Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent
aa r X i v : . [ m a t h . D S ] M a y ZERO TEMPERATURE LIMITS OF EQUILIBRIUM STATESFOR SUBADDITIVE POTENTIALS AND APPROXIMATION OFTHE MAXIMAL LYAPUNOV EXPONENT
REZA MOHAMMADPOUR
Abstract.
In this paper we study ergodic optimization problems for subaddi-tive sequences of functions on a topological dynamical system. We prove thatfor t → ∞ any accumulation point of a family of equilibrium states is a maximiz-ing measure. We show that the Lyapunov exponent and entropy of equilibriumstates converge in the limit t → ∞ to the maximum Lyapunov exponent andentropy of maximizing measures.In the particular case of matrix cocycles we prove that the maximal Lyapunovexponent can be approximated by Lyapunov exponents of periodic trajectoriesunder certain assumptions. Introduction and statement of the results
Throughout this paper X is a compact metric space that is endowed with themetric d . We call ( X, T ) a topological dynamical system (TDS), if T : X → X is acontinuous map on the compact metric space X . We say that Φ := { log φ n } ∞ n =1 isa subadditive potential if each φ n is a continuous non-negative-valued function on X such that ≤ φ n + m ( x ) ≤ φ n ( x ) φ m ( T n ( x )) ∀ x ∈ X, m, n ∈ N . Furthermore,
Φ = { log φ n } ∞ n =1 is said to be an almost additive potential if thereexists a constant C > such that for any m, n ∈ N , x ∈ X , we have C − φ n ( x ) φ m ( T n )( x ) ≤ φ n + m ( x ) ≤ Cφ n ( x ) φ m ( T n ( x )) . For any T − invariant measure µ such that log + φ ∈ L ( µ ) , the pointwise Lya-punov exponent χ ( x, Φ) := lim n →∞ n log φ n ( x ) ∈ [ −∞ , ∞ ) , exists for a.e. point x . Date : May 15, 2020.2010
Mathematics Subject Classification.
Key words and phrases. thermodynamic formalism, subadditive potentials, zero temperaturelimits, maximal Lyapunov exponent.
By Kingman’s subadditive ergodic theorem [30, Theorem 3.3], the
Lyapunovexponent of measure µ χ ( µ, Φ) := lim n →∞ n Z log φ n ( x ) dµ ( x ) exists. If µ is ergodic then χ ( x, Φ) = χ ( µ, Φ) for µ − a.e. point x . We remark thatFurstenberg and Kesten [15] first proved, under a suitable integrability assumption,the existence of maximal and minimal Lyapunov exponents for matrix cocycles.Indeed, their result is a straightforward consequence of the Kingman’s subadditiveergodic theorem, where it was proved later.In this paper, we are interested in the maximal Lyapunov exponent , defined as β (Φ) := lim n →∞ n log sup x ∈ X φ n ( x ) . We denote by M ( X, T ) the space of all T − invariant Borel probabilty measureson X. Morris [26] showed that one can define the maximal Lyapunov exponent asthe supremum of the Lyapunov exponents of measures over invariant measures.That means,(1.1) β (Φ) = sup µ ∈M ( X,T ) χ ( µ, Φ) . Feng and Huang [13] gave a different proof of it.Let us define the set of maximizing measures of Φ to be the set of measures on X given by M max (Φ) := { µ ∈ M ( X, T ) , β (Φ) = χ ( µ, Φ) } . In this paper, we study the behavior of the equilibrium measures ( µ t ) for thesubadditive potentials t Φ when t → ∞ . In the thermodynamic interpretation ofthe parameter t , it