, then the entropy map is upper semi-continuous.
As a corollary, if is expansive and is continuous, then it has an equilibrium state; the upper semi-continuous function must achieve its supremum on the compact space (endowed with the weak- topology). In fact, the ergodic decomposition, along with the fact that the entropy map is affine (, Theorem 8.1), implies that the extreme points of the simplex of equilibrium states are precisely the ergodic equilibrium states. In particular, any with multiple equilibrium states also has multiple ergodic equilibrium states.
Definition 2.23 ().
A potential on is called hyperbolic if every equilibrium state has .
Though this is not the original definition from , it was shown to be an equivalent one in their Proposition 3.1.
Given a dynamical system and a potential , the partial sum variations of at scale are given by
We say that has partial sum variation bounds at scale if whenever .
Our remaining definitions are for specification properties on . We first need a general notion of shadowing.
Given a dynamical system , , points , and integers , , , , , , we say that -shadows for iterates with gaps if for every and ,
A dynamical system has specification if for any , there exists a constant so that for any , any points , , , and any integers , , , , , , satisfying when , there exists a point which -shadows for iterates with gaps and for which .
A related property in the literature is weak specification, which is identical to the definition above except that no periodicity of is assumed. For expansive , this distinction is irrelevant; weak specification in fact implies specification (see , Lemma 9).
We now move to the mixing properties which we will consider in this work, both of which can be thought of as non-uniform generalizations of specification with no assumption of periodicity.
For an increasing function , a dynamical system has non-uniform specification with gap bounds at scale if for any , any points , , , , and any integers , , , , , , satisfying , there exists a point which -shadows for iterates with gaps .
This property is almost the same as the main property used by Marcus in  (which was not there given a name). There are two differences: the first is that Marcus required as part of his definition, and the second is that in Marcus’s definition was only assumed greater than or equal to . Essentially, non-uniform specification only guarantees the ability to shadow when gaps are large enough in comparison to the lengths of orbit segments being shadowed before and after the gap, and Marcus’s unnamed property requires only that gaps be large compared to the orbit segment before the gap.
We also consider the following significantly weaker property of non-uniform transitivity, which is weaker than non-uniform specification in two important ways. The first is that it only guarantees the ability to shadow two orbit segments, rather than arbitrarily many, and the second is that it guarantees the existence of only a single gap length which allows for shadowing, rather than guaranteeing that all gaps above a certain threshold suffice.
For an increasing function , a dynamical system has non-uniform transitivity with gap bounds at scale if for any and any points , there exists and which -shadows for iterates with gap .
We note that for with this property and any and , one can also find which -shadows for iterates with some gap . This is because one can instead start with the pair , and for any which -shadows for iterates with gap , it is immediate that -shadows for iterates with gap .
3. Preservation under factors, products, and changes of scale
In this section, we summarize some simple results illustrating preservation of various properties/quantities under expansive factors, products, and changes of scale.
If is expansive (with expansivity constant ) and has non-uniform specification (transitivity) at scale with gap bounds , then for every there exists a constant so that has non-uniform specification (transitivity) at scale with gap bounds .
We present only the proof for non-uniform specification, as the one for non-uniform transitivity is extremely similar. Choose as in the theorem, and any . By expansivity, there exists so that if for , then . Now, choose any , any , and any . Use non-uniform specification to choose which -shadows for iterates, with gaps . Then by definition of , -shadows for iterates, with gaps , proving the desired non-uniform specification at scale . ∎
If is expansive (with expansivity constant ), , and is a potential with partial sum variation bounds at scale , then there exists a constant so that for every , has partial sum variation bounds at scale .
Choose and as in the theorem, and any . By expansivity, there exists so that if for , then . For the second inequality, choose any and any for which for . Then by definition of , for , and so
Taking the supremum over such pairs completes the proof.
If is a factor map, then for all there exists such that if has non-uniform specification (transitivity) with gap bounds at scale , then has non-uniform specification (transitivity) with gap bounds at scale .
We give a proof only for non-uniform specification, as the proof for non-uniform transitivity is trivially similar. Given , use uniform continuity of to choose so that . Now, given any and , , , , choose arbitrary and use non-uniform specification of at scale to find which -shadows for iterates with gaps . It is immediate that -shadows for iterates with gaps , completing the proof. ∎
The following corollary follows immediately by using Lemma 3.1.
If is a factor map, and are expansive (with expansivity constants and respectively), and has non-uniform specification (transitivity) with gap bounds at scale where (), then there exists with () for which has non-uniform specification (transitivity) with gap bounds at scale .
