Unique equilibrium states for flows and homeomorphisms with nonuniform structure
Abstract.
Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker nonuniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and nonuniformly hyperbolic settings.
1. Introduction
Let be a compact metric space and a continuous flow on . Given a potential function , we study the question of existence and uniqueness of equilibrium states for – that is, invariant measures which maximize the quantity . We also study the same question for homeomorphisms . This problem has a long history [rB74, rB75, HK82, DKU90, oS99, IT10, PSZ, Pa15, CP16] and is connected with the study of global statistical properties for dynamical systems [dR76, yK90, PP90, BSS02, CRL11, vC15].
For homeomorphisms, Bowen showed [rB74] that has a unique equilibrium state whenever is an expansive system with specification and satisfies a certain regularity condition (the Bowen property). Bowen’s method was adapted to flows by Franco [eF77]. Previous work by the authors established similar uniqueness results for shift spaces with a broad class of potentials [CT, CT2], and for nonsymbolic discretetime systems in the case [CT3]. In this paper, we consider potential functions satisfying a nonuniform version of the Bowen property in both the discrete and continuoustime case.
While we do not explore applications of this theory in this paper, we emphasize that the results are developed with a view to novel applications in the setting of smooth dynamical systems beyond uniform hyperbolicity. In particular, the main theorems of this paper are applied to diffeomorphisms with weak forms of hyperbolicity in [CFT] and to geodesic flows in nonpositive curvature in [BCFT].
We review the main points of our techniques for proving uniqueness of equilibrium states for maps, referring the reader to [CT2, CT3] for details. Our approach is based on weakening each of the three hypotheses of Bowen’s theorem: expansivity, the specification property, and regularity of the potential. Instead of asking for specification and regularity to hold globally, we ask for these properties to hold on a suitable collection of orbit segments . Instead of asking for expansivity to hold globally, we ask that all measures with large enough free energy should observe expansive behavior.
These ideas lead naturally to a notion of orbit segments which are obstructions to specification and regularity, and measures which are obstructions to expansivity. The guiding principle of our approach is that if these obstructions have less topological pressure than the whole space, then a version of Bowen’s strategy can still be developed. Some of the main points are as follows:

For a discretetime dynamical system, we work with , which we think of as the space of orbit segments by identifying with . At the heart of our approach is the concept of a decomposition for . We ask for specification and regularity to hold on a collection of ‘good’ orbit segments , while the collections of are thought of as ‘bad’ orbit segments which are obstructions to specification and regularity. We ask that any orbit segment can be decomposed as a ‘good core’ that is preceded and succeeded by elements of and , respectively. More precisely, for any , there are numbers so that , and
The choice of the decomposition depends on the setting of any given application, and the dynamics of the situation are encoded in this choice.

We define a natural version of topological pressure for orbit segments, and we require that the topological pressure of , which we think of as the pressure of the obstructions to specification and regularity, is less than that of the whole space.

The positive expansivity property introduced in [CT3] is that for small , for almost every , for any ergodic with , where is a constant less than . We think of the smallest so that this is true as the entropy of obstructions to expansivity.
Under these hypotheses, our strategy is then inspired by Bowen’s: his main idea was to construct an equilibrium state with the Gibbs property, and to show that this rules out the existence of a mutually singular equilibrium state. We obtain a certain Gibbs property which only applies to orbit segments in , and then we have to work to show that this is still sufficient to prove uniqueness of the equilibrium state.
The above strategy was carried out in [CT, CT2, CT3] under the assumption that either is a shift space or . In this paper, we work in the setting of a continuous flow or homeomorphism on a compact metric space, and a continuous potential function. This necessitates several new developments, which we now describe. For homeomorphisms and flows, we develop a theory for potential functions which are regular only on ‘good’ orbit segments. The lack of global regularity introduces fundamental technical difficulties not present in the classical theory or the symbolic setting. For flows, which are the main focus of this paper, we work with the space , where the pair is thought of as the orbit segment . The main points addressed in this paper are:

Our potentials are not regular on the whole space, and this forces us to introduce and control nonstandard ‘twoscale’ partition sums throughout the proof (see §2.1).

For flows, expansivity issues can be subtle and require new ideas beyond the discretetime case. We introduce the notion of almost expansivity for a flowinvariant ergodic measure (§2.5), adapting a discretetime version of this definition which was used in [CT3]. We also introduce the notion of almost entropy expansivity (§3.1) for a mapinvariant ergodic measure. This is a natural analogue of entropy expansivity [rB72], adapted to apply to almost every point in the space. Measures which are almost expansive for the flow are almost entropy expansive for the time map. Almost entropy expansivity plays a crucial role in our proof via Theorem 3.2, a general ergodic theoretic result that strengthens [rB72, Theorem 3.5].