is the inverse temperature . The limits t → ∞ are called zerotemperature limits , and the accumulation points of the measures ( µ t ) as t → ∞ are called ground states .The topic of ergodic optimization revolves around realizing invariant measureswhich maximize the Lyapunov exponents. Zero temperature limits laws are alsorelated to ergodic optimization, because for t → ∞ any accumulation point of theequilibrium measure ( µ t ) will be a maximizing measure Φ (maximizing χ ( µ, Φ)) .We refer the reader to [3] and [18].The behavior of the equilibrium measure ( µ t ) as t → ∞ has also been analyzed.In particular, the continuities of zero temperature limits ( µ t ) t →∞ in the sense,(1.2) χ ( µ, Φ) = lim t →∞ χ ( µ t , Φ) , and(1.3) h µ ( T ) = lim t →∞ h µ t ( T ) , have been investigated by many authors [11], [17], [18], [19], [25], [31], [32]. ERO TEMPERATURE LIMITS OF EQUILIBRIUM STATES 3
In the non-compact space setting, (1.2) and (1.3) were proved by Jenkinson,Mauldin and Urbański [19], and Morris [25] on the additive potential ψ : X → R . In fact, the proof of Theorem 1.1 will be based on ideas from those works.Moreover, this kind of result is known for almost subadditive potentials by Zhao[32] under the specification property, upper semi-continuity of entropy and finitetopological entropy assumptions.The goal of this paper is to extend the above results for subadditive potentials.Note that even though we know the existence of an accumulation point for thesequence ( µ t ) (see Proposition 2.4), this does not imply that the lim t →∞ µ t exists.In fact, Chazottes and Hochman [10] constructed an example on compact sub-shifts of finite type and Hölder potentials, where there is no convergence. Formore information about zero temperature see [18].Our main results are Theorems . and . formulated as follows: Theorem 1.1.
Let ( X, T ) be a TDS such that the entropy map µ h µ ( T ) isupper semi-continuous and topological entropy h top ( T ) < ∞ . Suppose that
Φ = { log φ n } ∞ n =1 is a subadditive potential on the compact metric space X which satisfies β (Φ) > −∞ . Then any weak ∗ accumulation point µ of a family of equilibriummeasures ( µ t ) for potentials t Φ , where t > , is a Lyapunov maximizing measurefor Φ . Moreover, (i) χ ( µ, Φ) = lim t →∞ χ ( µ t , Φ) , (ii) h µ ( T ) = lim t →∞ h µ t ( T ) = max { h ν ( T ) , ν ∈ M max (Φ) } . Furthermore, β (Φ) can be approximated by Lyapunov exponents of equilibrium mea-sures of a subadditive potential t Φ . Let A : X → GL ( d, R ) be a measurable function. We can define a linear cocycle F : X × R d → X × R d as F ( x, v ) = ( T ( x ) , A ( x ) v ) . We say that F is generated by T and A , we will also denote by ( T, A ) . Observethat F n ( x, v ) = ( T n ( x ) , A n ( x ) v ) for each n ≥ , where A n ( x ) = A ( T n − ( x )) A ( T n − ( x )) . . . A ( x ) . If T is invertible then so is F . Moreover, F − n ( x ) = ( T − n ( x ) , A − n ( x ) v ) for each n ≥ , where A − n ( x ) = A ( T − n ( x )) − A ( T − n +1 ( x )) − . . . A ( T − ( x )) − . A special class of linear cocycles is a class of locally constant cocycles which isdefined as follows.
Example 1.2.
