Equilibrium states in dynamical systems via geometric measure theory
EEQUILIBRIUM STATES IN DYNAMICAL SYSTEMS VIA GEOMETRICMEASURE THEORY
VAUGHN CLIMENHAGA, YAKOV PESIN, AND AGNIESZKA ZELEROWICZ
Abstract.
Given a dynamical system with a uniformly hyperbolic (“chaotic”) attrac-tor, the physically relevant Sinai–Ruelle–Bowen (SRB) measure can be obtained as thelimit of the dynamical evolution of the leaf volume along local unstable manifolds. Weextend this geometric construction to the substantially broader class of equilibrium statescorresponding to H¨older continuous potentials; these states arise naturally in statisticalphysics and play a crucial role in studying stochastic behavior of dynamical systems. Thekey step in our construction is to replace leaf volume with a reference measure that isobtained from a Carath´eodory dimension structure via an analogue of the constructionof Hausdorff measure. In particular, we give a new proof of existence and uniquenessof equilibrium states that does not use standard techniques based on Markov partitionsor the specification property; our approach can be applied to systems that do not haveMarkov partitions and do not satisfy the specification property.
Contents
1. Introduction 12. Motivating examples 53. Equilibrium states and their relatives 84. Description of reference measures and main results 175. Carath´eodory dimension structure 256. Outline of proofs 29References 401.
Introduction
Systems with hyperbolic behavior. A smooth dynamical system with discrete time consists of a smooth manifold M – the phase space – and a diffeomorphism f : M → M .Each state of the system is represented by a point x ∈ M , whose orbit ( f n ( x )) n ∈ Z gives thetime evolution of that state. We are interested in the case when the dynamics of f exhibit hyperbolic behavior. Roughly speaking, this means that orbits of nearby points separateexponentially quickly in either forward or backward time; if the phase space is compact,this leads to the phenomenon popularly known as ‘chaos’. Date : October 26, 2018.2010
Mathematics Subject Classification.
Primary 37D35, 37C45; secondary 37C40, 37D20.V. C. is partially supported by NSF grants DMS-1362838 and DMS-1554794. Ya. P. and A. Z. arepartially supported by NSF grant DMS-1400027. a r X i v : . [ m a t h . D S ] O c t VAUGHN CLIMENHAGA, YAKOV PESIN, AND AGNIESZKA ZELEROWICZ
Hyperbolic behavior turns out to be quite common, and for such systems it is not feasibleto make specific forecasts of a single trajectory far into the future, because small initial errorsquickly grow large enough to spoil the prediction. On the other hand, one may hope tomake statistical predictions about the asymptotic behavior of orbits of f . A measurementof the system corresponds to a function ϕ : M → R ; the sequence ϕ , ϕ ◦ f , ϕ ◦ f , . . . represents the same observation made at successive times. When specific forecasts of ϕ ◦ f n are impossible, we can treat this sequence as a stochastic process and make predictionsabout its asymptotic behavior. For a more complete discussion of this point of view, see[ER85], [Ma˜n87, Chapter 1], and [Via97].1.2. Physical measures and equilibrium states.
To fully describe the stochastic pro-cess ( ϕ ◦ f n ) n ∈ Z , we need a probability measure µ on M that represents the likelihoodof finding the system in a given state at the present time. The measure f ∗ µ defined by (cid:82) ϕ d ( f ∗ µ ) = (cid:82) ϕ ◦ f dµ represents the distribution one unit of time into the future. An in-variant measure has µ = f ∗ µ , and hence µ = f n ∗ µ for all n , so the sequence of observationsbecomes a stationary stochastic process.In this paper we will consider uniformly hyperbolic systems , for which the tangent bundleadmits an invariant splitting T M = E u ⊕ E s such that E u is uniformly expanded and E s uniformly contracted by Df ; see § § M , or at least an invariant measure that is absolutely continuous with respectto volume. However, for dissipative systems such as the solenoid described in § Sinai–Ruelle–Bowen (SRB)measure , which we describe in §§ physically relevant ; it describesthe asymptotic statistical behavior of volume-typical trajectories.SRB measures can be constructed via the following “geometric approach”: let m benormalized Lebesgue measure (volume) for some Riemannian metric on M ; consider itsforward iterates f n ∗ m ; then average the first N of these and take a limit measure as N → ∞ .Another approach to SRB measures, which we recall in § thermodynamic for-malism , which imports mathematical tools from equilibrium statistical physics in order todescribe the behavior of large ensembles of trajectories. This program began in the late1950’s, when Kolmogorov and Sinai introduced the concept of entropy into dynamical sys-tems; see [Kat07] for a historical overview. Given a potential function ϕ : M → R , onestudies the equilibrium states associated to ϕ , which are invariant measures that maxi-mize the quantity h µ ( f ) + (cid:82) ϕ dµ , where h µ denotes the Kolmogorov–Sinai entropy. Themaximum value is called the topological pressure of ϕ and denoted P ( ϕ ).In the 1960s and 70s, it was shown by Sinai, Ruelle, and Bowen that for uniformlyhyperbolic systems, every H¨older continuous potential has a unique equilibrium state (see From the statistical physics point of view, the quantity E µ := − ( h µ ( f ) + (cid:82) ϕ dµ ) is the free energy ofthe system, so that an equilibrium state minimizes the free energy; see [Sar15, § QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 3
Section 3.1). Applying this result to the particular case of the geometric potential ϕ geo ( x ) = − log | det Df | E u ( x ) | , one has P ( ϕ geo ) = 0 and the equilibrium state is the SRB measuredescribed above; see § Different approaches to constructing equilibrium states.
There are two mainclassical approaches to thermodynamic formalism. The first uses
Markov partitions of themanifold M ; we recall the general idea in § specification property ,which we overview in § reference measure , which need not be invariant. For the physical SRBmeasure, this reference measure was Lebesgue; to extend this approach to other equilibriumstates, one must start by choosing a new reference measure. The definition of this referencemeasure, and its motivation and consequences, is the primary goal of this paper, and ourmain result can be roughly stated as follows: For every H¨older continuous potential ϕ , one can use the tools of geometric measuretheory to define a reference measure m ϕ for which the pushforwards f n ∗ m ϕ converge inaverage to the unique equilibrium state for ϕ . A precise statement of the result is given in §
4. An important motivation for this workis that the “geometric” approach can be applied to more general situations beyond theuniformly hyperbolic systems studied in this paper. For example, the geometric approachwas used in [CDP16] to construct SRB measures for some non-uniformly hyperbolic systems,and in [CPZ18] we use it to construct equilibrium states for some partially hyperbolicsystems. The first two approaches – Markov partitions and specification – have also beenextended beyond uniform hyperbolicity (see [CP17] for a survey of the literature), but theoverall theory in this generality is still very far from being complete, so it seems worthwhileto add another tool by developing the geometric approach as well.1.4.
Reference measures for general potentials.
In the geometric construction of thephysical SRB measure, one can take the reference measure to be either Lebesgue measure m on M or Lebesgue measure m W on any local unstable leaf W = V u loc ( x ). These leaves are d u -dimensional submanifolds of M that are tangent at each point to the unstable distribu-tion E u ( x ) ⊂ T x M ; they are expanded by the dynamics of f and have the property that f ( V u loc ( x )) ⊃ V u loc ( f ( x )) (see § W = V u loc ( x ),we will write m W or m ux for the leaf volume determined by the induced Riemannian metric.This has the following key properties.(1) m W = m ux is a finite nonzero Borel measure on W .(2) If W and W are local unstable leaves with non-trivial intersection, then m W and m W agree on the overlap. Here the determinant is taken with respect to any orthonormal bases for E u ( x ) and E u ( f ( x )). If themap f is of class of smoothness C α for some α >
0, then one can show that ϕ geo ( x ) is H¨older continuous. VAUGHN CLIMENHAGA, YAKOV PESIN, AND AGNIESZKA ZELEROWICZ (3) Under the dynamics of f , the leaf volumes scale by the rule(1.1) m uf ( x ) ( A ) = (cid:90) f − A | det Df | E u ( y ) | dm ux ( y ) . As mentioned in § ϕ geo ( x ) = − log | det Df | E u ( x ) | has P ( ϕ geo ) =0, so the integrand in (1.1) can be written as e P ( ϕ geo ) − ϕ geo ( y ) . In § ϕ we will construct on every local unstable leaf W = V u loc ( x ) a reference measure m C x satisfying similar properties to m ux , but with scaling rule (1.2) m C f ( x ) ( A ) = (cid:90) f − A e P ( ϕ ) − ϕ ( y ) dm C x ( y ) . The superscript C is shorthand for a Carath´eodory dimension structure determined by thepotential ϕ and a scale r >
0; see § P ( ϕ ) and m C x areanalogous to the definitions of Hausdorff dimension and Hausdorff measure, respectively,but take the dynamics into account. Recall that the latter definitions involve covers byballs of decreasing radius; the modification to obtain our quantities involves covering bydynamically defined balls, as explained in § Some history.
The idea of constructing dynamically significant measures for uni-formly hyperbolic maps by first finding measures on unstable leaves with certain scalingproperties goes at least as far back as work of Sinai [Sin68], which relies on Markov par-titions. For uniformly hyperbolic dynamical systems with continuous time (flows) and thepotential ϕ = 0 the corresponding equilibrium state, which is the measure of maximal en-tropy, was obtained by Margulis [Mar70]; he used a different construction of leaf measuresvia functional analysis of a special operator (induced by the dynamics) acting on the Ba-nach space of continuous functions with compact support on unstable leaves. These leafmeasures were studied further in [RS75, BM77].Hasselblatt gave a description of the Margulis measure in terms of Hausdorff dimension[Has89], generalizing a result obtained by Hamenst¨adt for geodesic flows on negativelycurved compact manifolds [Ham89]. In this geometric setting, where stable and unstableleaves are naturally identified with the ideal boundary of the universal cover, Kaimanovichobserved in [Kai90, Kai91] that these leaf measures could be identified with the measureson the ideal boundary introduced by Patterson [Pat76] and Sullivan [Sul79]. For geodesicflows in negative curvature, this approach was recently extended to nonzero potentials byPaulin, Pollicott, and Schapira [PPS15].For general hyperbolic systems and nonzero potential functions, families of leaf measureswith the appropriate scaling properties were constructed by Haydn [Hay94] and Leplaideur[Lep00], both using Markov partitions. The key innovation in the present paper is thatwe can construct these leaf measures directly, without using Markov partitions, by anapproach reminiscent of Hasselblatt’s from [Has89]. This requires us to interpret quantitiesin thermodynamic formalism by analogy with Hausdorff dimension, an idea which wasintroduced by Bowen for entropy [Bow73], developed by Pesin and Pitskel’ for pressure[PP84], and generalized further by Pesin [Pes88, Pes97]. Note that in general P ( ϕ ) (cid:54) = 0. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 5
Plan of the paper.
We describe some motivating examples in §
2, and give generalbackground definitions in §
3. These sections are addressed to a general mathematical audi-ence, and the reader who is already familiar with thermodynamic formalism for hyperbolicdynamical systems can safely skip to §
4, where we give the new definition of the refer-ence measures m C x and formulate our main results. In § §
6, and refer to [CPZ18] for complete details.
Acknowledgments.
This work had its genesis in workshops at ICERM (Brown Univer-sity) and ESI (Vienna) in March and April 2016, respectively. We are grateful to bothinstitutions for their hospitality and for creating a productive scientific environment.2.
Motivating examples
Before recalling general definitions about uniformly hyperbolic systems and their invari-ant measures in §
3, we describe three examples to motivate the idea of a ‘physical measure’.Our discussion here is meant to convey the overall picture and omits many details.2.1.
Hyperbolic toral automorphisms.
Our first example is the diffeomorphism f onthe torus T = R / Z induced by the linear action of the matrix L = (cid:0) (cid:1) on R , as shownin Figure 2.1. (cid:0) (cid:1) (mod Z ) E u E s Figure 2.1.
Lebesgue measure is preserved by f .This system is uniformly hyperbolic : The matrix L has two positive real eigenvalues λ s < < λ u , whose associated eigenspaces E s and E u give a Df -invariant splitting ofthe tangent bundle T T . The lines in R parallel to these subspaces project to f -invariantfoliations W s and W u of the torus.What about invariant measures? If p ∈ T has f n ( p ) = p , then the measure µ = n ( δ p + δ f ( p ) + · · · + δ f n − ( p ) ) is invariant. Every point with rational coordinates is periodicfor f , so this gives infinitely many f -invariant measures. Lebesgue measure is also invariantsince det Df = det L = 1. (This is far from a complete list, as we will see.)A measure µ is ergodic if every f -invariant function (every ϕ ∈ L ( µ ) with ϕ = ϕ ◦ f )is constant µ -almost everywhere. One can check easily that the periodic orbit measuresfrom above are ergodic, and with a little more work that Lebesgue measure is ergodic too. This can be proved either by Fourier analysis or by the more geometric
Hopf argument , see § VAUGHN CLIMENHAGA, YAKOV PESIN, AND AGNIESZKA ZELEROWICZ
Birkhoff’s ergodic theorem says that if µ is ergodic, then µ -almost everywhere orbit hasasymptotic behavior controlled by µ . More precisely, we say that the basin of attraction for µ is the set of initial conditions satisfying a law of large numbers governed by µ forcontinuous observables:(2.1) B µ = (cid:26) x ∈ T : 1 n n − (cid:88) k =0 ϕ ( f k x ) n →∞ −−−→ (cid:90) ϕ dµ for all ϕ ∈ C ( T , R ) (cid:27) . The ergodic theorem says that if µ is ergodic, then µ ( B µ ) = 1.For the periodic orbit measures, this says very little, since it leaves open the possibilitythat the measure µ only controls the asymptotic behavior of finitely many orbits. ForLebesgue measure m , however, this says quite a lot: m governs the statistical behavior ofLebesgue-almost every orbit, and in particular, a point chosen at random with respect toany volume form on T has a trajectory whose asymptotic behavior is controlled by m .This is the sense in which Lebesgue measure is the ‘physically relevant’ invariant measure,and we make the following definition. Definition 2.1.
An invariant measure µ for a diffeomorphism f is a physical measure ifits basin B µ has positive volume. Smale–Williams solenoid.
From Birkhoff’s ergodic theorem, we see that if µ is anergodic invariant measure that is equivalent to a volume form, then that volume form givesfull weight to the basin B µ , and so a volume-typical trajectory has asymptotic behaviorcontrolled by µ .The problem now is that there are many examples for which no such µ exists. Onesuch is the Smale–Williams solenoid studied in [Sma67, § I.9] and [Wil67]; see also [PC09,Lecture 29] for a gentle introduction and further discussion. This is a map from the opensolid torus U into itself. Abstractly, the solid torus is the direct product of a disc and acircle, so that one may use coordinates ( x, y, θ ) on U , where x and y give coordinates onthe disc and θ is the angular coordinate on the circle. Define a map f : U → U by(2.2) f ( x, y, θ ) := ( x + cos θ, y + sin θ, θ ) . Figure 2.2 shows two iterates of f , with half of the original torus for reference. Figure 2.2.