Adapting the framework introduced in [CT3] to the case of flows requires careful control of small differences in transition times, particularly in Lemma LABEL:lem:multiplicity.

The unique equilibrium state we construct admits a weak upper Gibbs bound, which in many cases we use to obtain the upper bound from the level2 large deviations principle, using results of Pfister and Sullivan (see §LABEL:sec:LDP).
We now state a version of our main result, which should be understood as a formalization of the strategy described previously. We introduce our notation, referring the reader to §2 for precise definitions: is the standard topological pressure; the quantity is the largest free energy of an ergodic measure which observes nonexpansive behavior; the specification property and Bowen property are versions of the classic properties which apply only on rather than globally; the expression is the topological pressure of the obstructions to specification and regularity.
Theorem A.
Let be a continuous flow on a compact metric space, and a continuous potential function. Suppose that and that admits a decomposition with the following properties:

has the weak specification property;

has the Bowen property on ;

.
Then has a unique equilibrium state.
In fact, we will prove a slightly more general result, of which Theorem A is a corollary. The more general version, Theorem 2.9, applies under slightly weaker versions of our hypotheses, which we discuss and motivate in §2.6.
We also develop versions of our results that apply for homeomorphisms. These discretetime arguments are analogous to, and easier than, the flow case, so we just outline the proof, highlighting any differences with the flow case. Our main results for homeomorphisms are Theorem LABEL:thm:mapssimple, which is the analogue of Theorem A, and Theorem LABEL:thm:mapsD, which is the analogue for homeomorphisms of Theorem 2.9. Finally, in Theorem LABEL:thm:ldp, we establish the upper level2 large deviations principle for the unique equilibrium states provided by Theorem LABEL:thm:mapssimple.
Structure of the paper
We collect our definitions, particularly for flows, in §2. Our main results for flows are proved in §§3–4. Our main results for maps are proved in §§LABEL:sec:maps–LABEL:sec:mapspf. In §LABEL:sec:LDP, we prove the large deviations results of Theorem LABEL:thm:ldp. In §LABEL:sec:aee, we prove Theorem 3.2, which is a selfcontained result about measuretheoretic entropy for almost entropy expansive measures.
2. Definitions
In this section we give the relevant definitions for flows; the corresponding definitions for maps are given in §LABEL:sec:maps.
2.1. Partition sums and topological pressure
Throughout, will denote a compact metric space and will denote a continuous flow on . We write for the set of Borel invariant probability measures on . Given , , and we define the Bowen metric
(2.1) 
and the Bowen balls
(2.2)  
Given , , and , we say that is separated if for every distinct we have . Writing , we view as the space of finite orbit segments for by associating to each pair the orbit segment . Our convention is that is identified with the empty set rather than the point . Given and we write .
Now we fix a continuous potential function . Given a fixed scale , we use to assign a weight to every finite orbit segment by putting
(2.3) 
In particular, . The general relationship between and is that
(2.4) 
where .
Given and , we consider the partition function
(2.5) 
We will often suppress the function from the notation, since it is fixed throughout the paper, and simply write . When is the entire system, we will simply write or . We call a separated set that attains the supremum in (2.5) maximizing for . We are only guaranteed the existence of such sets when , since otherwise may not be compact.
The pressure of on at scales is given by
(2.6) 
Note that is monotonic in both and , but in different directions; thus the same is true of . Again, we write in place of to agree with more standard notation, and we let
(2.7) 
When is the entire space of orbit segments, the topological pressure reduces to the usual notion of topological pressure on the entire system, and we write in place of , and in place of . The variational principle for flows [BR75] states that , where is the usual measuretheoretic entropy of the time map of the flow. A measure achieving the supremum is called an equilibrium state.
Remark 2.1.
The most obvious definition of partition function would be to take so that the weight given to each orbit segment is determined by the integral of the potential function along that exact orbit segment, rather than by nearby ones. To match more standard notation, we often write in place of . The partition sums arise throughout this paper, particularly in §4.1 and §LABEL:sec:adapted. The relationship between the two quantities can be summarised as follows.

If is expansive at scale , then .

If is Bowen at scale , then the two pressures above are equal, and moreover the ratio between and is bounded away from and .