Let X = { , . . . , k } Z be a symbolic space. Let T : X → X be ashift map, i.e. T ( x l ) l = ( x l +1 ) l . Given a finite set of matrices A = { A , . . . , A k } ⊂ GL ( d, R ) , we define the function A : X → GL ( d, R ) by A ( x l ) l = A x . REZA MOHAMMADPOUR
We say that a homeomorphism T satisfies the Anosov closing property if thereexists
C, ε, δ > such that for any x ∈ X and n ∈ N with d ( x, T n ( x )) < ε thereexists a point p ∈ X with T n ( p ) = p such that the orbits O + ( T ( x )) := { T k ( x ) , k ∈ N } , and O + ( T ( p )) := { T k ( p ) , k ∈ N } are exponentially close, i.e. d ( T i ( x ) , T i ( p )) ≤ Ce − δ min { i,n − i } d ( T n ( x ) , x ) for every i = 0 , . . . , n. Note that shifts of finite type, Axiom A diffeomorphisms, and hyperbolic home-omorphisms are particular systems satisfying the Anosov closing property. See formore information [22].Kalinin and Sadovskaya [21] proved that if a homeomorphism T satisfies theAnosov closing property, and A : X → GL ( d, R ) is a Hölder continuous Banachcocycle, then the maximal Lyapunov exponent can be approximated by Lyapunovexponents of measures supported on periodic orbits. For a locally constant cocycle ( T, A ) , where A : X → GL (2 , R ) , we show that the maximal Lyapunov exponentcan be approximated by Lyapunov exponents of measures supported on periodicorbits. Theorem 1.3 is implied by the Kalinin and Sadovskaya’s result. However,the methods of proof are different.We write φ n := kA n k , where kk is the operator norm. Theorem 1.3.
Let ( T, A ) be a locally constant cocycle satisfying the Anosov clos-ing property. Then the maximal Lyapunov exponent β (Φ) can be approximated byLyapunov exponents of measures supported on periodic orbits. In general, Kalinin [20] showed that for a Hölder continuous map A : X → GL ( d, R ) , Lyapunov exponents can be approximated by Lyapunov exponents ofmeasures supported on periodic orbits under an assumption slightly stronger thanthe Anosov closing property.This paper is organized as follows. In Section 2, we recall some preliminarymaterial regarding convex functions as well as some results in thermodynamicformalism for subadditive setting. In Section 3, we prove Theorem 1.1. In Section4, we state a theorem about the continuity of Lyapunov exponents for locallyconstant cocycles, and we prove Theorem 1.3. Acknowledgements.
The author thanks M. Rams for his careful reading ofan earlier version of this paper and many helpful suggestions. The author waspartially supported by the National Science Center grant 2014/13/B/ST1/01033(Poland). 2.
Preliminaries
Convex functions.
We first give some notation and basic facts in convexanalysis. For details, one is referred to [16].
ERO TEMPERATURE LIMITS OF EQUILIBRIUM STATES 5
Let x, y ∈ R n , the line segment connecting x and y is the set [ x, y ] formallygiven by [ x, y ] = { βx + (1 − β ) y β ∈ [0 , } . We say that a set X ⊂ R n is convex when for any two points x, y ∈ X , the linesegment [ x, y ] also belongs to the set X , i.e., βx + (1 − β ) y ∈ X for any x, y ∈ X and β ∈ (0 , . Let C be a convex subset of R n . A point x ∈ C is called an extremepoint of C if whenever x = βy + (1 − β ) z for some y, z ∈ C and < β < , then x = y = z . We denote by ext( C ) the set of extreme points of C .A function f : R n → R is a convex function if its domain dom( f ) is a convexset and for all x , y ∈ dom( f ) and β ∈ (0 , , the following relation holds f ( βx + (1 − β ) y )) ≤ βf ( x ) + (1 − β ) f ( y ) . In other words, a function f : R n → R is convex when for every segment [ x , x ] ,as the vector x β = βx +(1 − β ) x varies within the line segment [ x , x ] , the points ( x β , f ( x β )) on the graph { ( x, f ( x )) | x ∈ R n } lie below the segment connecting ( x , f ( x )) and ( x , f ( x )) , as illustrated in Figure 1. ( x , f ( x )) ( x , f ( x ))( x β , f ( x β )) Figure 1.