No absolutely continuous invariant measure.Every invariant measure is supported on the attractor
Λ = (cid:84) n ≥ f n ( U ), which has zerovolume. In particular, there is no invariant measure that is absolutely continuous with In fact B µ is infinite, being a union of leaves of the stable foliation W s . Recall that two measures µ and ν are equivalent if µ (cid:28) ν and ν (cid:28) µ , in which case we write µ ∼ ν . QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 7 respect to volume. Nevertheless, it is still possible to find an invariant measure that is‘physically relevant’ in the sense given above. To do this, first observe that since thesolenoid map f contracts distances along each cross-section D × { θ } , any two points inthe same cross-section have orbits with the same (forward) asymptotic behavior: given aninvariant measure µ , the basin B µ is a union of such cross-sections. Figure 2.3.
A cross-section of the attractor.This fact suggests that we should look for an invariant measure that is absolutely contin-uous in the direction of the circle coordinate θ , which is expanded by f . To construct sucha measure, observe that each cross-section intersects the images f n ( U ) in a nested sequenceof unions of discs, as shown in Figure 2.3, so the attractor Λ intersects this cross-section ina Cantor set. Thus Λ is locally the direct product of an interval in the expanding directionand a Cantor set in the contracting directions. Let m u be Lebesgue measure on the circle,and let µ be the measure on Λ that projects to m u and gives equal weight to each of the 2 n pieces at the n th level of the Cantor set construction in Figure 2.3. One can show withouttoo much difficulty that µ is invariant and ergodic, and that moreover B µ has full volume inthe solid torus U . Thus even though µ is singular, it is still the physically relevant invariantmeasure due to its absolute continuity in the expanding direction.2.3. Smale’s horseshoe.
Finally, we recall an example for which no physical measureexists – the horseshoe introduced by Smale in the early 1960s; see [Sma67, § I.5] and [PC09,Lecture 31] for more details, see also [Sma98] for more history. Consider a map f : R → R which acts on the square R := [0 , as shown in Figure 2.4: first the square is contractedvertically by a factor of α < / β >
2; then it isbent and positioned so that f ( R ) ∩ R consists of two rectangles of height α and length 1. β αR f ( R ) f ( R ) f ( R ) f ( R ) R R R R f ( R ) Figure 2.4.
No physical measure.Observe that a part of the square R is mapped to the complement of R . Consequently, f is not defined on the whole square R , but only on the union of two vertical strips in VAUGHN CLIMENHAGA, YAKOV PESIN, AND AGNIESZKA ZELEROWICZ R . The set where f is defined is the union of four vertical strips, two inside each of theprevious ones, and so on; there is a Cantor set C u ⊂ [0 ,
1] such that every point outside C u × [0 ,
1] can be iterated only finitely many times before leaving R . In particular, every f -invariant measure has B µ ⊂ C u × [0 , B µ is Lebesgue-null, so there is nophysical measure.Note that the argument in the previous paragraph did not consider the stable (vertical)direction at all. For completeness, observe that there is a Cantor set C s ⊂ [0 ,
1] such that (cid:84) n ≥ f n ( R ) = [0 , × C s , and that the maximal f -invariant set Λ := (cid:84) n ∈ Z f n ( R ) is a directproduct C u × C s .2.4. Main ideas.
The three examples discussed so far have certain features in common,which are representative of uniformly hyperbolic systems.First: every invariant measure lives on a compact invariant set Λ that is locally thedirect product of two sets, one contracted by the dynamics and one expanded. For thehyperbolic toral automorphism (cid:0) (cid:1) , Λ = T and both of these sets were intervals; forthe solenoid, there was an interval in the expanding direction and a Cantor set in thecontracting direction; for the horseshoe, both were Cantor sets.Second: the physically relevant invariant measure (when it existed) could also be ex-pressed as a direct product. For the hyperbolic toral automorphism, it was a product ofLebesgue measure on the two intervals. For the solenoid, it was a product of Lebesguemeasure on the interval (the expanding circle coordinate) and a ( , ) -Bernoulli measure on the contracting Cantor set.Third, and most crucially for our purposes: in identifying the physical measure, it isenough to look at how invariant measures behave along the expanding (unstable) direction .We will make this precise in § conditional measures , and this ideawill motivate our main construction in § n → + ∞ . If we would instead consider theasymptotics as n → −∞ , then the roles of stable and unstable objects would be reversed.We should also stress an important difference between the case when the invariant set Λ isan attractor (as in the second example) and the case when it is a Cantor set (as in the thirdexample): in the former case the trajectories that start near Λ exhibit chaotic behavior forall time t > persistent chaos ), while in the latter case thechaotic behavior occurs for a limited period of time whenever the trajectory passes by in avicinity of Λ (the phenomenon known as intermittent chaos ).3. Equilibrium states and their relatives
Hyperbolic sets.
Now we make our discussion more precise and more general. Weconsider a smooth Riemannian manifold M and a C α diffeomorphism f : M → M , andrestrict our attention to the dynamics of f on a locally maximal hyperbolic set . We recallhere the basic definition and most relevant properties, referring the reader to the book ofKatok and Hasselblatt [KH95, Chapter 6] for a more complete account. In what follows itis useful to keep in mind the three examples discussed above. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 9 A hyperbolic set for f is a compact set Λ ⊂ M with f (Λ) = Λ such that for every x ∈ Λ, the tangent space admits a decomposition T x M = E s ( x ) ⊕ E u ( x ) with the followingproperties.(1) The splitting is Df -invariant: Df x ( E σ ( x )) = E σ ( f x ) for σ = s, u .(2) The stable subspace E s ( x ) is uniformly contracting and the unstable subspace E u ( x )is uniformly expanding: there are constants C ≥ < χ < n ≥ v s,u ∈ E s,u ( x ), we have (cid:107) Df n v s (cid:107) ≤ Cχ n (cid:107) v s (cid:107) and (cid:107) Df − n v u (cid:107) ≤ Cχ n (cid:107) v u (cid:107) . Replacing the original Riemannian metric with an adapted metric , we can (andwill) take C = 1.In the case Λ = M the map f is called an Anosov diffeomorphism .Of course there are some diffeomorphisms that do not have any hyperbolic sets (thinkof isometries), but it turns out that a very large class of diffeomorphisms do, including theexamples from the previous section. These examples also have the property that Λ is locallymaximal , meaning that there is an open set U ⊃ Λ for which any invariant set Λ (cid:48) ⊂ U iscontained in Λ; in other words, Λ = (cid:84) n ∈ Z f n ( U ). In this case every C -perturbationof f also has a locally maximal hyperbolic set contained in U ; in particular, the set ofdiffeomorphisms possessing a locally maximal hyperbolic set is open in the C -topology.A number of properties follow from the definition of a hyperbolic set. First, the subspaces E s,u ( x ) depend continuously on x ∈ Λ; in particular, the angle between them is uniformlyaway from zero. In fact, since f is C α , the dependence on x is H¨older continuous:(3.1) ρ ( E s,u ( x ) , E s,u ( y )) ≤ Kd ( x, y ) β , where ρ is the Grassmannian distance between the subspaces, d is the distance in M generated by the (adapted) Riemannian metric, and K, β > Proposition 3.1 ([KH95, Theorem 6.2.3]) . The subspaces E s,u can be “integrated” locally:for every x ∈ Λ there exist local stable and unstable submanifolds V s,u loc ( x ) given via thegraphs of C α functions ψ s,ux : B s,ux (0 , τ ) → E u,s ( x ) , for which we have: (1) V s,u loc ( x ) = exp x { v + ψ s,ux ( v ) : v ∈ B s,ux (0 , τ ) } ; (2) x ∈ V s,u loc ( x ) and T x V s,u loc ( x ) = E s,u ( x ) ; (3) f ( V u loc ( x )) ⊃ V u loc ( f ( x )) and f ( V s loc ( x )) ⊂ V s loc ( f ( x )) ; (4) there is λ ∈ ( χ, such that d ( f ( y ) , f ( z )) ≤ λd ( y, z ) for all y, z ∈ V s loc ( x ) and d ( f − ( y ) , f − ( z )) ≤ λd ( y, z ) for all y, z ∈ V u loc ( x ) ; (5) there is C > such that the H¨older semi-norm satisfies | Dψ s,ux | α ≤ C for all x ∈ Λ . The number τ is the size of the local manifolds and will be fixed at a sufficiently smallvalue to guarantee various estimates (such as the last item in the list above); note thatthe properties listed above remain true if τ is decreased. The manifolds V s,u loc ( x ) dependcontinuously on x ∈ Λ. This metric may not be smooth, but will be at least C γ for some γ >
0, which is sufficient for ourpurposes. Here B s,ux (0 , τ ) is the ball in E s,u ( x ) ⊂ T x M of radius τ centered at 0. This means that the local unstable manifold for f is the local stable manifold for f − . Given a hyperbolic set Λ, there is ε > x, y ∈ Λ with d ( x, y ) < ε ,the intersection V s loc ( x ) ∩ V u loc ( y ) consists of a single point, denoted by [ x, y ] and called the Smale bracket of x and y . One can show that Λ is locally maximal if and only if [ x, y ] ∈ Λfor all such x, y ; this is the local product structure referred to in § Definition 3.2.
A closed set R ⊂ Λ is called a rectangle if [ x, y ] is defined and lies in R for all x, y ∈ R . Given p ∈ R we write V s,uR ( p ) = V s,u loc ( p ) ∩ R for the parts of the localmanifolds that lie in R . R xy [ x, y ] V sR ( x ) V uR ( y ) Figure 3.1.
A rectangle in the case when Λ = M .Given a rectangle R and a point p ∈ R , let A = V uR ( p ) ⊂ V u loc ( p ) and B = V sR ( p ) ⊂ V s loc ( p ).Then [ x, y ] is defined for all x ∈ A and y ∈ B , and(3.2) R = [ A, B ] := { [ x, y ] : x ∈ A, y ∈ B } . Conversely, it is not hard to show that (3.2) defines a rectangle whenever p ∈ Λ and theclosed sets A ⊂ V u loc ( p ) ∩ Λ and B ⊂ V s loc ( p ) ∩ Λ are contained in a sufficiently smallneighborhood of p .For the hyperbolic toral automorphism from § A and B to be intervalsaround p in the stable and unstable directions respectively; then [ A, B ] is the direct productof two intervals, consistent with our usual picture of a rectangle. However, in general, wecould just as easily let A and B be Cantor sets, and thus obtain a dynamical rectanglethat does not look like the picture we are familiar with. For the solenoid and horseshoe,this is the only option; in these examples the hyperbolic set Λ has zero volume and emptyinterior, and we see that rectangles are not even connected.Indeed, there is a general dichotomy: given a C α diffeomorphism f and a locally maxi-mal hyperbolic set Λ, we either have Λ = M (in which case f is an Anosov diffeomorphism),or Λ has zero volume. Even when m (Λ) = 0, the dynamics on Λ still influences the behav-ior of nearby trajectories, as is most apparent when Λ is an attractor , meaning that thereis an open set U ⊃ Λ (called a trapping region ) such that f ( U ) ⊂ U and Λ = (cid:84) n ∈ N f n ( U ),as was the case for the solenoid. In this case every trajectory that enters U is shadowed bysome trajectory in Λ, and Λ is a union of unstable manifolds: V u loc ( x ) ⊂ Λ for every x ∈ Λ.One final comment on the topological dynamics of hyperbolic sets is in order. Recallthat if X is a compact metric space and f : X → X is continuous, then the system ( X, f )is called topologically transitive if for every open sets
U, V ⊂ X there is n ∈ N such that This dichotomy can fail if f is only C ; see [Bow75b]. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 11 U ∩ f − n ( V ) (cid:54) = ∅ , and topologically mixing if for every such U, V there is N ∈ N suchthat U ∩ f − n ( V ) (cid:54) = ∅ for all n ≥ N . Every locally maximal hyperbolic set Λ admits a spectral decomposition [Sma67]: it can be written as a union of disjoint closed invariantsubsets Λ , . . . , Λ k ⊂ Λ such that each f | Λ i is topologically transitive, and moreover eachΛ i is a union of disjoint closed invariant subsets Λ i, , . . . , Λ i,n i such that f (Λ i,j ) = Λ i,j +1 for 1 ≤ j < n i , f (Λ i,n i ) = Λ i, , and each f n i | Λ i,j is topologically mixing. For this reasonthere is no real loss of generality in restricting our attention to topologically mixing locallymaximal hyperbolic sets.3.2. Conditional measures.
Now we consider measures on Λ, writing M (Λ) for the setof all Borel probability measures on Λ, and M ( f, Λ) for the set of all such measures thatare f -invariant. We briefly mention several basic facts that play an important role in theproofs (see [EW11, Chapter 4] for details): the set M ( f, Λ) is convex, and its extreme pointsare precisely the ergodic measures; every µ ∈ M ( f, Λ) has a unique ergodic decomposition µ = (cid:82) M e ( f, Λ) ν dζ ( ν ), where ζ is a probability measure on the space of ergodic measures M e ( f, Λ); and finally, M ( f, Λ) is compact in the weak* topology.As suggested by the discussion in § µ governs the forward asymptotic behavior of trajectories, we should study how µ behaves“along the unstable direction”. To make this precise, we now recall the notion of conditionalmeasures ; for more details, see [Roh52] or [EW11, § µ ∈ M ( f, Λ), consider a rectangle R ⊂ Λ with µ ( R ) >
0. Let ξ be the partition of R by local unstable sets V uR ( x ) = V u loc ( x ) ∩ R , x ∈ R ; these depend continuously on the point x , so the partition ξ is measurable . This implies that the measure µ can be disintegratedwith respect to ξ : for µ -almost every x ∈ R , there is a conditional measure µ ξV uR ( x ) on thepartition element V uR ( x ) for any Borel subset E ⊂ R , we have (3.3) µ ( E ) = (cid:90) R (cid:90) V uR ( x ) E ( y ) dµ ξV uR ( x ) ( y ) dµ ( x ) . Since µ ξV uR ( x ) = µ ξV uR ( x (cid:48) ) whenever x (cid:48) ∈ V uR ( x ), the outer integral in (3.3) can also be writtenas an integral over the quotient space R/ξ , which inherits a factor measure ˜ µ from µ | R inthe natural way. By the local product structure of R , we can also fix p ∈ R and identify R/ξ with V sR ( p ). Then ˜ µ gives a measure on V sR ( p ) by ˜ µ ( A ) = µ ( (cid:83) x ∈ A V uR ( x )). Writing µ ux = µ ξV uR ( x ) for the conditional measure on the leaf through x , we can rewrite (3.3) as (3.4) µ ( E ) = (cid:90) V sR ( p ) (cid:90) V uR ( x ) E ( y ) dµ ux ( y ) d ˜ µ ( x ) . This disintegration is unique under the assumption that the conditional measures are nor-malized. Although the definition depends on R , in fact choosing a different rectangle R (cid:48) merely has the effect of multiplying µ ux by a constant factor on R ∩ R (cid:48) [CPZ18, Lemma 2.5]. For a finite partition, the obvious way to define a conditional measure on a partition element A with µ ( A ) > µ A ( E ) = µ ( E ∩ A ) /µ ( A ). Roughly speaking, measurability of ξ guarantees that it can bewritten as a limit of finite partitions, and the conditional measures in (3.3) are the limits of the conditionalmeasures for the finite partitions; see Proposition 6.7 for a precise statement. Note that µ ux depends on R , although this is suppressed in the notation. One can similarly define a system of conditional measures { µ sx } on V sR ( x ) for x ∈ V uR ( p );it is then natural to ask whether the conditional measure µ sp agrees with the measure ˜ µ on V sR ( p ) from (3.4), and we will return to this question in § absolutecontinuity and the Hopf argument .3.3.