In the absence of regularity or expansivity assumptions, we have the relationship
and thus . By continuity of , this establishes that as , but does not give us the conclusions of (1) or (2).
Because our versions of expansivity and the Bowen property do not hold globally, we are in case (3) above, so a priori we cannot replace with in the proofs.
Remark 2.2.
We can restrict to separated sets of maximal cardinality in the definition of pressure: these always exist, even when is noncompact, since the possible values for the cardinality are finite (by compactness of ). If were not of maximal cardinality, we could just add in another point, which would increase the partition sum (2.5). Furthermore, a separated set of maximal cardinality is spanning in the sense that . If this were not so then we could add another point to and increase the cardinality.
2.2. Decompositions
We introduce the notion of a decomposition for a subcollection of the space of orbit segments.
Definition 2.3.
A decomposition for consists of three collections and three functions such that for every , the values , , and satisfy , and
(2.8) 
If , we say that is a decomposition for . Given a decomposition and , we write for the set of orbit segments for which and .
We make a standing assumption that to allow for orbit segments to be decomposed in ‘trivial’ ways; for example, can belong ‘purely’ to one of the collections , , or or can transition directly from to – note that formally the symbols are identified with the empty set. This is implicit in our earlier work [CT, CT2, CT3].
We will be interested in decompositions where has specification, has the Bowen property on , and carries smaller pressure than the entire system. In the case of flows, a priori we must replace the collections and that appear in the decomposition with a related and slightly larger collection , where given we write
(2.9) 
Passing from to ensures that the decomposition is well behaved with respect to replacing continuous time with discrete time. This issue occurs in Lemma LABEL:lem:manyinG.
2.3. Specification
We say that has weak specification at scale if there exists such that for every there exists a point and a sequence of “gluing times” with such that for and , we have (see Figure 1)
(2.10) 
We say that has weak specification at scale with maximum gap size if we want to declare a value of that plays the role described above. We say that has weak specification if it has weak specification at every scale .
Remark 2.4.
We often write (W)specification as an abbreviation for weak specification. Furthermore, since (W)specification is the only version of the specification property considered in this paper, we henceforth use the term specification as shorthand for this property.
Intuitively, (2.10) means that there is some point whose orbit shadows the orbit of for time , then after a “gap” of length at most , shadows the orbit of for time , and so on. Note that is the time spent for the orbit to shadow the orbit segments up to . Note that we differ from Franco [eF77] in allowing to take any value in , not just one that is close to . This difference is analogous in the discrete time case to the difference between (S)specification where we take the transition times exactly , or (W)specification where the transition times are bounded above by . Franco also asks that the shadowing orbit can be taken to be periodic, and that the gluing time does not depend on any of the orbit segments with .
We can weaken the definition of specification so that it only applies to elements of that are sufficiently long. This gives us some useful additional flexibility which we exploit in Lemma 2.10.
Definition 2.5.
We say that has tail (W)specification at scale if there exists so that has weak specification at scale ; i.e. the specification property holds for the collection of orbit segments
We also sometimes write “ has (W)specification at scale for ” to describe this property.
2.4. The Bowen property
The Bowen property was first defined for maps in [rB74], and extended to flows by Franco [eF77]. We give a version of this definition for a collection of orbit segments .
Definition 2.6.
Given , a potential has the Bowen property on at scale if there exists so that
(2.11) 
We say has the Bowen property on if there exists so that has the Bowen property on at scale .
In particular, we say that has the Bowen property if has the Bowen property on ; this agrees with the original definition of Bowen and Franco. This dynamicallydefined regularity property is central to Bowen’s proof of uniqueness of equilibrium states. For a uniformly hyperbolic system, every Hölder potential has the Bowen property. This is no longer true in nonuniform hyperbolicity; for example, the geometric potential for the Manneville–Pomeau map is a natural potential which is Hölder but not Bowen. Asking for the Bowen property to hold on a collection rather than globally allows us to deal with nonuniformly hyperbolic systems where one only expects this kind of regularity to hold for those orbit segments which experience a definite amount of hyperbolicity, and where it may not be known whether natural potentials such as the geometric potential are Hölder [CFT, BCFT].
We sometimes call the distortion constant for the Bowen property. Note that if has the Bowen property at scale on with distortion constant , then for any , has the Bowen property at scale on with distortion constant given by .
2.5. Almost expansivity
Given and , consider the set
(2.12) 
which can be thought of as a twosided Bowen ball of infinite order for the flow. Note that is compact for every .
Expansivity for flows was defined by Bowen and Walters; their definition, details of which can be found in [BW72], implies that for every , there exists such that
(2.13) 
for every . Since points on a small segment of orbit always stay close for all time, (2.13) essentially says that the set is the smallest possible. Thus, we declare the set of nonexpansive points to be those where (2.13) fails. We want to consider measures that witness expansive behaviour, so we declare an almost expansive measure to be one that gives zero measure to the nonexpansive points. This is the content of the next definition.
Definition 2.7.
Given , the set of nonexpansive points at scale for a flow is the set
We say that an invariant measure is almost expansive at scale if .
A measure which is almost expansive at scale gives full measure to the set of points for which there exists for which (2.13) holds. We remark that in contrast to the BowenWalters definition, we allow to be large or even unbounded. Furthermore, our hypotheses do not preclude the existence of fixed points for the flow; for expansive flows, fixed points can only be isolated [BW72, Lemma 1] and can hence be disregarded.
The following definition gives a quantity which captures the largest possible free energy of a nonexpansive ergodic measure.
Definition 2.8.
Given a potential , the pressure of obstructions to expansivity at scale is
We define a scalefree quantity by
Note that is nonincreasing as , which is why the limit in the above definition exists. It is essential that the measures in the first supremum are ergodic. If we took this supremum over invariant measures, and a nonexpansive measure existed, we would include measures that are a convex combination of a nonexpansive measure and a measure with large free energy, so the supremum would equal the topological pressure.
2.6. Main results for flows
Theorem 2.9.
Let be a continuous flow on a compact metric space, and a continuous potential function. Suppose there are with such that and there exists which admits a decomposition with the following properties:

For every , has tail (W)specification at scale ;

has the Bowen property at scale on ;

.
Then has a unique equilibrium state.
These hypotheses weaken those of Theorem A in two main directions.

The hypotheses of Theorem A require knowledge of the system at all scales: in particular, the specification condition 1 in Theorem A requires specification to hold at every scale . Here, we require a specification property to be verified only at a fixed scale , and all other hypotheses to be verified at a larger fixed scale . An example where this is useful is the Bonatti–Viana family of diffeomorphisms, where in [CFT] we are able to verify the discretetime version of these hypotheses at suitably chosen scales, but establishing them for arbitrarily small scales is difficult, and perhaps impossible.
We make a few more remarks on these hypotheses. By Remark 2.1, we can guarantee 3 by checking the bound
(2.14) 
We do not claim that the relationship is sharp, but we do not expect that it can be significantly improved using these methods. The number does not have any special significance but it is unavoidable that we control the Bowen property and expansivity at a larger scale than where specification is assumed.
If we assume the hypotheses of Theorem A, we can verify the hypotheses of Theorem 2.9 by taking , and any suitably small with . The only hypothesis which is not immediate to verify from the hypotheses of Theorem A is 1, and this is verified by the following lemma.
Lemma 2.10.
Proof.
Given , let be such that implies that for every . (Positivity of follows from continuity of the flow and compactness of .) Now let be such that has specification at scale . Given any with , we must have . Thus if is any collection of orbit segments in with , then there are and such that . Since we can use the specification property on to get an orbit that shadows each to within (with transition times at most ). By our choice of , this orbit shadows each to within (with transition times at most ). We conclude that has tail specification at scale . ∎
3. Weak expansivity and generating for adapted partitions
In this section, we develop some general preparatory results on generating properties of partitions in the presence of weak expansivity properties.
3.1. Almost entropy expansivity
It is well known that the time map of an expansive flow is entropy expansive. We develop an analogue of entropy expansivity for measures called almost entropy expansivity, which has the property that if is almost expansive for a flow, then it is almost entropy expansive for the time map of the flow. This property plays an important role in our proof, as entropy expansivity does for Franco, and is key to obtaining a number of results on generating for partitions.
Let be a compact metric space and a homeomorphism. Let be an ergodic invariant Borel probability measure. For a set , let denote the (upper capacity) entropy of . That is, corresponds to for as defined in §LABEL:sec:maps, which is the natural analogue for maps of (2.5)–(2.7).
Given , consider the set
(3.1) 
Recall from [rB72] that the map is said to be entropy expansive if for every . We will need the following weaker notion.
Definition 3.1.
We say that is almost entropy expansive at scale (in the metric ) with respect to if for a.e. .
Our notation emphasizes the role of the metric because later in the paper we will need to use this notion relative to various metrics . Bowen proved that if is entropy expansive at scale , then every partition with diameter smaller than has . This result was obtained as an immediate consequence of the main part of [rB72, Theorem 3.5], which shows that for any and any partition with , we have
(3.2) 
Clearly, is entropy expansive if the supremum is 0. Similarly, one sees immediately that is almost entropy expansive at scale if and only if the essential supremum
(3.3) 
vanishes, and we strengthen Bowen’s result by showing that one can use the essential supremum in (3.2). The following theorem is proved in §LABEL:sec:aee.
Theorem 3.2.
Let be a compact metric space and a homeomorphism. Let be an ergodic invariant Borel probability measure. If is any partition with in the metric , then
(3.4) 
In particular, if is almost entropy expansive at scale , then every partition with diameter smaller than has .