Convex lineLet U be an open convex subset of R n and f be a real continuous convex functionon U . We say a vector a ∈ R n is a subgradient of f at x if for all z ∈ U , f ( z ) ≥ f ( x ) + a T ( z − x ) , where the right hand side is the scalar product.For each x ∈ R n set the subdifferential of f at a point x to be ∂f ( x ) := { a : a is a subgradient for f at x } . REZA MOHAMMADPOUR
For x ∈ U , the subdifferential ∂f ( x ) is a nonempty convex compact set. Define ∂ e f ( x ) := ext { ∂f ( x ) } . In the case n = 1 , ∂ e f ( x ) = { f ′ ( x − ) , f ′ ( x + ) } , where f ′ ( x − ) (resp. f ′ ( x + )) denotes the left (resp. right) derivative. We say that f is differentiable at x when ∂ e f ( x ) = { a } .We define(2.1) ∂f ( U ) = ∪ x ∈ U ∂f ( x ) and ∂ e f ( U ) = ∪ x ∈ U ∂ e f ( x ) . In the case n = 1 , Lebesgue’s theorem on the differentiability of monotone func-tions says ∂ e f is differentiable almost everywhere. The case n = 2 was proven byH. Busemann and W. Feller [8]. The general case was settled by A. D. Alexandrov[1]. The following result is well known (cf. [29, Theorem 7.9]). Theorem 2.1.
Let f be a continuous function defined on an open interval thathas a derivative at each point of R except on a countable set, and f ′ ≤ Lebesguealmost everywhere, then f is a non-increasing function. Thermodynamic formalism for a subadditive potential.
We requiresome elements from the subadditive thermodynamic formalism. The additive the-ory of thermodynamic formalism extends to the subadditive theory with suitablegeneralizations. Let ( X, T ) be a TDS and let Φ = { log φ n } ∞ n =1 be a subadditivepotential on ( X, T ) .We introduce the topological pressure of Φ as follows. The space X is endowedwith the metric d . For any n ∈ N , one can define a new metric d n on X by d n ( x, y ) = max (cid:8) d ( T k ( x ) , T k ( y )) : k = 0 , . . . , n − (cid:9) . For any ε > a set E ⊂ X is said to be a ( n, ε ) - separated subset of X if d n ( x, y ) > ε for any two different points x, y ∈ E . We define for Φ P n ( T, Φ , ε ) = sup (X x ∈ E φ n ( x ) : E is ( n, ε ) -separated subset of X ) . Since P n ( T, Φ , ε ) is a decreasing function of ε , we define P ( T, Φ , ε ) = lim sup n →∞ n log P n ( T, Φ , ε ) , and P ( T, Φ) = lim ε → P ( T, Φ , ε ) . We call P ( T, Φ) the topological pressure of Φ . We define h top ( T ) := P ( T, . Bowen [7] showed that for any Hölder continuous ψ : X → R on a mixinghyperbolic system, there exists a unique equilibrium measure (which is also aGibbs state) for ψ .Feng and Käenmäki [14] extended the Bowen’s result for the subadditive poten-tials t Φ on a locally constant cocycle under the assumption that the matrices in A do not preserve a common proper subspace of R d (i.e. ( T, A ) is irreducible). ERO TEMPERATURE LIMITS OF EQUILIBRIUM STATES 7
Recently, Park [28] showed the continuity of the topological pressure, and theuniqueness of the equilibrium measure for general cocycles under generic assump-tions. In [24] the continuity of the topological pressure was proven under someassumption which is weaker than Park’s assumptions.Let ( X, τ, µ ) be a Borel probability space, and T : X → X be a measurepreserving transformation.A partition of ( X, τ, µ ) is a subfamily of τ consisting of mutually disjoint ele-ments whose union is X . We denote by α and β the countable partition of X .Let α = { A i , i ≥ } , where A i ∈ τ . We define H µ ( α ) = − X A ∈ α µ ( A ) log µ ( A ) to be the entropy of α (with the convention 0 log 0 = 0).We denote by α ∨ β the joint partition { A ∩ B | A ∈ α, B ∈ β } . Let T − ( α ) = { T − ( A ) | A ∈ α } . We define h ( µ, α ) = lim n →∞ n H µ ( n − _ j =0 T − ( α )) to be the entropy of T relative to α .Then the metric entropy of µ is defined as h µ ( T ) = sup h ( µ, α ) , where the supremum is taken over all countable partitions α with H µ ( α ) < ∞ . We can define the topological pressure by the following variational principle . Itwas proved by Cao, Feng and Huang [9].