SRB measures.
Now suppose that Λ is a hyperbolic attractor and hence, containsthe local unstable leaf V u loc ( x ) for every x ∈ Λ. Definition 3.3.
Given a hyperbolic attractor Λ for f and a point y ∈ Λ with a localunstable leaf W = V u loc ( y ) , let m W be the leaf volume on W generated by the restrictionof the Riemannian metric to W . An invariant measure µ is a Sinai–Ruelle–Bowen (SRB)measure if for every rectangle R ⊂ Λ with µ ( R ) > , the conditional measures µ ux areabsolutely continuous with respect to the leaf volumes m ux for ˜ µ -almost every x . One of the major goals in the study of systems with some hyperbolicity is to constructSRB measures. In the uniformly hyperbolic setting, this was done by Sinai, Ruelle, andBowen.
Theorem 3.4 ([Sin68, Bow75a, Rue76]) . Let Λ be a topologically transitive hyperbolicattractor for a C α diffeomorphism f . Then there is a unique SRB measure for f | Λ . As suggested by the discussion in §
2, it is not hard to show that SRB measures are physical in the sense of Definition 2.1. In fact, one can prove that for hyperbolic attractors,SRB measures are the only physical measures. In addition to this physicality property, it was shown in [Sin68, Rue76] that the SRBmeasure µ has the property that(3.5) µ = lim n →∞ n n − (cid:88) k =0 f k ∗ m | U , where m | U is normalized volume on the trapping region U ⊃ Λ. In [PS82], this idea wasused in order to construct
SRB measures with m | U replaced by leaf volume m V u loc ( x ) . Werefer to this as the “geometric construction” of SRB measures, and will return to it whenwe discuss our main results. First, though, we observe that the original constructions ofSRB measures followed a different approach and used mathematical tools borrowed fromstatistical physics, as we discuss in the next section.3.4. Equilibrium states.
It turns out that it is possible to relate the absolute continuityrequirement in Definition 3.3 to a variational problem. The Margulis–Ruelle inequality[Rue78a] (see also [BP13, § µ supported An example due to Bowen and Katok [Kat80, § In fact, they proved the stronger property that f n ∗ m | U → µ . More precisely, [PS82] considered the partially hyperbolic setting and used this approach to constructinvariant measures that are absolutely continuous along unstable leaves; SRB measures are a special caseof this when the center bundle is trivial.
QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 13 on a hyperbolic set Λ, we have the following upper bound for the Kolmogorov–Sinaientropy:(3.6) h µ ( f ) ≤ (cid:90) log | det Df | E u ( x ) | dµ. Recall that h µ ( f ) can be interpreted as the average asymptotic rate at which informationis gained if we observe a stochastic process distributed according to µ ; (3.6) says that thisrate can never exceed the average rate of expansion in the unstable direction.Pesin’s entropy formula [Pes77] states that equality holds in (3.6) if µ is absolutelycontinuous with respect to volume. In fact, Ledrappier and Strelcyn proved that it issufficient for µ to have conditional measures on local unstable manifolds that are absolutelycontinuous with respect to leaf volume [LS82], and Ledrappier proved that this conditionis also necessary [Led84]. In other words, equality holds in (3.6) if and only if µ is an SRBmeasure.Since every hyperbolic attractor Λ has an SRB measure, we conclude that the function ϕ geo ( x ) = − log | det Df | E u ( x ) | has the property thatsup µ ∈M ( f, Λ) (cid:18) h µ ( f ) + (cid:90) ϕ geo dµ (cid:19) = 0 , and the SRB measure for f is the unique measure achieving the supremum, as claimed in § Definition 3.5.
Let ϕ : M → R be a continuous function, which we call a potential . An equilbrium state (or equilibrium measure ) for ϕ is a measure µ achieving the supremum (3.7) sup µ ∈M ( f, Λ) (cid:18) h µ ( f ) + (cid:90) ϕ dµ (cid:19) . Thus SRB measures are equilibrium states for the geometric potential ϕ geo , which isH¨older continuous on every hyperbolic set as long as f is C α , by (3.1). This means thatexistence and uniqueness of SRB measures is a special case of the following classical result. Theorem 3.6 ([Sin72, Bow75a, Rue78b]) . Let Λ be a locally maximal hyperbolic set for a C α diffeomorphism f and ϕ : Λ → R a H¨older continuous potential. Assume that f | Λ istopologically transitive. Then there exists a unique equilibrium state µ for ϕ . In §§ f | Λ as a topological Markov chain or on the specificationproperty of f | Λ . In § µ (cid:55)→ h µ ( f ) + (cid:82) ϕ dµ is affine; it follows that the unique equilibrium state µ must be ergodic, otherwise every element of its ergodic decomposition would also be anequilibrium state. In fact, it has many good ergodic properties: one can prove that it isBernoulli, has exponential decay of correlations, and satisfies the Central Limit Theorem[Bow75a]. There is a more general version of this inequality that holds without the assumption that µ is supportedon a hyperbolic set, but it requires the notion of Lyapunov exponents , which are beyond the scope of thispaper.
The fundamental result of thermodynamic formalism is the variational principle , whichestablishes that the supremum in (3.7) is equal to the topological pressure of ϕ , which canbe defined as follows without reference to invariant measures. Definition 3.7.
Given an integer n ≥ , consider the dynamical metric of order n (3.8) d n ( x, y ) = max { d ( f k x, f k y ) : 0 ≤ k < n } and the associated Bowen balls B n ( x, r ) = { y : d n ( x, y ) < r } for each r > . We say that E ⊂ Λ is ( n, r )-separated if d n ( x, y ) ≥ r for all x (cid:54) = y ∈ E , and that E is ( n, r )-spanning for X ⊂ Λ if X ⊂ (cid:83) x ∈ E B n ( x, r ) .Writing S n ϕ ( x ) = (cid:80) n − k =0 ϕ ( f k x ) for the n th Birkhoff sum along the orbit of x , the partition sum of ϕ on a set X ⊂ Λ refers to one of the following two quantities: Z span n ( X, ϕ, r ) := inf (cid:110) (cid:88) x ∈ E e S n ϕ ( x ) : E ⊂ X is ( n, r ) -spanning for X (cid:111) ,Z sep n ( X, ϕ, r ) := sup (cid:110) (cid:88) x ∈ E e S n ϕ ( x ) : E ⊂ X is ( n, r ) -separated (cid:111) . Then the topological pressure is given by (3.9) P ( ϕ ) = lim r → lim n →∞ n log Z span n (Λ , ϕ, r ) = lim r → lim n →∞ n log Z sep n (Λ , ϕ, r ) . (One gets the same value if lim is replaced by lim .) It is worth noting at this point that the definition of P ( ϕ ) bears a certain similarityto the definition of box dimension: one covers X by a collection of balls at a given scale,associates a certain weight to this collection, and then computes the growth rate of thisweight as the balls in the cover are refined. The difference is that here the refinement isdone dynamically rather than statically, and different balls carry different weight accordingto the ergodic sum S n ϕ ( x ); we will discuss this point further in § §
5. When ϕ = 0,we obtain the topological entropy h top ( f ) = P (0), which gives the asymptotic growth rateof the cardinality of an ( n, r )-spanning or ( n, r )-separated set; one can show that this isalso the asymptotic growth rate of the number of periodic orbits in Λ of length n .Now the variational principle [Wal82, Theorem 9.10] can be stated as follows:(3.10) P ( ϕ ) = sup µ ∈M ( f, Λ) (cid:18) h µ ( f ) + (cid:90) ϕ dµ (cid:19) . The discussion at the beginning of this section shows that P ( ϕ geo ) = 0. Given a potential ϕ , we see that an equilibrium state for ϕ is an invariant measure µ ϕ such that P ( ϕ ) = h µ ϕ ( f ) + (cid:82) ϕ dµ ϕ . For the potential function ϕ = 0, the equilibrium state µ ϕ = µ is the measure of maximal entropy . The fact that the limits coincide is given by an elementary argument comparing Z span n and Z sep n . Infact, the limit in r can be removed due to expansivity of f | Λ ; see Definition 3.9 and [Wal82, Theorem 9.6]. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 15
First proof of Theorem 3.6: symbolic representation of f | Λ . The original proofof Theorem 3.6 uses a symbolic coding of the dynamics on Λ. If Λ = X ∪ · · · ∪ X p , then wesay that a bi-infinite sequence ω ∈ Ω p := { , . . . , p } Z codes the orbit of x ∈ Λ if f n x ∈ X ω n for all n ∈ Z . When Λ is a locally maximal hyperbolic set, it is not hard to show that every ω ∈ Ω p codes the orbit of at most one x ∈ Λ; if such an x exists, call it π ( ω ). Let Σ ⊂ Ω p be the set of all sequences that code the orbit of some x ∈ Λ; then Σ is invariant under theshift map σ : Ω p → Ω p defined by ( σω ) n = ω n +1 , and the map π : Σ → Λ is a topologicalsemi-conjugacy , meaning that the following diagram commutes.(3.11) Σ σ (cid:47) (cid:47) π (cid:15) (cid:15) Σ π (cid:15) (cid:15) Λ f (cid:47) (cid:47) ΛIf the sets X i overlap, then the coding map may fail to be injective. One would like toproduce a coding space Σ with a ‘nice’ structure for which the failure of injectivity is‘small’. This was accomplished by Sinai when Λ = M [Sin68] and by Bowen in the generalsetting [Bow70, Bow75a]; they showed that things can be arranged so that Σ is defined by anearest-neighbor condition, with the failure of injectivity confined to sets that are invisiblefrom the point of view of equilibrium states. Theorem 3.8 ([Bow75a]) . If Λ is a locally maximal hyperbolic set for a diffeomorphism f ,then there is a Markov partition Λ = X ∪ · · · ∪ X p such that each X i is a rectangle thatis the closure of its interior (in the induced topology on Λ ) and the corresponding codingspace Σ is a topological Markov chain(3.12) Σ = { ω ∈ Ω p : f (int X ω n ) ∩ int X ω n +1 (cid:54) = ∅ for all n ∈ Z } , and there is a set Λ (cid:48) ⊂ Λ such that (1) every x ∈ Λ (cid:48) has a unique preimage under π , and (2) if µ ∈ M ( f, Λ) is an equilibrium state for a H¨older continuous potential ϕ : Λ → R ,then µ (Λ (cid:48) ) = 1 . With this result in hand, the problem of existence and uniqueness of equilibrium statescan be transferred from the smooth system (Λ , f, ϕ ) to the symbolic system (Σ , σ, ˜ ϕ := ϕ ◦ π ),where tools from statistical mechanics and Gibbs distributions can be used; we recall herethe most important ideas, referring to [Sin72, Bow75a, Rue78b] for full details.Give Σ the metric d ( ω, ω (cid:48) ) = 2 − min {| n | : ω n (cid:54) = ω (cid:48) n } , so that two sequences are close if theyagree on a long interval of integers around the origin. The coding map π is H¨older continuousin this metric, so ˜ ϕ is also H¨older continuous. Fixing r ∈ ( , B n ( ω, r ) = { ω (cid:48) ∈ Σ : ω (cid:48) i = ω i for all 0 ≤ i < n } =: C n ( ω ) , which we call the n -cylinder of ω . Let E n ⊂ Σ contain exactly one point from each n -cylinder; then E n is both ( n, r )-spanning and ( n, r )-separated, and writing Z n (Σ , ˜ ϕ ) = (cid:80) ω ∈ E n e S n ˜ ϕ ( ω ) , one obtains P ( ϕ ) = P ( ˜ ϕ ) = lim n →∞ n log Z n (Σ , ˜ ϕ ) . To understand what an equilibrium state for ˜ ϕ should look like, recall that the Kolmogorov–Sinai entropy of a σ -invariant measure ˜ µ is defined as h ˜ µ ( σ ) = lim n →∞ n (cid:88) ω ∈ E n − ˜ µ ( C n ( ω )) log ˜ µ ( C n ( ω )) . A short exercise using invariance of ˜ µ and continuity of ϕ shows that (cid:90) ˜ ϕ d ˜ µ = lim n →∞ (cid:88) ω ∈ E n (cid:90) C n ( ω ) ˜ ϕ d ˜ µ = lim n →∞ (cid:88) ω ∈ E n ˜ µ ( C n ( ω )) · n S n ˜ ϕ ( ω ) . Thus maximizing h ˜ µ ( σ ) + (cid:82) ˜ ϕ d ˜ µ involves maximizing the limit of a sequence of expressionsof the form F ( p , . . . , p N ) = (cid:80) Ni =1 p i ( − log p i + a i ), where N ( n ) = E n and p i , a i are givenby ˜ µ ( C n ( ω )) and S n ˜ ϕ ( ω ), so that p i ≥ (cid:80) i p i = 1. It is a calculus exercise to showthat with a i fixed, F achieves its maximum value of F = log (cid:80) e a j = log Z n (Σ , ˜ ϕ ) ≈ nP ( ϕ )when p i = e a i / (cid:80) j e a j ≈ e a i e − nP ( ϕ ) .This last relation can be rewritten as ˜ µ ( C n ( ω )) ≈ e S n ˜ ϕ ( ω ) e − nP ( ϕ ) . With this in mind,one can use tools from functional analysis and statistical mechanics to show that there is a σ -invariant ergodic measure ˜ µ on Σ which has the Gibbs property with respect to ˜ ϕ : thereis Q > ω ∈ Σ and n ∈ N , we have(3.14) Q − ≤ ˜ µ ( C n ( ω ))exp( − P ( ϕ ) n + S n ˜ ϕ ( ω )) ≤ Q, By a general result that we will state momentarily, this is enough to guarantee that ˜ µ is theunique equilibrium state for (Σ , σ, ˜ ϕ ), and hence by Theorem 3.8, its projection µ = π ∗ ˜ µ isthe unique equilibrium state for (Λ , f, ϕ ).To formulate the link between the Gibbs property and equilibrium states, we first recallthe following more general definitions. Definition 3.9.
Given a compact metric space X , a homeomorphism f : X → X is saidto be expansive if there is ε > such that every x (cid:54) = y ∈ X have d ( f n x, f n y ) > ε for some n ∈ Z . Definition 3.10.