To apply Theorem 3.2 in the setting of our main results, we first relate almost expansivity for the flow with almost entropy expansivity for the time map of the flow.
Proposition 3.3.
If is almost expansive at scale , then is almost entropy expansive (at scale in the metric ) with respect to the time map .
Proof.
It is immediate from the definitions that . Thus, if is almost expansive for , then for a.e. , the set is contained in for some . Fix such an and let . In what follows, we will show that . This shows that , and since this argument applies to almost every , it follows that is almost entropy expansive for .
So, it just remains to show that the entropy of the finite orbit segment is with respect to . Let be sufficiently small that for all (this is possible by continuity of the flow and compactness of the space). Given , fix large enough such that . Let , and note that for all and all . Thus, for every , the set is spanning under for . It follows that, in the metric , is spanning under , which gives . ∎
The following proposition, which plays a similar role as [CT3, Proposition 2.6], is a consequence of Theorem 3.2 and Proposition 3.3.
Proposition 3.4.
If is almost expansive at scale and is a finite measurable partition of with diameter less than in the metric for some , then the timet map satisfies .
3.2. Adapted partitions and results on generating
We extend Proposition 3.4 to some useful results on generating using the notion of an adapted partition. This terminology was introduced in [CT3], although the concept goes back to Bowen [rB73].
Definition 3.5.
Let be a separated set of maximal cardinality. A partition of is adapted to if for every there is such that .
Adapted partitions exist for any separated set of maximal cardinality since the sets are disjoint and the sets cover .
Lemma 3.6.
If is almost expansive at scale , and is an adapted partition for a separated set of maximal cardinality, then .
Proof.
For any , there exists so that ; this shows that in the metric . By Proposition 3.4, we have . ∎
The proof of the following proposition requires both Lemma 3.6 and a careful use of the almost expansivity property to take a crucial step of replacing a term of the form with .
Proposition 3.7.
If , then for every .
Proof.
Given an ergodic , write for convenience. We prove the proposition by showing that for every ergodic with . We do this by relating both and to an adapted partition. In order to carry this out we first introduce a technical lemma that will be used both here and in the proof of Lemma LABEL:lem:posfores.
Given a finite partition and an invariant measure , for each with we define a function by
(3.5) 
Given , write .
Lemma 3.8.
Suppose is almost expansive at scale , and let . Let be an adapted partition for a maximizing separated set for . Let be a union of elements of . Then for every we have
where , and is as in (3.5).
Proof.
Abramov’s formula [lA59] gives for all , and Lemma 3.6 gives , so
Let , and write . Breaking up the above sum and normalizing, we have
Recall that for nonnegative with and arbitrary we have ; the conclusion of Lemma 3.8 follows by applying this to the first sum with , , and the second sum with , . ∎
Now we return to the proof of Proposition 3.7. Let be as in the hypothesis, and let be ergodic with , so that is almost expansive at scale . Fix . Given , consider the set
We have , so there is such that .
Now, we fix , and for an arbitrary , we write
For any , we have
In particular, given as above and , we see that is an open set which contains , so there is so that . Now, for , let
We have , so we can fix sufficiently large so that . We now pass to the set of points whose orbits spend a large proportion of time in . Given , consider the set
and note that by the Birkhoff ergodic theorem. Take large enough that for all . The following lemma gives us a regularity property for the potential for points in .
Lemma 3.9.
Given and , we have
(3.6) 
Proof.
Let and choose such that (here we use that ). Define iteratively as follows: let , and then given , choose any , and put .
It follows from the definition of and properties of that

for every ;

; and

for every .
Since , the third property gives for some ; see Figure 2. Thus
The first two properties give
and putting it all together we have