Theorem 2.2 ([9, Theorem 1.1]) . Let ( X, T ) be a TDS such that h top ( T ) < ∞ .Suppose that Φ = { log φ n } ∞ n =1 is a subadditive potential on the compact metricspace X . Then P ( T, Φ) = sup { h µ ( T ) + χ ( µ, Φ): µ ∈ M ( X, T ) , χ ( µ, Φ) = −∞} . For t ∈ R + , let us denote P ( T, t
Φ) = P ( t ) . Theorem 2.3 ([13, Theorem 1.2]) . Let ( X, T ) be a TDS such that h top ( T ) < ∞ .Assume that Φ = { log φ n } ∞ n =1 is a subadditive potential on the compact metricspace X which satisfies β (Φ) > −∞ .Then the pressure function P ( t ) is a continuous real convex function on (0 , ∞ ) .Furthermore, P ′ ( ∞ ) := lim t →∞ P ( t ) t = β (Φ) . Limits exist by subadditivity.
REZA MOHAMMADPOUR
Let t ∈ R + , we denote by Eq( t ) the collection of invariant measures µ such that h µ ( T ) + t.χ ( µ, Φ) = P ( t ) . If Eq( t ) = ∅ , then each element Eq( t ) is called an equilibrium state for t Φ . Proposition 2.4 ([13, Theorem 3.3]) . Let ( X, T ) be a TDS such that the entropymap µ h µ ( T ) is upper semi-continuous and h top ( T ) < ∞ . Suppose that Φ = { log φ n } ∞ n =1 is a subadditive potential on the compact metric space X which satisfies β (Φ) > −∞ . For any t > , Eq( t ) is a non-empty compact convex subset of M ( X, T ) , and every extreme point of Eq( t ) is an ergodic measure. Moreover, ∂P ( t ) = { χ ( µ t , Φ) : µ t ∈ Eq( t ) } . Theorem 2.5 ([13, Proposition 3.2]) . Suppose that
Φ = { log φ n } ∞ n =1 is a subad-ditive potential on a TDS ( X, T ) . Assume that h top ( T ) < ∞ and β (Φ) > −∞ . Then ∂P ( R + ) ⊆ ( −∞ , β (Φ)] , where ∂P ( R + ) defined in (2.1). We denote by M ( X ) the space of all Borel probability measure on X with weak ∗ topology. Theorem 2.6 ([9, Lemma 2.3]) . Suppose { ν n } ∞ n =1 is a sequence in M ( X ) and Φ = { log φ n } ∞ n =1 is a subadditive potential on a TDS ( X, T ) . We form the newsequence { µ n } ∞ n =1 by µ n = n P n − i =0 ν n oT i . Assume that µ n i converges to µ in M ( X ) for some subsequence { n i } of natural numbers. Then µ ∈ M ( X, T ) and (2.2) lim sup i →∞ n i Z log φ n i ( x ) dν i ( x ) ≤ χ ( µ, Φ) . Proof of the Theorem 1.1
We start the proof of Theorem 1.1 ( i ) with the key Proposition 2.4 which tellsus that the subderivative of the topological pressure for a subadditive potential isequal to the Lyapunov exponent of the equilibrium state. Let ( X, T ) be a TDSsuch that the entropy map µ h µ ( T ) is upper semi-continuous and h top ( T ) < ∞ .Suppose that Φ = { log Φ n } ∞ n =1 is a subadditive potential on the compact metricspace X which satisfies β (Φ) > −∞ . We write Theorem 1.1 ( i ) as follows. Theorem 3.1.