A measure µ on X is a Gibbs measure for ϕ : X → R if for every small r > there is Q = Q ( r ) > such that for every x ∈ X and n ∈ N , we have (3.15) Q − ≤ µ ( B n ( x, r ))exp( − P ( ϕ ) n + S n ϕ ( x )) ≤ Q. Note that σ : Σ → Σ is expansive, and that (3.14) implies (3.15) in this symbolic setting.Then uniqueness of the equilibrium state is a consequence of the following general result.
Proposition 3.11 ([Bow75, Lemma 8]) . If X is a compact metric space, f : X → X is anexpansive homeomorphism, and µ is an ergodic f -invariant Gibbs measure for ϕ : X → R ,then µ is the unique equilibrium state for ϕ . We remark that (3.15) does not require the Gibbs measure to be invariant. Indeed,one can separate the problem of finding a unique equilibrium state into two parts: firstconstruct a Gibbs measure without worrying about whether or not it is invariant, then finda density function (bounded away from 0 and ∞ ) that produces an ergodic invariant Gibbsmeasure, which is the unique equilibrium state by Proposition 3.11. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 17
Second proof of Theorem 3.6: specification property.
There is another proofof Theorem 3.6 which is due to Bowen [Bow75] and avoids symbolic dynamics. Instead, ituses the fact that f satisfies the following specification property on a topologically mixinglocally maximal hyperbolic set Λ: for each δ > p ( δ ) such that givenany points x , . . . , x n ∈ Λ and intervals of integers I , . . . I n ⊂ [ a, b ] with d ( I i , I j ) ≥ p ( δ ) for i (cid:54) = j , there is a point x ∈ Λ with f b − a + p ( δ ) ( x ) = x and d ( f k ( x ) , f k ( x i )) < δ for k ∈ I i .Roughly speaking, f satisfies specification if for every finite number of orbit segments onecan find a single periodic orbit that consecutively approximates each segment with a fixedprecision δ >
0, and such that transition times are bounded by p ( δ ). This property allowsone to study some topological and statistical properties of f by only analyzing periodicorbits.The construction of the Gibbs measure ˜ µ in the first approach uses eigendata of a certainlinear operator acting on an appropriately chosen Banach space of functions on Λ. Thespecification property allows one to use a more elementary construction and obtain a Gibbsmeasure on Λ as a weak* limit point of measures supported on periodic orbits. Let Per n := { x ∈ Λ : f n x = x } and Z per n ( ϕ ) := (cid:80) x ∈ Per n e S n ϕ ( x ) (compare this to Z span n and Z sep n fromDefinition 3.7); then consider the f -invariant Borel probability measures given by µ n := 1 Z per n ( ϕ ) (cid:88) x ∈ Per n e S n ϕ ( x ) δ x , where δ x is the atomic probability measure with δ x ( { x } ) = 1.Using some counting estimates on the partition sums Z per n ( ϕ ) provided by the specifica-tion property, one can prove that every weak* limit point µ of the sequence µ n is an ergodicGibbs measure as in (3.15). Then Proposition 3.11 shows that µ is the unique equilibriumstate for ϕ ; a posteriori , the sequence µ n converges.4. Description of reference measures and main results
In this section, and especially in Theorem 4.11, we describe a new proof of Theorem3.6 that avoids Markov partitions and the specification property, and instead mimics thegeometric construction of SRB measures in § ϕ : Λ → R , we define for each x ∈ Λ a measure m C x on X = V u loc ( x ) ∩ Λ such that the sequence of measures(4.1) µ n = 1 n n − (cid:88) k =0 f k ∗ m C x m C x ( X )converges to the unique equilibrium state. In § m C x to have; then in § § m C x , including how these measures can be used to prove Theorem 3.6. In § To be more precise we need first to extend m C x from X to a measure on Λ by assigning to any Borelset E ⊂ Λ the value m C x ( E ∩ X ). We shall always assume that in (4.1) m C x is extended in this way. Conditional measures as reference measures.
We start with the observation thatif we were already in possession of the equilibrium state µ , then the conditional measuresof µ would immediately define reference measures for which the construction just describedproduces µ . Indeed, suppose X is a compact topological space, f : X → X a continuousmap, and µ a finite f -invariant ergodic Borel probability measure on X . Given Y ⊂ X with µ ( Y ) > ξ of Y ; let ˜ µ be the corresponding factor-measure on˜ Y := Y /ξ , and { µ ξW : W ∈ ξ } the conditional measures on partition elements. We provethe following result in § Theorem 4.1.
For ˜ µ -almost every W ∈ ξ , any probability measure ν on W such that ν (cid:28) µ ξW has the property that ν n := n (cid:80) n − k =0 f k ∗ ν converges in the weak ∗ topology to themeasure µ . Of course, Theorem 4.1 is not much help in finding the equilibrium state µ , because weneed to know µ to obtain the conditional measures µ ξW . We must construct the referencemeasure m C x independently, without using any knowledge of existence of equilibrium states.Once we have done this, we will eventually show that m C x is equivalent to the conditionalmeasure of the constructed equilibrium state, so our approach not only allows us to developa new way of constructing equilibrium states, but also describes their conditional measures.4.2. Conditions to be satisfied by reference measures.
To motivate the propertiesthat our reference measures must have, we first consider the specific case when Λ is anattractor and outline the steps in constructing SRB measures.(1) Given a local unstable leaf W = V u loc ( x ) through x ∈ Λ and n ≥
0, the image W n := f n ( W ) is contained in the union of local leaves V u loc ( y i ) for some points y , . . . , y s ∈ W n , and leaf volume m ux is pushed forward to a measure f n ∗ m ux suchthat ( f n ∗ m ux ) | V u loc ( y i ) (cid:28) m uy i for each i .(2) Each µ n := n (cid:80) n − k =0 f k ∗ m ux can be written as a convex combination of measureswith the form ρ in dm uy i for some functions ρ in : V u loc ( y i ) → [0 , ∞ ) that are uniformlybounded away from 0 and ∞ .(3) To show that any limit measure µ = lim j →∞ µ n j has absolutely continuous con-ditional measures on unstable leaves, first observe that given a rectangle R , thepartition ξ into local unstable leaves can be approximated by a refining sequenceof finite partitions ξ (cid:96) , and the conditional measures µ ξx are the weak* limits of theconditional measures µ ξ (cid:96) x as (cid:96) → ∞ .(4) The bounds on the density functions ρ in allow us to control the conditional measures µ ξ (cid:96) x , and hence to control µ ξx as well; in particular, these measures are absolutelycontinuous with respect to leaf volume, and thus µ is an SRB measure.Now we describe two crucial properties of the leaf volumes m ux , which we will eventuallyneed to mimic with our reference measures m C x . The first of these already appeared in (1.1),and describes how m ux scales under iteration by f ; this will let us conclude that the SRB Note that ξ is not assumed to have any dynamical significance; in particular it need not be a partitioninto local unstable leaves, although this is the most relevant partition for our purposes. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 19 measure µ is an equilibrium state for ϕ geo . The second property describes how m ux behaveswhen we ‘slide along stable leaves’ via a holonomy map ; this issue has so far been ignoredin our discussion, but plays a key role in the proof that the SRB measure µ is ergodic, andhence is the unique equilibrium state for ϕ geo .4.2.1. Scaling under iteration.
Given any x, y ∈ Λ and A ⊂ f ( V u loc ( x )) ∩ V u loc ( y ), we have f ∗ m ux ( A ) = m ux ( f − A ) = (cid:90) A | det f − | E u ( z ) | dm uy ( z ) = (cid:90) A e ϕ geo ( f − z ) dm uy ( z )and so the Radon–Nikodym derivative comparing the family of measures m ux to their push-forwards is given in terms of the geometric potential:(4.2) d ( f ∗ m ux ) dm uy ( z ) = e ϕ geo ( f − ( z )) . Iterating this, we see that given A ⊂ f n ( V u loc ( x )) ∩ V u loc ( y ) we have(4.3) f n ∗ m ux ( A ) = m ux ( f − n A ) = (cid:90) A e (cid:80) nk =1 ϕ geo ( f − k z ) dm uy ( z ) . By H¨older continuity of ϕ geo and the fact that f − contracts uniformly along each V u loc , onecan easily show that(4.4) (cid:12)(cid:12)(cid:12) n (cid:88) k =1 (cid:0) ϕ geo ( f − k z ) − ϕ geo ( f − k z ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ Q u for all z , z ∈ V u loc ( y ) , where Q u is a constant independent of y ∈ Λ, z , z ∈ V u loc ( y ), and n ∈ N (see Lemma 6.6for details). Together with (4.3), this gives(4.5) e − Q u ≤ m ux ( f − n A ) e S n ϕ geo ( x ) m uy ( A ) ≤ e Q u for all A ⊂ f n ( V u loc ( x )) ∩ V u loc ( y ) . In particular, writing B u ( y, r ) = B ( y, r ) ∩ V u loc ( y ), we observe that for each r > K = K ( r ) > m uy ( B u ( y, r )) ∈ [ K − , K ] for all y ∈ Λ, and deduce from(4.5) that the u -Bowen ball B un ( x, r ) := { z ∈ V u loc ( x ) : d n ( x, z ) < r } = f − n B u ( f n x, r )admits the following leaf volume estimate:(4.6) K − e − Q u ≤ m ux ( B un ( x, r )) e S n ϕ geo ( x ) ≤ Ke Q u . Definition 4.2.
Consider a family of measures { µ x : x ∈ Λ } such that µ x is supported on V u loc ( x ) . We say that this family has the u -Gibbs property with respect to the potentialfunction ϕ : Λ → R if there is Q = Q ( r ) > such that for all x ∈ Λ and n ∈ N , we have (4.7) Q − ≤ µ x ( B un ( x, r )) e − nP ( ϕ )+ S n ϕ ( x ) ≤ Q . Note that this is a different notion than the idea of u -Gibbs state from [PS82]. In particular, (4.6) says that m ux has the u -Gibbs property with respect to the potentialfunction ϕ geo . Since the SRB measure µ constructed above has conditional measures thatare given by multiplying the leaf volumes m ux by ‘nice’ density functions, one can use (4.6)to ensure that the conditional measures of µ also have the u -Gibbs property; integratingthese conditional measures gives the Gibbs property for µ , and then some straightforwardestimates involving Z span n (Λ , ϕ, r ) demonstrate that µ is an equilibrium state correspondingto the function ϕ geo .4.2.2. Sliding along stable leaves.
It remains, then, to show that µ is the unique equilibriumstate for ϕ geo ; this will follow from Proposition 3.11 if µ is proved to be ergodic. To establishergodicity we use the Hopf argument , which goes back to E. Hopf’s work on geodesic flowover surfaces [Hop39]. The first step is to observe that if µ is any invariant measure, thenby Birkhoff’s ergodic theorem, for every ψ ∈ L ( µ ), the forward and backward ergodicaverages exist and agree for µ -a.e. x :(4.8) lim n →∞ n n − (cid:88) k =0 ψ ( f k x ) = lim n →∞ n n − (cid:88) k =0 ψ ( f − k x ) . Let
B ⊂
Λ be the set of points where the limits in (4.8) exist and agree for every continuous ψ : Λ → R ; such points are called Birkhoff regular . For each x ∈ B , write ψ ( x ) for thecommon value of these limits; note that ψ is defined µ -a.e. It is not hard to prove that µ is ergodic if and only if the function ψ : B → R is constant µ -a.e. for every continuous ψ : Λ → R . By topological transitivity and the fact that ψ ◦ f = ψ on B , one obtains thefollowing standard result, whose proof we omit. Lemma 4.3. An f -invariant measure µ is ergodic if and only if for every continuous ψ : Λ → R and every rectangle R ⊂ Λ , the function ψ : B ∩ R → R is constant µ -a.e. Now comes the central idea of the Hopf argument: given ψ ∈ C (Λ), if ψ ( x ) exists thena short argument using the left-hand side of (4.8) gives ψ ( y ) = ψ ( x ) for all y ∈ B ∩ V sR ( x ).Similarly, ψ is constant on B ∩ V uR ( x ) using the right-hand side of (4.8).We want to conclude the proof of ergodicity by saying something like the following: “Since B has full measure in R , it has full measure in almost every stable and unstable leafin R ; thus there is p ∈ R such that B p := (cid:83) x ∈B∩ V uR ( p ) B ∩ V sR ( x ) has full measure in R , andby the previous paragraph, ψ is constant on B p , so Lemma 4.3 applies.” There is a subtlety involved in making this step rigorous. To begin with, the term“full measure” is used in two different ways: “ B has full measure in R ” means that itscomplement B c = R \ B has µ ( R \ B ) = 0, while “ B has full measure in the stable leaf V sR ( x )” means that µ sx ( B c ) = 0, where µ sx is the conditional measure of µ along the stableleaf. Using the analogue of (3.3)–(3.4) for the decomposition into stable leaves, we have(4.9) µ ( B c ) = (cid:90) R µ sx ( B c ) dµ ( x ) = (cid:90) V uR ( p ) µ sx ( B c ) d ˜ µ p ( x ) , This is a more general version of the ergodic theorem than the one we mentioned in § µ is not ergodic, but does not require that the limits in (4.8) are equal to (cid:82) ψ dµ ; instead,one obtains (cid:82) ψ dµ = (cid:82) ψ dµ , which implies the earlier version in the case when ψ is constant µ -a.e. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 21 where ˜ µ p is the measure on V uR ( p ) defined by˜ µ p ( A ) = µ (cid:0) (cid:91) x ∈ A V sR ( x ) (cid:1) . Let B (cid:48) = { x ∈ R : µ sx ( B c ) = 0 } . It follows that0 = µ ( B c ) = (cid:90) R µ sx ( B c ) dµ ( x ) = (cid:90) R \B µ sx ( B c ) dµ ( x ) , and since µ sx ( B c ) > x ∈ R \ B by definition, we conclude that µ ( R \ B (cid:48) ) = 0; inother words, µ sx ( B c ) = 0 for µ -a.e. x ∈ R . A similar argument produces B (cid:48)(cid:48) ⊂ B (cid:48) such that µ ( R \ B (cid:48)(cid:48) ) = 0 and µ ux ( B c ) = µ sx ( B c ) = 0 for every x ∈ B (cid:48)(cid:48) .So far, things are behaving as we expect. Now can we conclude that µ ( B p ) = µ ( R ) for p ∈ B (cid:48)(cid:48) , thus completing the proof of ergodicity? Using (4.9), we have µ ( B p ) = (cid:90) V uR ( p ) µ sx (cid:18) (cid:91) y ∈B∩ V uR ( p ) B ∩ V sR ( y ) (cid:19) d ˜ µ p ( x ) ≥ (cid:90) B (cid:48)(cid:48) ∩ V uR ( p ) µ sx ( B ) d ˜ µ p ( x ) = ˜ µ p ( B (cid:48)(cid:48) ) . We would like to say that ˜ µ p ( B (cid:48)(cid:48) ) = ˜ µ p ( V uR ( p )) = µ ( R ), and conclude that B p has full µ -measure in R . We know that µ up ( B (cid:48)(cid:48) ) = µ up ( V uR ( p )), and so the proof will be complete ifthe answer to the following question is “yes”. Question.