We now move to products, with the goal of proving Corollary 1.6. All products of metric spaces will be endowed with the metric defined by . The proofs of the following results are left to the reader.
If , are expansive dynamical systems with the same expansivity constant and which have non-uniform specification with gap bounds and (at scale ) respectively, then is an expansive dynamical system with expansivity constant which has non-uniform specification with gap bounds at scale .
If and are expansive dynamical systems with the same expansivity constant and and are potentials with partial sum variation bounds and (at scale ) respectively, then the potential on defined by has partial sum variation bounds at scale .
If and are expansive dynamical systems with non-uniform gap specification with gap bounds and respectively, and if and are potentials with partial sum variation bounds and respectively (all at scales equal to the relevant expansivity constants), and if there exists a sequence so that , and the potential on is defined by , then satisfies the hypotheses of Theorem 1.2.
The reason no assumption need be made on the equality of expansivity constants in Corollary 3.8 is that one can always render them equal (say to ) by normalizing the metrics.
We can now apply the following result of Ledrappier.
Theorem 3.10 ().
If is an expansive dynamical system, is a potential, and has a unique equilibrium state for the potential defined by , then has a unique equilibrium state for , and is a K-system.
Corollary 1.6 is now implied by Corollary 3.8 and Theorem 3.10. Corollary 1.8 follows as well: the closure under expansive factors comes from Corollary 3.4, Hölder potentials are Bowen (i.e. have bounded partial sum variation bounds) for subshifts, and positive entropy of the unique equilibrium state comes from Corollary 1.6 since K-systems have positive entropy.
Finally, we need some technical results about behavior of separated sets/partition functions under changes of scale. The proof of the following lemma is motivated by arguments from .
If is expansive (with expansivity constant ) and , then there exists a constant so that for every , every -separated set can be written as a disjoint union of sets which are each -separated.
Choose and as in the theorem. By expansiveness, there exists so that if for , then . Since is a homeomorphism, there exists so that if , then for . Take any partition of by sets of diameter less than .
Now, choose any and -separated set . For each , define . We claim that each is -separated, which will complete the proof for . To see this, fix any and any . Since and is -separated, there exists so that . Since , , and so for . Similarly, since , , and so for . Therefore, . However, by definition of , this means that there exists so that , completing the proof.
If is expansive (with expansivity constant ) and , then for any potential ,
where is as in Lemma 3.11.
Consider an -separated set for which . Then by Lemma 3.11, we can write , where each is -separated. Then
Under the hypotheses of Theorem 1.2, there exists a sequence so that . Then if we take , , so satisfies the conclusion of the lemma.
Similarly, under the hypotheses of Theorem 1.3, , so also, so taking satisfies the conclusion of the lemma. ∎
A sequence satisfying the conclusion of Lemma 4.1 is called an anchor sequence for and .
The next tools that we’ll need are some upper bounds on the partition function for expansive systems with non-uniform specification, which generalize the well-known upper bound of of times a constant when specification is assumed (e.g. Lemma 3 of ).
If is an expansive dynamical system (with expansivity constant ) and non-uniform specification with gap bounds at scale , and is a potential with partial sum variation bounds at scale , then for all ,
Suppose that and are as in the theorem, denote , and fix any . Then, choose any and an -separated set so that . Then, for any , we can use non-uniform specification to choose a point which -shadows for iterates, with gaps . Then
We note that the set is -separated by definition, and so
Taking logarithms, dividing by , and letting yields
(The first equality comes from Lemma 2.9.) Now, solving for completes the proof.
If satisfies the hypotheses of Theorem 1.2 and is an anchor sequence, then for every , there exists so that for all and ,
By definition of anchor sequence, we can take and to be gap bounds and partial sum variation bounds at scale , and can choose so that for , . Then, for any such and , Theorem 4.3 (along with monotonicity of ) implies
We now prove a somewhat similar bound under the assumption of non-uniform transitivity, which requires information about for all large rather than a single value.
If is an expansive dynamical system (with expansivity constant ) and non-uniform transitivity with gap bounds at scale , is a potential with partial sum variation bounds , and satisfy for all , then for all ,
where are constants depending only on and .
Suppose that , , , , and are as in the theorem, and fix any . We use to denote . For every , choose a -separated set for which . We will use non-uniform transitivity to give lower bounds on for a recursively defined sequence . Define , and for define . Note that all , and so