For each t > , the family of equilibrium measures ( µ t ) , has aweak ∗ accumulation point µ as t → ∞ . Any such accumulation point is a Lyapunovmaximizing measure for Φ . Moreover, χ ( µ, Φ) = lim t →∞ χ ( µ t , Φ) . Proof.
It is obvious that ( µ t ) has at least one accumulation point, let us call it µ .By Theorem . , P ( t ) is convex , then we have ∂P ( t ) = { χ ( µ t , Φ) } by Proposition2.4. Moreover, since P ( t ) is convex for t > , t χ ( µ t , Φ) is non-decreasing and ERO TEMPERATURE LIMITS OF EQUILIBRIUM STATES 9 bounded above .It follows that lim t →∞ ∂P ( t ) = lim t →∞ χ ( µ t , Φ) exists and is finite . Since Lyapunov exponents are upper semi-continuous lim t →∞ χ ( µ t , Φ) ≤ χ ( µ, Φ) . By the definition of
Eq( t ) ,(3.1) χ ( µ t , Φ) + h µ t ( T ) t ≥ χ ( µ, Φ) + h µ ( T ) t . Since the TDS ( X, T ) has finite topological entropy, so when t → ∞ , (3.1)implies lim t →∞ χ ( µ t , Φ) ≥ χ ( µ, Φ) . Now, we shall show that µ is a Lyapunov maximizing measure.By contradiction, let us assume that there exists ν with χ ( ν, Φ) − χ ( µ, Φ) = κ > . One can define the affine map T ν : R + → R by T ν ( t ) = h ν ( T ) + tχ ( ν, Φ) . Weknow that t χ ( µ t , Φ) is a function which increases to its limit χ ( µ, Φ) , so χ ( µ, Φ) ≥ χ ( µ t ∗ , Φ) = ∂ e P ( t ∗ ) , where t ∗ = t − or t + , and T ′ ν ( t ) = χ ( ν, Φ) = χ ( µ, Φ) + κ ≥ ∂ e P ( t ∗ ) + κ. Consequently, h ν ( T ) + tχ ( ν, Φ) > P ( t ) for all sufficiently large t > that contra-dicts our assumption. So, µ is a Lyapunov maximizing measure.Moreover, our proof implies that β (Φ) can be approximated by Lyapunov expo-nents of equilibrium measures of a subadditive potential t Φ . (cid:3) Theorem 1.1 ( ii ) is obtained by combining Lemmas 3.2 and 3.3 below. Lemma 3.2.
The maps t h µ t ( T ) and t P ( t Φ − tβ (Φ)) are non-increasingand bounded below on the interval (0 , ∞ ) . Moreover, we have lim t →∞ h µ t ( T ) = lim t →∞ P ( t Φ − tβ (Φ)) ≥ sup ν ∈M max (Φ) h ν ( T ) . Proof.