Are the measures ˜ µ p and µ up on V uR ( p ) equivalent? Note that the measures µ up are defined in terms of the foliation V uR , while the measures˜ µ p are defined in terms of the foliation V sR . We can write the measures ˜ µ p in terms of µ up as follows: given A ⊂ V uR ( p ), we have(4.10) ˜ µ p ( A ) = µ (cid:18) (cid:91) x ∈ A V sR ( x ) (cid:19) = (cid:90) R µ uy { V uR ( y ) ∩ V sR ( x ) : x ∈ A } dµ ( y ) . For each p, y ∈ R , consider the (stable) holonomy map π py : V uR ( p ) → V uR ( y ) defined by π py ( x ) = V uR ( y ) ∩ V sR ( x ), which maps one unstable leaf to another by sliding along stableleaves ; see Figure 4.1. Then (4.10) becomes˜ µ p ( A ) = (cid:90) R ( µ uy ◦ π py )( A ) dµ ( y ) . In other words, ˜ µ p is the average of the conditional measures π ∗ py µ uy = µ uy ◦ π py taken overall y ∈ R . xy π py ( x ) π py ( A ) V sR ( x ) V uR ( y ) A p V uR ( p ) Figure 4.1.
The stable holonomy map from V uR ( p ) to V uR ( y ). Definition 4.4.
Let { ν x } x ∈ R be a family of measures on R with the property that each ν x is supported on V uR ( x ) , and ν x = ν y whenever y ∈ V uR ( x ) . We say that the family { ν x } is absolutely continuous with respect to stable holonomies if π ∗ xy ν y (cid:28) ν x for all x, y ∈ R . The preceding arguments lead to the following result; full proofs (and further discussion)can be found in [CHT16].
Proposition 4.5.
Let Λ be a topologically transitive hyperbolic set for a C α diffeomor-phism f , and let µ be an f -invariant measure on Λ . Suppose that for every rectangle R ⊂ Λ with µ ( R ) > , the unstable conditional measures µ ux are absolutely continuous with respectto stable holonomies. Then µ is ergodic. Construction of reference measures.
In light of the previous section, our goal isto construct for each potential ϕ a reference measure m C x on each leaf V u loc ( x ) satisfying aproperty analogous to (4.2) with ϕ in place of ϕ geo , together with the absolute continuityproperty from Definition 4.4.From now on we fix a local unstable manifold W = V u loc ( x ) of size τ and consider the set X = W ∩ Λ on which we will build our reference measure. Before treating general potentials,we start with the geometric potential ϕ geo , and we assume that Λ is an attractor for f , sothat W ⊂ Λ. This is necessary for the moment since the measure we build will be supportedon W ∩ Λ, and the support of m W is all of W ; in the general construction below we willnot require Λ to be an attractor. For the geometric potential ϕ geo , we know the referencemeasure m C x should be equivalent to leaf volume m W on W . Leaf volume is equivalent tothe Hausdorff measure m H ( · , α ) with α = dim E u , which is defined by(4.11) m H ( Z, α ) := lim ε → inf ∞ (cid:88) i =1 (diam U i ) α , where the infimum is taken over all collections { U i } of open sets U i ⊂ W with diam U i ≤ ε which cover Z ⊂ W .We want to describe a measure that is equivalent to m H ( · , α ) but whose definition usesthe dynamics of f . In (4.11), the covers used to measure Z were refined geometrically bysending ε →
0. We consider instead covers that refine dynamically : we restrict the sets U i to be u -Bowen balls B un ( x, r ) = B n ( x, r ) ∩ W , and refine the covers by requiring n tobe large rather than by requiring r to be small. Note that if U i is a metric ball B ( x, ε ),then (diam U i ) α ≈ m W ( U i ) up to a multiplicative factor that is bounded away from 0 and ∞ . For a u -Bowen ball, on the other hand, (4.6) gives m W ( B un ( x, r )) ≈ e S n ϕ geo ( x ) , and sowe use this quantity to compute the weight of the cover. This suggests that we should fix r > Z ⊂ W by(4.12) m ϕ geo x ( Z ) := lim N →∞ inf ∞ (cid:88) i =1 e S ni ϕ geo ( x i ) , There is a related, but distinct, notion of absolute continuity of a foliation (with respect to volume),which also plays a key role in smooth ergodic theory; see [BP07, § From Theorem 4.1 we see that the equivalence class of the measure is the crucial thing for the geometricconstruction to work.
QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 23 where the infimum is taken over all collections { B un i ( x i , r ) } of u -Bowen balls with x i ∈ W , n i ≥ N , which cover Z . It is relatively straightforward to derive property (4.2) from (4.12).Now it is nearly apparent what the definition should be for a general potential; we wantto replace ϕ geo with ϕ in (4.12). There is one small subtlety, though. First, Definition 3.7gives P ( ϕ + c ) = P ( ϕ ) + c for c ∈ R . This along with the definition of equilibrium stateand the variational principle (3.10) shows that adding a constant to ϕ does not changeits equilibrium states, and thus we should also expect that ϕ and ϕ + c produce the samereference measure on W ∩ Λ. For this to happen, we need to modify (4.12) so that addinga constant to ϕ does not affect the value. This can be achieved by multiplying each termin the sum by e − n i P ( ϕ ) ; note that since P ( ϕ geo ) = 0 this does not modify (4.12). Thus wemake the following definition. Definition 4.6.
Let < r < τ . We define a measure on X := W ∩ Λ by (4.13) m C x ( Z ) := lim N →∞ inf (cid:88) i e − n i P ( ϕ ) e S ni ϕ ( x i ) , where the infimum is taken over all collections { B un i ( x i , r ) } of u -Bowen balls with x i ∈ X , n i ≥ N , which cover Z , and for convenience we write C = ( ϕ, r ) to keep track of the dataon which the reference measure depends. Both definitions (4.12) and (4.13) are specific cases of the Carath´eodory measure pro-duced by a dynamically defined Carath´eodory dimension structure, which we discuss atgreater length in §
5; this is the Pesin–Pitskel’ definition of topological pressure [PP84] thatgeneralized Bowen’s definition of topological entropy for non-compact sets [Bow73]. Inparticular, Proposition 5.4 establishes the crucial property that every local unstable leafcarries the same topological pressure as the entire set Λ.4.4.
Statements of main results.
Now we state the most important properties of m C x and show how it can be used as a reference measure to construct the equilibrium state for ϕ . All results in this section are proved in detail in [CPZ18]; we outline the proofs in § m C x is finite and nonzero. Theorem 4.7. [CPZ18, Theorem 4.2]
Let Λ be a topologically transitive locally maximalhyperbolic set for a C α diffeomorphism f , and let ϕ : Λ → R be H¨older continuous. Fix r as in Definition 4.6, and for each x ∈ Λ , let m C x be given by (4.13) , where C = ( ϕ, r ) .Then there is K > such that for every x ∈ Λ , m C x is a Borel measure on V u loc ( x ) ∩ Λ with m C x ( V u loc ( x ) ∩ Λ) ∈ [ K − , K ] . If V u loc ( x ) ∩ V u loc ( y ) ∩ Λ (cid:54) = ∅ , then m C x and m C y agree on theintersection. As described in § m C x transform under (1)the dynamics of f and (2) sliding along stable leaves via holonomy. For the first of theseproperties, the following result gives the necessary scaling property analogous to (4.2). Theorem 4.8. [CPZ18, Theorem 4.4]
Under the hypotheses of Theorem 4.7, for every x ∈ Λ , we have f ∗ m C f ( x ) := m C f ( x ) ◦ f (cid:28) m C x , with Radon–Nikodym derivative e P ( ϕ ) − ϕ , sothat (1.2) holds. The numbering of references within [CPZ18] refers to the first arXiv version; it is possible that thenumbering will change between this and the final published version.
Corollary 4.9. [CPZ18, Corollary 4.5]
Under the hypotheses of Theorem 4.7, the familyof measures { m C x } x ∈ Λ has the u -Gibbs property from Definition 4.2. In particular, for everyrelatively open U ⊂ V u loc ( x ) ∩ Λ , we have m C x ( U ) > . The final crucial property of the reference measures is that they are absolutely continuousunder holonomy.
Theorem 4.10. [CPZ18, Theorem 4.6]
Under the hypotheses of Theorem 4.7, there is aconstant Q > such that for every rectangle R ⊂ Λ and every y, z ∈ R , the measures π ∗ yz m C z = m C z ◦ π yz and m C y are equivalent on V uR ( y ) , with Q − ≤ dπ ∗ yz m C z dm C y ≤ Q . Note that Theorem 4.10 in particular shows that given a rectangle R ⊂ Λ, if m C x ( V uR ( x )) > x ∈ R , then the same is true for every x ∈ R ; moreover, by Corollary 4.9 thishappens whenever R is the closure of its interior (relative to Λ).Using these properties of the measures m C x , we can carry out the geometric constructionof equilibrium states; see § Theorem 4.11.
Under the hypotheses of Theorem 4.7, the following are true. (1)
For every x ∈ Λ , the sequence of measures µ n := n (cid:80) n − k =0 f k ∗ m C x /m C x ( V u loc ( x )) from (4.1) is weak* convergent as n → ∞ to a probability measure µ ϕ that is independentof x . (2) The measure µ ϕ is ergodic, gives positive weight to every open set in Λ , has theGibbs property (3.15) and is the unique equilibrium state for (Λ , f, ϕ ) . (3) For every rectangle R ⊂ Λ with µ ϕ ( R ) > , the conditional measures µ uy generated by µ ϕ on unstable sets V uR ( y ) are equivalent for µ ϕ -almost every y ∈ R to the referencemeasures m C y | V uR ( y ) . Moreover, there exists Q > , independent of R and y , suchthat for µ ϕ -almost every y ∈ R we have (4.14) Q − ≤ dµ uy dm C y ( z ) m C y ( R ) ≤ Q for µ uy -a.e. z ∈ V uR ( y ) . Theorems 4.10 and 4.11(3) allow us to show that the equilibrium state µ ϕ has localproduct structure, as follows. Consider a rectangle R ⊂ Λ with µ ϕ ( R ) >
0, and a systemof conditional measures µ ux with respect to the partition ξ of R into local unstable leaves.Given p ∈ R , define a measure ˜ µ p on V sR ( p ) by ˜ µ p ( A ) = µ ϕ ( (cid:83) x ∈ A V uR ( x )) as in the paragraphpreceding (3.4). Since R is homeomorphic to the direct product of V uR ( p ) and V sR ( p ), theproduct of the measures µ up and ˜ µ p gives a measure on R that we denote by µ up ⊗ ˜ µ p . Thefollowing local product structure result is a consequence of Theorem 4.10, Theorem 4.11(3),and (3.4); see § Corollary 4.12.
For every rectangle R and µ ϕ -almost every p ∈ R , we have π ∗ py µ uy ∼ µ up for ˜ µ p − almost every y ∈ V sR ( p ) , and thus µ ϕ ∼ µ up ⊗ ˜ µ p . Moreover, it follows that ˜ µ p is It is reasonable to expect, based on analogy with the case of SRB measure, that the Radon–Nikodymderivative in (4.14) is in fact H¨older continuous and given by an explicit formula; at present we can onlyprove this for a modified version of m C x , whose definition we omit here. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 25 equivalent to µ sp , the conditional measure on V sR ( p ) with respect to the partition into stableleaves, for µ ϕ -a.e. p ∈ R . We remark that Corollary 4.12 was also proved by Leplaideur [Lep00]. His proof usesMarkov partitions to construct families of leaf measures with the properties given in Theo-rems 4.7 and 4.10. Historically, this description of µ ϕ in terms of its direct product structuredates back to Margulis [Mar70], who described the unique measure of maximal entropy fora transitive Anosov flow as a direct product of leafwise measures satisfying the continuous-time analogue of (1.2) for ϕ = 0. In this specific case the equivalences in Corollary 4.12can be strengthened to equalities.5. Carath´eodory dimension structure
The definition of the measures m C x in (4.13) is a specific instance of the Carath´eodorydimension construction introduced by the second author in [Pes88] (see also [Pes97, § Carath´eodory dimension and measure. A Carath´eodory dimension structure , or C -structure , on a set X is given by the following data.(1) An indexed collection of subsets of X , denoted F = { U s : s ∈ S} .(2) Functions ξ, η, ψ : S → [0 , ∞ ) satisfying the following conditions: A1. if U s = ∅ then η ( s ) = ψ ( s ) = 0; if U s (cid:54) = ∅ , then η ( s ) > ψ ( s ) > A2. for any δ > ε > η ( s ) ≤ δ for any s ∈ S with ψ ( s ) ≤ ε ; A3. for any ε >
G ⊂ S that covers X (meaning that (cid:83) s ∈G U s ⊃ X ) and has ψ ( G ) := sup { ψ ( s ) : s ∈ S} ≤ ε .Note that no conditions are placed on ξ ( s ), which we interpret as the weight of U s . Thevalues η ( s ) and ψ ( s ) can each be interpreted as a size or scale of U s ; we allow these functionsto be different from each other.The C -structure ( S , F , ξ, η, ψ ) determines a one-parameter family of outer measures on X as follows. Fix a nonempty set Z ⊂ X and consider some G ⊂ S that covers Z (meaningthat (cid:83) s ∈G U s ⊃ Z ). Then ψ ( G ) is interpreted as the largest size of sets in the cover, andwe set for each α ∈ R ,(5.1) m C ( Z, α ) := lim ε → inf G (cid:88) s ∈G ξ ( s ) η ( s ) α , where the infimum is taken over all finite or countable G ⊂ S covering Z with ψ ( G ) ≤ ε .Defining m C ( ∅ , α ) := 0, it follows from [Pes97, Proposition 1.1] that m C ( · , α ) is an outermeasure. The measure induced by m C ( · , α ) on the σ -algebra of measurable sets is the α -Carath´eodory measure ; it need not be σ -finite or non-trivial. Proposition 5.1 ([Pes97, Proposition 1.2]) . For any set Z ⊂ X there exists a critical value α C ∈ R such that m C ( Z, α ) = ∞ for α < α C and m C ( Z, α ) = 0 for α > α C . In [Pes97], condition
A1. includes the requirement that there is s ∈ S such that U s = ∅ ; here weremove this assumption and instead define m C ( ∅ , α ) := 0, which is equivalent. We call dim C Z = α C the Carath´eodory dimension of the set Z associated to the C -structure ( S , F , ξ, η, ψ ). By Proposition 5.1, α = dim C X is the only value of α for which(5.1) can possibly produce a non-zero finite measure on X , though it is still possible that m C ( X, dim C X ) is equal to 0 or ∞ .5.2. Examples of C -structures. The C -structures in which we are interested are gener-ated by other structures on the set X .5.2.1. Hausdorff dimension and measure. If X is a metric space, then consider the C -structure given by S := X × (0 , ∞ ) and F := { B ( x, r ) : x ∈ X, r > } , ξ ( x, r ) = 1 , η ( x, r ) = ψ ( x, r ) = r. Comparing (4.11) and (5.1), we see that m C ( Z, α ) = m H ( Z, α ) for every Z ⊂ X , andthe Hausdorff dimension dim H ( Z ) is the critical value such that m H ( Z, α ) is infinite for α < dim H ( Z ) and 0 for α > dim H ( Z ). Thus dim C Z = dim H Z , and the outer measure m C ( · , dim H Z ) on Z is the (dim H Z )-dimensional spherical Hausdorff measure.It is useful to understand when an outer measure defines a Borel measure on a metricspace. Recall that an outer measure m on a metric space ( X, d ) is a metric outer measure if m ( E ∪ F ) = m ( E ) + m ( F ) whenever d ( E, F ) := inf { d ( x, y ) : x ∈ E, y ∈ F ) } > Proposition 5.2 ([Fed69, § . If X is a metric space and m is a metric outermeasure on X , then every Borel set in X is m -measurable, and so m defines a Borelmeasure on X . Given any
E, F ⊂ X , with d ( E, F ) >
0, we see that any cover
G ⊂ F of E ∪ F with ψ ( G ) ≤ d ( E, F ) / E and a cover of F ;using this it is easy to show that m H ( E ∪ F, α ) = m H ( E, α ) + m H ( F, α ), so m H ( · , α ) is ametric outer measure. By Proposition 5.2, this defines a Borel measure on X .5.2.2. Topological pressure as a Carath´eodory dimension.
Let f be a continuous map of acompact metric space X , and ϕ : X → R a continuous function. Then as described alreadyin § r > x, n ) ∈ X × N , associate the Bowen ball B n ( x, r ). Let F be thecollection of all such Bowen balls, and let S = X × N , so s = ( x, n ) has U s = B n ( x, r ). Nowput(5.2) ξ ( x, n ) = e S n ϕ ( x ) , η ( x, n ) = e − n , ψ ( x, n ) = n . It is easy to see that ( S , F , ξ, η, ψ ) satisfies A1. – A3. , so this defines a C -structure. Theassociated outer measure is given by(5.3) m C ( Z, α ) = lim N →∞ inf G (cid:88) ( x,n ) ∈G e S n ϕ ( x ) e − nα , where the infimum is over all G ⊂ S such that (cid:83) ( x,n ) ∈G B n ( x, r ) ⊃ Z and n ≥ N for all( x, n ) ∈ G . QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 27
Remark 5.3.
The measure m C ( · , α ) is not necessarily a metric outer measure, since theremay be x (cid:54) = y ∈ X such that y ∈ B n ( x, r ) for all n ∈ N . Thus Borel sets in X need notbe m C ( · , α ) -measurable. Writing dim rC Z for the critical value of α , where the superscript emphasizes the depen-dence on r , the quantity P Z ( ϕ ) = lim r → dim rC Z is called the the topological pressure of ϕ on the set Z . Observe that this notion of thetopological pressure is more general than the one introduced in Definition 3.7 as it is moresuited to arbitrary subsets Z (which need not be compact or invariant); both definitionsagree when Z = X [Pes97, Theorem 11.5].5.2.3. A C -structure on local unstable leaves. Now consider the setting of Theorems 4.7–4.10: Λ is a hyperbolic set for a C α diffeomorphism f , and ϕ : Λ → R is H¨older continuous.Fix r > C -structure on X = V u loc ( x ) ∩ Λ, which depends on ϕ , in the followingway. To each ( x, n ) ∈ X × N , associate the Bowen ball B n ( x, r ). Let F be the collection ofall such balls, and let S = X × N , so s = ( x, n ) has U s = B n ( x, r ). Now put(5.4) ξ ( x, n ) = e S n ϕ ( x ) , η ( x, n ) = e − n , ψ ( x, n ) = n . Again, ( S , F , ξ, η, ψ ) satisfies A1. – A3. and defines a C -structure, whose associated outermeasure is given by(5.5) m C ( Z, α ) = lim N →∞ inf G (cid:88) ( x,n ) ∈G e S n ϕ ( x ) e − nα , where the infimum is over all G ⊂ S such that (cid:83) ( x,n ) ∈G B n ( x, r ) ⊃ Z and n ≥ N for all( x, n ) ∈ G .Given x ∈ Λ we are interested in two things:(1) the Carath´eodory dimension of X , as determined by this C -structure; and(2) the (outer) measure on X defined by (5.3) at α = dim C ( X ).The first of these is settled by the following, which is proved in [CPZ18, Theorem 4.2(1)]. Proposition 5.4.
With Λ , f, ϕ, r as above, and the C -structure defined on X = V u loc ( x ) ∩ Λ by Bowen balls B n ( x, r ) and (5.4) , we have dim rC ( X ) = P ( ϕ ) for every x ∈ Λ . In particular,this implies that P X ( ϕ ) = P Λ ( ϕ ) . Note that on each X = V u loc ( x ) ∩ Λ, covers by Bowen balls B n ( x, r ) are the same thing ascovers by u -Bowen balls B un ( x, r ) = B n ( x, r ) ∩ V u loc ( x ), which we used in § α = P ( ϕ ), we see that (5.5) agrees with (4.13) for every Z ⊂ X , and in particular, thequantity m C x ( Z ) defined in (4.13) is the outer measure on X associated to the C -structureabove and the parameter value α = P ( ϕ ).One must still do some work to show that this outer measure is finite and nonzero; thisis done in [CPZ18], and the idea of the argument is given in § metric outer measure. Indeed, given any x ∈ Λ and y ∈ V u loc ( x ) ∩ Λ, wehave diam B un ( y, r ) ≤ rλ n for all n ∈ N by Proposition 3.1, so if E, F ⊂ X have d ( E, F ) > In fact m C ( · , α ) is an outer measure if and only if f is positively expansive to scale r . then there is N ∈ N such that B n ( y, r ) ∩ B k ( z, r ) = ∅ whenever y ∈ E , z ∈ F , and k, n ≥ N .Then for N sufficiently large, any G as in (5.5) has the property that it splits into disjointcovers of E and F , and thus m C x ( E ∪ F ) = m C x ( E ) + m C x ( F ). By Proposition 5.2, m C x definesa Borel measure on X , as claimed in Theorem 4.7.5.3. An application: measures of maximal dimension. If X is a measurable spacewith a measure µ , and dim C is a Carath´eodory dimension on X , then the quantitydim C µ = inf { dim C Z : µ ( Z ) = 1 } = lim δ → inf { dim C Z : µ ( Z ) > − δ } is called the Carath´eodory dimension of µ . We say that µ is a measure of maximalCarath´eodory dimension if dim C µ = dim C X . Note that if the Carath´eodory measure m C ( X, α ) at dimension α = dim C X is finite and positive, then this measure is a measureof maximal Carath´eodory dimension.With f : Λ → Λ as in Theorem 4.7, we consider a particular but important family ofpotential functions ϕ geo t ( x ) on Λ, called the geometric t -potentials : for any t ∈ R ϕ geo t ( x ) := − t log | det Df | E u ( x ) | . Since the subspace E u ( x ) depends H¨older continuously on x ∈ Λ (see (3.1)), for each t ∈ R the function ϕ geo t ( x ) is H¨older continuous and hence, it admits a unique equilibrium state µ t := µ ϕ geo t .We consider the function P ( t ) := P ( ϕ geo t ) called the pressure function . One can showthat this function is monotonically decreasing, convex and real analytic in t . Moreover, P ( t ) → + ∞ as t → −∞ and P ( t ) → −∞ as t → + ∞ with P (1) ≤
0. Therefore, there isa number 0 < t ≤ Bowen’s equation P ( t ) = 0. We shallshow that given x ∈ Λ, there is a C -structure on the set X = V u loc ( x ) ∩ Λ with respect towhich t is the Carath´eodory dimension of the set X . Indeed, since P ( t ) = 0, the measure m t x := m C x , given by (4.13) for C = ( ϕ geo t , r ), can be written as(5.6) m t x ( Z ) = lim N →∞ inf (cid:40)(cid:88) i (cid:18) n i − (cid:89) k =0 det (cid:0) Df | E u ( f k ( x i )) (cid:1)(cid:19) − t (cid:41) , where the infimum is taken over all collections { B un i ( x i , r ) } of u -Bowen balls with x i ∈ X , n i ≥ N , which cover Z .Relation (5.6) shows that the measure m t x is the Carath´eodory measure generated bythe C -structure τ (cid:48) = ( S , F , ξ (cid:48) , η (cid:48) , ψ ), where ξ (cid:48) ( x, n ) := 1 , η (cid:48) ( x, n ) := n i − (cid:89) k =0 det (cid:0) Df | E u ( f k ( x i )) (cid:1) − . It is easy to see that with respect to the C -structure τ (cid:48) we have that dim C,τ (cid:48) X = t andthe measure m t x = m C,τ (cid:48) ( · , t ) is the measure of maximal Carath´eodory dimension. Inparticular, the Carath´eodory dimension of X = V u loc ( x ) ∩ Λ does not depend on the choiceof the point x ∈ Λ. It is also clear that the number t depends continuously on f in the C topology and hence, so does the Carath´eodory dimension dim C,τ (cid:48) X . QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 29
We consider the particular case when the map f is u -conformal; that is Df | E u ( x ) = a ( x )Isom x for all x ∈ Λ, where Isom x is an isometry. The direct calculation involving (5.6)shows that in this case m t x is a measure of full Hausdorff dimension and that t dim E u =dim H X .Given a locally maximal hyperbolic set Λ, it has been a long-standing open problemto compute the Hausdorff dimension of the set X = V u loc ( x ) ∩ Λ and to find an invariantmeasure whose conditional measures on unstable leaves have maximal Hausdorff dimension,provided such a measure exists. The above result solves this problem for u -conformal dif-feomorphisms. The reader can find the original proof and relevant references in [Pes97]. Itwas recently proved that without the assumption of u -conformality, there are examples forwhich there is no invariant measure whose conditionals have full Hausdorff dimension; see[DS17]. Theorem 4.11 provides one way to settle the issue in the non-conformal case by re-placing ‘measure of maximal Hausdorff dimension’ with ‘measure of maximal Carath´eodorydimension’ with respect to the C -structure τ (cid:48) just described.6. Outline of proofs In §§ § § Reference measures are nonzero and finite.
Recall that Λ is a locally maximalhyperbolic set for f , on which each x has local stable and unstable manifolds of size τ >
0. We assume that f | Λ is topologically transitive. In what follows we occasionally usethe following notation: given A, B, C, a ≥
0, we write A = C ± a B as shorthand to mean C − a B ≤ A ≤ C a B . The key to the proof of Theorem 4.7 is the following result. Proposition 6.1.
For every r ∈ (0 , τ ) and r ∈ (0 , τ / there is C > such that for every x ∈ Λ and n ∈ N we have (6.1) Z span n ( B u Λ ( x, r ) , ϕ, r ) = C ± e nP ( ϕ ) . Similar partition sum bounds are obtained in Bowen’s paper [Bow75], where they areproved for all of Λ instead of for a single unstable leaf. For the full proof of Proposition6.1, see [CPZ18, § § § Z span n , and in § Elementary counting lemmas.
Lemma 6.2. If Z n > is a sequence of numbers satisfying Z n + m ≤ Z n Z m for all m, n ,then P = lim n →∞ n log Z n exists and is equal to inf n ∈ N n log Z n . In particular, Z n ≥ e nP for every n .Proof. Fix n ∈ N ; then for all m ∈ N we can write m = an + b where a ∈ N and b ∈{ , , . . . , n − } , and iterate the submultiplicativity property to obtain Z m ≤ Z an Z b . Taking logs and dividing by m gives1 m log Z m ≤ am log Z n + log Z b m ≤ anm · n log Z n + max { log Z , . . . log Z n − } m . Sending m → ∞ we see that anm →
1, so(6.2) lim m →∞ m log Z m ≤ n log Z n . Since n was arbitrary we deduce thatlim m →∞ m log Z m ≤ inf n ∈ N n log Z n ≤ lim n →∞ n log Z n , whence all three terms are equal and the limit exists. Now (6.2) implies that Z n ≥ e nP . (cid:3) Lemma 6.3. If Z n > is a sequence of numbers satisfying Z n + m ≤ CZ n Z m for all m, n ,where C > is independent of m, n , then P = lim n →∞ n log Z n exists and is equal to inf n ∈ N n log( CZ n ) . In particular, Z n ≥ C − e nP for all n .Proof. Follows by applying Lemma 6.2 to the sequence Y n = CZ n , which satisfies Y n + m = CZ n + m ≤ C Z n Z m = Y n Y m . (cid:3) Lemma 6.4. If Z n > is a sequence of numbers satisfying Z n + m ≥ C − Z n Z m for all m, n , where C > is independent of m, n , then P = lim n →∞ n log Z n exists and is equal to sup n ∈ N n log( Z n /C ) . In particular, Z n ≤ Ce nP for all n .Proof. Follows by applying Lemma 6.2 to the sequence Y n = C/Z n , which satisfies Y n + m = C/Z n + m ≤ C / ( Z n Z m ) = Y n Y m . (cid:3) Partition sums are nearly multiplicative.
In light of Lemmas 6.3 and 6.4, Proposition6.1 can be proved by showing that the partition sums Z span n ( B u Λ ( x, r ) , ϕ, r ) are ‘nearlymultiplicative’: Z span n + m = C ± Z span n Z span m . A short argument given in [CPZ18, Lemma 6.3]shows that Z sep n = e ± Q u Z span n , and thus it suffices to show that Z span n + m ≤ CZ span n Z span m , Z sep n + m ≥ C − Z sep n Z sep m , where we are being deliberately vague about the arguments of Z span and Z sep . xy f n y f n y f n y f n y f n y f n B u ( x, r ) B u ( f n y , r ) z z Figure 6.1.
Proving that Z sep n + m ≥ C − Z sep n Z sep m . QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 31
Figure 6.1 illustrates the idea driving the estimate for Z sep : if E n = { y , . . . , y a } ⊂ B u Λ ( x, r ) is a maximal ( n, r )-separated set of points, and to each 1 ≤ i ≤ a we associate amaximal ( m, r )-separated set E im = { z i , . . . , z ib i } ⊂ B u Λ ( f n y i , r ), then pulling back all thepoints z ij gives an ( m + n, r )-separated set E m + n = a (cid:91) i =1 f − n ( E im ) = { f − n z ij : 1 ≤ i ≤ a, ≤ j ≤ b i } ⊂ B u Λ ( x, r ) , so we expect to get an estimate along the lines of(6.3) Z sep m + n ≥ (cid:88) p ∈ E m + n e S m + n ϕ ( p ) = (cid:88) p ∈ E m + n e S n ϕ ( p ) e S m ϕ ( f n ( p )) (cid:39) a (cid:88) i =1 b i (cid:88) j =1 e S n ϕ ( y i ) e S m ϕ ( z ij ) = a (cid:88) i =1 e S n ϕ ( y i ) (cid:18) b i (cid:88) j =1 e S m ϕ ( z ij ) (cid:19) ≈ (cid:18) a (cid:88) i =1 e S n ϕ ( y i ) (cid:19) Z sep m ≈ Z sep m Z sep n , where we continue to be deliberately vague about the arguments of Z sep . If we can makethis rigorous, then a similar argument with spanning sets instead of separated sets will leadto Z span m + n ≤ CZ span m Z span n , which will prove Proposition 6.1.But how do we make (6.3) rigorous? There are two sources of error which are hinted atby the “ ≈ ” symbols.(1) Given p ∈ E m + n and the corresponding y ∈ E n (where f n ( p ) ∈ B u Λ ( f n y, r )), theapproximation on the first line of (6.3) requires us to compare the ergodic sums S n ϕ ( p ) and S n ϕ ( y ). In particular, we must find a constant Q u (independent of y, p, n ) such that | S n ϕ ( p ) − S n ϕ ( y ) | ≤ Q u whenever p ∈ B un ( y, r ).(2) The omission of the arguments for Z sep n obscures the fact that Z sep m + n and Z sep m in(6.3) both refer to ( n, r )-separated subsets of B u Λ ( x, r ), while Z sep n refers to ( n, r )-separated subsets of B u Λ ( f n y i , r ). Thus we must control how Z sep n ( B u Λ ( x, r ) , ϕ, r )changes when we fix n and let x, r , r vary; in particular, we must find for each r , r (cid:48) , r , r (cid:48) a constant C such that for every n, x, y , we have Z sep n ( B u Λ ( x, r ) , ϕ, r ) = C ± Z sep n ( B u Λ ( y, r (cid:48) ) , ϕ, r (cid:48) ) . The first source of error described above can be controlled by establishing a generalizedversion of property (4.4).
Definition 6.5.
We say that a potential ϕ : Λ → R has the u -Bowen property if there is Q u > such that for every x ∈ Λ , n ≥ , and y ∈ B un ( x, τ ) ∩ Λ , we have | S n ϕ ( x ) − S n ϕ ( y ) | ≤ Q u . We also say that ϕ has the s -Bowen property if there is Q s > such that for every x ∈ Λ , n ≥ , and y ∈ B s Λ ( x, τ ) = B s ( x, τ ) ∩ Λ , we have | S n ϕ ( x ) − S n ϕ ( y ) | ≤ Q s . The asymmetry in the definition comes because S n ϕ is a forward Birkhoff sum and B un ( x, τ ) is definedin terms of forward iterates; one could equivalently define the s -Bowen property in terms of backwardBirkhoff sums and s -Bowen balls. The s -Bowen property is needed to control the second source of errordescribed above. Lemma 6.6. If ϕ : Λ → R is H¨older continuous, then ϕ has the u -Bowen property and the s -Bowen property. Proof.
We prove the u -Bowen property; the proof of the s -Bowen property is similar. Given y ∈ B un ( x, τ ), for every 0 ≤ k < n we have d ( x, y ) ≤ τ λ n − k where 0 < λ < β for the H¨older exponent of ϕ , we have | S n ϕ ( x ) − S n ϕ ( y ) | ≤ n − (cid:88) k =0 | ϕ ( f k x ) − ϕ ( f k y ) | ≤ n − (cid:88) k =0 | ϕ | β d ( f k x, f k y ) β ≤ | ϕ | β τ β n − (cid:88) k =0 λ β ( n − k ) < | ϕ | β τ β (1 − λ β ) − =: Q u . (cid:3) To control the second source of error described above, the first main idea is that topolog-ical transitivity guarantees that for every δ >
0, the images f k ( B u Λ ( y, δ )) eventually comewithin δ of x , and that the k for which this occurs admits an upper bound that dependsonly on δ . Then given a spanning set E ⊂ B u Λ ( y, δ ), the part of the image f k ( E ) thatlies near x can be moved by holonomy along stable manifolds to give a spanning set inthe unstable leaf of x . This is made precise in [CPZ18, Lemma 6.4]. One can use similararguments to change the scales r , r ; for example, if x, y are on the same local unstableleaf and have orbits that remain within r of each other until time n , then they remainwithin r λ k of each other until time n − k . See [CPZ18, §
6] for full details.6.1.3.
Proving Theorem 4.7.
Fix x ∈ Λ and set X := V u loc ( x ) ∩ Λ. We showed in § m C x defines a metric outer measure on X , and hence gives a Borel measure. Note thatthe final claim in Theorem 4.7 about agreement on intersections is immediate from thedefinition. Thus it remains to prove that m C x ( X ) ∈ [ K − , K ], where K is independent of x ;this will complete the proof of Theorem 4.7, and will also prove Proposition 5.4.For full details, see [CPZ18, § r >
0, we have m C x ( B u Λ ( x, r )) uniformly bounded away from 0 and ∞ , since each V u loc ( x )can be covered with a uniformly finite number of balls B u Λ ( y, r ). The upper bound is easierto prove since it only requires that we exhibit a cover satisfying the desired inequality;this is provided by Proposition 6.1, which guarantees existence of an ( n, r )-spanning set E n ⊂ B u Λ ( x, r ) such that (cid:88) y ∈ E n e S n ϕ ( y ) ≤ Ce nP ( ϕ ) , and thus (4.13) gives m C x ( B u Λ ( x, r )) ≤ lim n →∞ (cid:88) y ∈ E n e − nP ( ϕ ) e S n ϕ ( y ) ≤ C. The lower bound is a little trickier since we must obtain a lower bound for an arbitrary cover by u -Bowen balls as in (4.13), which are allowed to be of different orders, so we donot immediately get an ( n, r )-spanning set for some particular n . This can be resolved byobserving that any open cover of B u Λ ( x, r ) has a finite subcover, so to bound m C x ( B u Λ ( x, r )) This is the only place where H¨older continuity is used; in particular, H¨older continuity could be replacedby the u - and s -Bowen properties in all our main results. QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 33 it suffices to consider covers of the form { B un i ( y i , r ) : 1 ≤ i ≤ a, n i ≥ N } . Given such acover, one can take n = max( n , . . . , n a ) and use arguments similar to those in the proof ofProposition 6.1 to cover each B un i ( y i , r ) by a union of u -Bowen balls B un ( z ji , r ) (1 ≤ j ≤ b i )satisfying (cid:88) j e S n ϕ ( z ji ) ≤ C (cid:48) e ( n − n i ) P ( ϕ ) e S ni ϕ ( y i ) for some constant C (cid:48) that is independent of our choice of covers. Then the set E = { z ji :1 ≤ i ≤ a, ≤ j ≤ b i } is ( n, r )-spanning for B u Λ ( x, r ) and satisfies C − e nP ( ϕ ) ≤ (cid:88) z ∈ E e S n ϕ ( z ) ≤ a (cid:88) i =1 C (cid:48) e ( n − n i ) P ( ϕ ) e S ni ϕ ( y i ) . Dividing through by e nP ( ϕ ) , taking an infimum over all covers, and sending N → ∞ gives C − ≤ C (cid:48) m C x ( B u Λ ( x, r )). Again, full details are in [CPZ18, § Behavior of reference measures under iteration and holonomy.
Iteration and the u -Gibbs property. The simplest case of Theorem 4.8 occurs when ϕ = 0, so the claim is that m C f ( x ) = e P (0) m C x ◦ f − , which is exactly the scaling propertysatisfied by the Margulis measures on unstable leaves. Given E ⊂ V u loc ( f ( x )), we seefrom the relationship f − B un ( y, r ) = B un +1 ( f − y, r ) that any cover { B un i ( y i , r ) } of E leadsimmediately to a cover { B un i +1 ( f − y i , r ) } of f − E , and vice versa. Using this bijection inthe definition of the reference measures in (4.13), we get m C f ( x ) ( E ) = lim N →∞ inf (cid:88) i e − n i P (0) = e P (0) lim N →∞ inf (cid:88) i e − ( n i +1) P (0) = e P (0) m C x ( f − E ) . For nonzero potentials one must account for the factor of e S ni ϕ ( x i ) in (4.13). This can bedone by partitioning E into subsets E , . . . , E T on which ϕ is nearly constant, and repeatingthe above argument on each E i to get an approximate result that improves to the desiredresult as T → ∞ ; see [CPZ18, § m C f n ( x ) ( A ) = (cid:90) f − n ( A ) e nP ( ϕ ) − S n ϕ ( y ) dm C x ( y )for all A ⊂ V u loc ( f n ( x )). Applying this to A = B u ( f n ( x ) , δ ) = f n ( B un ( x, δ )) and usingTheorem 4.7 gives a constant Q = Q ( δ ) such that m C x ( B un ( x, δ )) e nP ( ϕ ) − S n ϕ ( x ) = e ± Q u (cid:90) B un ( x,δ ) e nP ( ϕ ) − S n ϕ ( y ) dm C x ( y )= e ± Q u m C f n ( x ) ( B u ( f n ( x ) , δ )) = e ± Q u Q ± , for every x, n , where the first estimate uses the u -Bowen property from Lemma 6.6. Thisestablishes the u -Gibbs property for m C x with Q = Q e Q u and proves Corollary 4.9. Holonomy maps.
Given nearby points y, z and sets E y ⊂ V u loc ( y ), E z ⊂ V u loc ( z ) suchthat π yz ( E y ) = E z (with respect to some rectangle), we observe that every cover of E y by u -Bowen balls { B un i ( x i , r ) } produces a cover of E z by the images { π yz B un i ( x i , r ) } . If y, z areclose enough to each other to guarantee that(6.5) π yz B un i ( x i , r ) ⊂ B un i ( x i , r )for each i , then we get E z ⊂ (cid:83) i B un i ( x i , r ). Fixing k ∈ N such that each x ∈ Λ has B u Λ ( x, r ) ⊂ (cid:83) kj =1 B u Λ ( x j , r ) for some x , . . . , x k , we see that E z ⊂ (cid:83) i,j B un i ( x ji , r ), and thus(4.13) gives m C z ( E z ) ≤ (cid:88) i,j e − n i P ( ϕ ) e S ni ϕ ( x ji ) ≤ k (cid:88) i e − n i P ( ϕ ) e S ni ϕ ( x i )+ Q u ;taking an infimum and then a limit gives m C z ( E z ) ≤ ke Q u m C y ( E y ).In general, if y, z lie close enough for holonomy maps to be defined, but not close enoughfor (6.5) to hold, then we can iterate E y , E z forward until some time n at which f n y, f n z areclose enough for the previous part to work, and use Theorem 4.8 to get (assuming withoutloss of generality that E y ⊂ B un ( y, τ ), and similarly for E z ) m C z ( E z ) = (cid:90) f n ( E z ) e − nP ( ϕ )+ S n ϕ ( f − n x ) dm C f n z ( x ) ≤ ke Q u (cid:90) f n ( E y ) e − nP ( ϕ )+ S n ϕ ( f − n x (cid:48) )+ Q s dm C f n y ( x (cid:48) ) = ke Q u + Q s m C y ( E y ) , where the inequality uses the result from the previous paragraph. Since the roles of y, z were symmetric, this proves Theorem 4.10 with Q = ke Q u + Q s . See [CPZ18, § Geometric construction of equilibrium states.
Now that we have established thebasic properties of the reference measures m C x associated to a H¨older continuous potentialfunction ϕ , the steps in the geometric construction of the unique equilibrium state µ ϕ areas follows.(1) Prove that every weak*-limit point µ of the sequence of probability measures µ n = n (cid:80) n − k =0 f k ∗ m C x /m C x ( V u loc ( x )) is an invariant measure whose conditional measures sat-isfy part (3) of Theorem 4.11; in particular, they are equivalent to the referencemeasures m C y .(2) Use this to deduce that any such µ satisfies part (2) of Theorem 4.11, namely:(a) the conditional measures of µ are absolutely continuous with respect to stableholonomies, and therefore µ is ergodic by the Hopf argument (Proposition 4.5);(b) µ gives positive weight to every open set in Λ;(c) the u -Gibbs property of the reference measures implies the Gibbs property(3.15) for µ ; and(d) µ is the unique equilibrium state for ϕ by Proposition 3.11.(3) Observe that each µ n is a Borel probability measure on Λ, and thus every sub-sequence has a subsubsequence that converges in the weak*-topology to a Borel QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 35 probability measure µ , which must be the unique equilibrium state µ ϕ by the pre-vious step. Since every subsequence of µ n has a subsubsequence converging to µ ϕ ,it follows that the sequence itself converges to this limit, which establishes part (1)of Theorem 4.11.The first step takes most of the work; once it is done, parts (a)–(c) of the second steponly require short arguments that leverage the properties already established, part (d) ofthe second step merely consists of observing that µ satisfies the hypotheses of Proposition3.11, and the third step is completely contained in the paragraph above. Thus we outlinehere the argument for the first step and parts (a)–(c) of the second step, referring oncemore to [CPZ18] for complete details.6.3.1. Conditional measures of limiting measures.
In order to understand the conditionalmeasures of µ = lim k →∞ µ n k , we start by studying the conditional measures of µ n . Given x ∈ Λ and n ∈ N , the iterate f n ∗ m C x is supported on W n = f n ( V u loc ( x ) ∩ Λ), and given any y ∈ W n , we can iterate the formula from Theorem 4.8 and obtain(6.6) d (( f n ∗ m C x ) | V u loc ( y ) ) dm C y ( z ) = e − nP ( ϕ )+ S n ϕ ( f − n z ) =: g n ( z )for every z ∈ W n ∩ V u loc ( y ). One can show that g n → n → ∞ , so it is convenient towrite ρ yn ( z ) := g n ( z ) /g n ( y ), and Lemma 6.6 gives(6.7) ρ yn ( z ) = e S n ϕ ( f − n z ) − S n ϕ ( f − n y ) ∈ [ e − Q u , e Q u ] . These functions describe the conditional measures of f n ∗ m C x . Indeed, given a rectangle R ,choose y , . . . , y a ∈ f n ( V u loc ( x )) ∩ R such that f n ( V u loc ( x )) ∩ R ⊂ (cid:83) ai =1 V u loc ( y i ), as in Figure6.2. Then for every Borel set E ⊂ R , we have (6.8) f n ∗ m C x ( E ) = a (cid:88) i =1 (cid:90) E g n ( z ) dm C y i ( z ) = a (cid:88) i =1 g n ( y i ) (cid:90) E ρ y i n ( z ) dm C y i ( z ) . In other words, one can write f n ∗ m C x | R as a linear combination of the measures ρ y i n dm C y i associated to the standard pairs ( V u loc ( y i ) , ρ y i n ), with coefficients given by g n ( y i ). Thisimmediately implies that the conditional measures of µ n on local unstable leaves are abso-lutely continuous with respect to the reference measures m C x , with densities bounded awayfrom 0 and ∞ .To go further, we need the following characterization of the conditional measures, whichis an immediate consequence of [EW11, Corollary 5.21]. There is a small technical issue here, namely that there may be some y i at which W n does not cross R completely, and so the integral in (6.8) actually gives too large a value. However, this can only occur if z i = f − n ( y i ) is very close to the boundary of V u loc ( x ), and the contribution of such points is negligible inthe limit; see [CPZ18]. Standard pairs consisting of a local leaf V u loc ( y ) and a density function ρ were introduced by Chernov andDolgopyat in [CD09] to study stochastic properties of dynamical systems; they are also used in constructingSRB measures for some dynamical systems with weak hyperbolicity, see [CDP16]. See [PS82] for the analogous argument controlling the conditionals of µ when ϕ is the geometricpotential and the reference measure is leaf volume. In that setting, the role of Proposition 6.7 here is playedby [PS82, Lemma 13]. xV u loc ( x ) z z z z z Ry y y y y f n ( z i = f − n ( y i )) Figure 6.2.
Studying f n ∗ m C x on a rectangle R . Proposition 6.7.
Let µ be a finite Borel measure on Λ and let R ⊂ Λ be a rectangle with µ ( R ) > . Let { ξ (cid:96) } (cid:96) ∈ N be a refining sequence of finite partitions of R that converge to thepartition ξ into local unstable sets V uR ( y ) = V u loc ( y ) ∩ R . Then there is a set R (cid:48) ⊂ R with µ ( R (cid:48) ) = µ ( R ) such that for every y ∈ R (cid:48) and every continuous ψ : R → R , we have (6.9) (cid:90) V uR ( y ) ψ ( z ) dµ uy ( z ) = lim (cid:96) →∞ µ ( ξ (cid:96) ( y )) (cid:90) ξ (cid:96) ( y ) ψ ( z ) dµ ( z ) , where ξ n ( y ) denotes the element of the partition ξ (cid:96) that contains y . ξ R ξ ξ ξ Figure 6.3.
A refining sequence of partitions of R .Now the proof of Part (3) of Theorem 4.11 goes as follows. Given a rectangle R ⊂ Λwith µ ( R ) >
0, let ξ (cid:96) be a refining sequence of finite partitions of R such that for every y ∈ R and (cid:96) ∈ N , the set ξ (cid:96) ( y ) is a rectangle, and (cid:84) (cid:96) ∈ N ξ (cid:96) ( y ) = V uR ( y ), as in Figure 6.3. Let R (cid:48) ⊂ R be the set given by Proposition 6.7. We prove that (4.14) holds for each y ∈ R (cid:48) byshowing that for every positive continuous function ψ : R → R , we have(6.10) (cid:90) V uR ( y ) ψ dµ uy = Q ± m C y ( V uR ( y )) (cid:90) V uR ( y ) ψ dm C y , where Q is a constant that is independent of ψ . To this end, we need to compare µ ( ξ (cid:96) ( y ))and (cid:82) ξ (cid:96) ( y ) ψ ( z ) dµ ( z ) and then apply (6.9). We see from (6.7) and (6.8) that for each j ∈ N there is a finite set Y j ⊂ R such that(6.11) (cid:90) ξ (cid:96) ( y ) ψ d ( f n ∗ m C x ) = (cid:88) p ∈ Y j g j ( p ) (cid:90) V uR ( p ) ψ ( z ) e ± Q u dm C p ( z ) . Given p ∈ ξ (cid:96) ( y ), Theorem 4.10 gives (cid:90) V uR ( p ) ψ dm C p = Q ± (cid:90) V uR ( y ) ψ ( π py z (cid:48) ) dm C y ( z (cid:48) ) = (2 Q ) ± (cid:90) V uR ( y ) ψ dm C y QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 37 whenever p, y are sufficiently close that ψ ( π py z (cid:48) ) = 2 ± ψ ( z (cid:48) ) for all z (cid:48) ∈ V uR ( y ). Thus for allsufficiently large (cid:96) , (6.11) gives (cid:90) ξ (cid:96) ( y ) ψ d ( f n ∗ m C x ) = e ± Q u (2 Q ) ± (cid:18) (cid:88) p ∈ Y j g j ( p ) (cid:19) (cid:90) V uR ( y ) ψ dm C y . Averaging over 0 ≤ j < n k and sending k → ∞ gives(6.12) (cid:90) ξ (cid:96) ( y ) ψ dµ = (2 Q e Q u ) ± (cid:16) lim k →∞ n k n k − (cid:88) j =0 (cid:88) p ∈ Y j g k ( p ) (cid:17) (cid:90) V uR ( y ) ψ dm C y . When ψ ≡ µ ( ξ (cid:96) ( y )) = (2 Q e Q u ) ± (cid:16) lim k →∞ n k n k − (cid:88) j =0 (cid:88) p ∈ Y j g k ( p ) (cid:17) m C y ( V uR ( y )) . Dividing (6.12) by (6.13), sending (cid:96) → ∞ , and using (6.9) yields (cid:90) V uR ( y ) ψ dµ uy = (2 Q e Q u ) ± m C y ( V uR ( y )) (cid:90) V uR ( y ) ψ dm C y . Since ψ > V u loc ( x ) that are dealt with in [CPZ18].6.3.2. Other properties of limiting measures.
Throughout this section, µ will denote anarbitrary f -invariant Borel probability measure on Λ that satisfies (4.14), so that the con-ditional measures µ uy are equivalent to the reference measures m C y . By the previous section,this includes every limit point of the sequence µ n .We first observe that by Theorem 4.10 and (4.14), for every rectangle R with µ ( R ) > µ -a.e. y, z ∈ R , and every A ⊂ V uR ( z ), we have(6.14) µ uy ( π zy A ) = Q ± m C y ( π zy A ) /m C y ( R ) = Q ± Q ± m C z ( A ) /m C z ( R ) = ( Q Q ) ± µ uz ( A ) . In particular, holonomy maps along stable manifolds are absolutely continuous with respectto the conditional measures µ uy , and thus by the standard Hopf argument (Proposition 4.5), µ is ergodic.Now we prove that µ is fully supported and satisfies the Gibbs property. Let δ > x ∈ Λ, the rectangle R ( x, δ ) := [ B u Λ ( x, δ ) , B s Λ ( x, δ )] = { [ y, z ] : y ∈ B u Λ ( x, δ ) , z ∈ B s Λ ( x, δ ) } is well-defined, as in (3.2). Given n ∈ N , consider the rectangle R n ( x, δ ) := [ B un ( x, δ ) ∩ Λ , B s Λ ( x, δ )] ⊂ R ( x, δ ) . It is shown in [CPZ18, Lemma 8.3] that for every δ >
0, there are δ , δ > R n ( x, δ ) ⊂ B n ( x, δ ) ⊂ R n ( x, δ )for every x ∈ Λ and n ∈ N ; thus to prove the Gibbs property (3.14) it suffices to establishthe corresponding bounds on µ ( R n ( x, δ )). Lemma 6.8.
Given δ > , there is Q > such that for every x, δ, n as above, we have (6.16) µ ( R n ( x, δ )) = Q ± e − nP ( ϕ )+ S n ϕ ( x ) µ ( R ( x, δ )) . Proof.
Writing µ uy for the conditional measures of µ on unstable leaves in R ( z, δ ), we have µ ( R n ( x, δ )) = (cid:90) R ( x,δ ) µ uy ( R n ( x, δ )) dµ ( y ) = Q ± (cid:90) R ( x,δ ) m C y ( R n ( x, δ )) m C y ( R ( x, δ )) dµ ( y )= ( KQ ) ± (cid:90) R ( x,δ ) m C y ( π xy B un ( x, δ )) dµ ( y ) = ( KQ Q ) ± m C x ( B un ( x, δ )) µ ( R ( x, δ )) , where the first equality uses the definition of conditional measures, the second uses (4.14),the third uses Theorem 4.7, and the fourth uses Theorem 4.10. Since B un ( x, δ ) ⊂ B un ( x, δ ) ⊂ B un ( x, δ ), the result follows from the u -Gibbs property of m C x . (cid:3) Lemma 6.9 ([CPZ18, Lemma 8.4]) . For every sufficiently small δ > , there is δ (cid:48) > suchthat for every z ∈ Λ and x ∈ R ( z, δ (cid:48) ) , we have R ( z, δ (cid:48) ) ⊂ R ( x, δ ) . Lemma 6.10. If y ∈ Λ has a backwards orbit that is dense in Λ , then µ ( R ( y, δ )) > forall δ > .Proof. Let δ (cid:48) > E ⊂ Λ suchthat (cid:83) z ∈ E R ( z, δ (cid:48) ) = Λ, and thus there is z ∈ E with µ ( R ( z, δ (cid:48) )) >
0. Since the backwardsorbit of y is dense, there is n ≥ x := f − n ( y ) ∈ R ( z, δ (cid:48) ). By Lemma 6.9 and ourchoice of x , we have µ ( R ( x, δ )) ≥ µ ( R ( z, δ (cid:48) )) > . By Lemma 6.8, we conclude that µ ( R n ( x, δ )) >
0. Moreover, we have f n R n ( x, δ ) = f n [ B un ( x, δ ) ∩ Λ , B s Λ ( x, δ )] ⊂ [ B u Λ ( y, δ ) , B s Λ ( y, δ )] = R ( y, δ ) , where we use the fact that (cid:107) Df | E s (cid:107) ≤
1. Since µ is f -invariant, this gives µ ( R ( y, δ )) ≥ µ ( R n ( x, δ )) > (cid:3) Since f is topologically transitive on Λ, every (relatively) open set in Λ contains a set ofthe form R ( y, δ ) where y has a dense backwards orbit. Thus Lemma 6.10 implies that µ isfully supported on Λ.Finally, we deduce the Gibbs property (3.15) for µ as follows. Given δ >
0, let δ (cid:48) > E ⊂ Λ be a finite set with (cid:83) z ∈ E R ( z, δ (cid:48) ) = Λ. Since µ is fully supported, we have η := min z ∈ E µ ( R ( z, δ (cid:48) )) >
0. Now given any x ∈ Λ, we have x ∈ R ( z, δ (cid:48) ) for some z ∈ E , and thus Lemma 6.9 gives µ ( R ( x, δ )) ≥ µ ( R ( z, δ (cid:48) )) ≥ η. In particular, η ≤ µ ( R ( x, δ )) ≤ x ∈ Λ, and then the Gibbs property (3.15)follows immediately from Lemma 6.8 and (6.15).
QUILIBRIUM STATES IN DYNAMICS VIA GEOMETRIC MEASURE THEORY 39
Local product structure.
To prove Corollary 4.12, we first observe that (6.14) gives π ∗ yp µ up ∼ µ uy for µ -a.e. p, y ∈ R , which is the first claim. For the second claim, we define afunction h : R → (0 , ∞ ) by h ( z ) = dµ uz d ( π ∗ zp µ up ) ( z ), so that (3.4) gives(6.17) µ ( E ) = (cid:90) V sR ( p ) (cid:90) V uR ( y ) E ( z ) dµ uy ( z ) d ˜ µ p ( y )= (cid:90) V sR ( p ) (cid:90) V uR ( y ) E ( z ) h ( z ) d ( π ∗ zp µ up )( z ) d ˜ µ p ( y ) = (cid:90) E h ( z ) d ( µ up ⊗ ˜ µ p )( z ) . for every measurable E ⊂ R . For the third claim, we observe that (6.17) gives µ ( E ) = (cid:90) E h ( z ) d ( µ up ⊗ ˜ µ p )( z ) = (cid:90) V uR ( p ) (cid:90) V sR ( y ) E ( z ) h ( z ) d ( π ∗ yp ˜ µ p )( z ) dµ up ( y ) , and since µ sy is uniquely determined up to a scalar (for µ -a.e. y ) by the condition that µ ( E ) = (cid:90) V uR ( p ) (cid:90) V sR ( y ) E ( z ) dµ sy ( z ) dν ( y )for some measure ν on V uR ( p ), we conclude that dµ sy = h d ( π ∗ yp ˜ µ p ).6.4. Proof of Theorem 4.1.
We have that for any Borel subset E ⊂ Y (6.18) µ ( E ) = (cid:90) ˜ Y (cid:90) W E ( z ) dµ ξW ( z ) d ˜ µ ( W ) . Without loss of generality we may assume that µ ξW is normalized, so that µ ξW ( W ) = 1.Consider the set B of all Birkhoff generic points x ∈ X , for which(6.19) lim n →∞ n n − (cid:88) k =0 h ( f k ( x )) = (cid:90) X h dµ for every continuous function h on X . Since µ is ergodic we have that B has full measurein Y . By (6.18), there is a set D ⊂ ˜ Y such that ˜ µ ( ˜ Y \ D ) = 0 and for every W ∈ D wehave µ ξW ( W \ B ) = 0. Given any such W and any measure ν (cid:28) µ ξW , we have ν ( X \ B ) = 0.Then Theorem 4.1 is a consequence of the following general result. Proposition 6.11.
Let X be a compact metric space, f : X → X a continuous map, and µ an f -invariant Borel probability measure on X . Let B be the set of Birkhoff generic pointsfor µ and let ν be any probability measure on X such that ν ( B ) = 1 . Then the sequence ofmeasures ν n := n (cid:80) n − k =0 f k ∗ ν converges in the weak* topology to the measure µ . Before proving Proposition 6.11, we note that µ is not required to be ergodic. In the casewhen µ is ergodic, Birkhoff’s theorem gives µ ( B ) = 1, so that in particular B is nonempty.For non-ergodic µ , the set B can be either empty or nonempty. Proof of Proposition 6.11.
Let κ be a weak* limit point of the sequence ν n , so that thereis a subsequence { n (cid:96) } (cid:96) ∈ N such that for every continuous function h on X , we have(6.20) (cid:90) X h dκ = lim (cid:96) →∞ (cid:90) X h dν n (cid:96) = lim (cid:96) →∞ (cid:90) n (cid:96) n (cid:96) − (cid:88) k =0 h ◦ f k dν. We show that κ ≤ µ , which implies that κ = µ since both are probability measures. Itsuffices to show that (cid:82) h dκ ≤ (cid:82) h dµ for every nonnegative continuous function h .Fix h as above. Given N ∈ N and ε >
0, let B N ( ε ) := (cid:26) x ∈ B : (cid:12)(cid:12)(cid:12) n S n h ( x ) − (cid:90) h dµ (cid:12)(cid:12)(cid:12) < ε for all n ≥ N (cid:27) . Then for every ε > (cid:83) N ∈ N B N ( ε ) = B , hence there is N ε such that ν ( B \ B N ( ε )) <ε . By (6.20) we can choose n (cid:96) > N ε such that (cid:90) X h dκ ≤ ε + (cid:90) n (cid:96) S n (cid:96) h dν = ε + (cid:90) B N ( ε ) n (cid:96) S n (cid:96) h dν + (cid:90) B \ B N ( ε ) n (cid:96) S n (cid:96) h dν ≤ ε + (cid:90) h dµ + ν ( B \ B N ( ε )) (cid:107) h (cid:107) < ε (2 + (cid:107) h (cid:107) ) + (cid:90) h dµ. Since ε > (cid:3)
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