The map t P ( t Φ − tβ (Φ)) is convex. By the definition of β (Φ) , χ ( µ t , Φ) ≤ β (Φ) for all µ t ∈ Eq( t ) . We assume that P ( t ) = P ( t Φ) . By the definition of the topological pressure, P ( t Φ − tβ (Φ)) = P ( t Φ) − tβ (Φ) . Then, ∂ e P ( t ∗ Φ − t ∗ β (Φ)) = ∂ e P ( t ∗ Φ) − β (Φ) = χ ( µ t ∗ , Φ) − β (Φ) ≤ , This follows from subadditivity. where t ∗ = t − or t + . Thus, P ( t Φ − tβ (Φ)) is non-increasing by Theorem 2.1. Weare going to show that t h µ t ( T ) is non-increasing. Since µ t is an equilibriummeasure, h µ t ∗ ( T ) = P ( t ) − t∂ e P ( t ∗ ) . For < x < y we have ∂ e P ( x ∗ ) ≤ P ( y ) − P ( x ) y − x ≤ ∂ e P ( y ∗ ) , so P ( y ) − P ( x ) ≤ y∂ e P ( y ∗ ) − x∂ e P ( y ∗ ) ≤ y∂ e P ( y ∗ ) − x∂ e P ( x ∗ ) , and then P ( y ) − y∂ e P ( y ∗ ) ≤ P ( x ) − x∂ e P ( x ∗ ) . Since t h µ t ( T ) and t P ( t Φ − tβ (Φ)) ≥ are non-increasing and non-negative, we conclude that lim t →∞ h µ t ( T ) and lim t →∞ P ( t Φ − tβ (Φ)) both exist.This implies that the limit lim t →∞ t∂ e P ( t ) − tβ (Φ) = lim t →∞ ( P ( t Φ − tβ (Φ)) − h µ t ( T )) exists. Then, lim t →∞ h µ t ( T ) = lim t →∞ P ( t Φ − tβ (Φ)) . The last part follows from the variational principle. (cid:3)
Lemma 3.3. M max (Φ) is compact, convex and nonempty, and its extreme pointsare precisely its ergodic elements.Proof. See [27, Appendix A]. (cid:3)
We write Theorem 1.1 ( ii ) as follows. Theorem 3.4. h µ ( T ) = lim t →∞ h µ t ( T ) = max { h ν ( T ) , ν ∈ M max (Φ) } .Proof. By Theorem 3.1 and Lemmas 3.2 and 3.3, h µ ( T ) ≤ max ν ∈M max (Φ) h ν ( T ) ≤ lim t →∞ h µ t ( T ) , the reverse inequality follows from upper semi-continuity of entropy. (cid:3) Remark 1.
Let ( T, A ) be a locally constant cocycle. Then, one can prove Theorem1.1 for Gibbs measures under the assumption that ( T, A ) is irreducible (see [14] ).Moreover, if T : X → X is a mixing subshift of finite type and A : X → GL ( d, R ) is a Hölder continuous function, then one can prove Theorem 1.1 for Gibbs mea-sures under the generic assumption on ( T, A ) (see [28] ). Remark 2.
Let ~q = ( q , ..., q d ) ∈ R d + , and ~ Φ = (Φ , ..., Φ d ) = ( { log φ n, } ∞ n =1 , ..., { log φ n,d } ∞ n =1 ) . Assume that ~q.~ Φ = P di =1 q i Φ i is a subadditive potential { q i log φ n,i } ∞ n =1 . ERO TEMPERATURE LIMITS OF EQUILIBRIUM STATES 11
We can write topological pressure, and maximal Lyapunov exponent of ~ Φ , respec-tively P ( ~q ) = P ( T, ~q.~ Φ) , β ( ~ Φ) = β ( d X i =1 Φ i ) . Feng and Huang [13] proved the higher dimensional versions of Theorem 2.3,Proposition 2.4, and Theorem 2.5. So, one can obtain the higher dimensionalversions of Theorem 1.1 by using [13] . Proof of the Theorem 1.3
In this section, we consider locally constant cocycles and we prove Theorem 1.3.We use the continuity of Lyapunov exponents (Theorem 4.1), the Anosov closingproperty and Theorem 2.6 for the proof.Let ( T, A ) be the locally constant cocycle which is defined in Example 1.2 and A : X → GL (2 , R ) . We denote χ ( µ, A ) := χ ( µ, Φ) , where φ n = kA n k .Bocker and Viana [4] proved the continuity of Lyapunov exponents for twodimensional locally constant cocycles. In order to state the result of Bocker andViana, we denote by △ k the collection of strictly positive probability vectors in R k for k ≥ . We denote by X the full shift space over k symbols. For p =( p , ..., p k ) ∈ △ k , let µ be the associated Bernoulli product measure on X . Theorem 4.1 ([4, Theorem B]) . For every ε > there exist δ > and a weak ∗ neighborhood V of µ in the space of probability measures on GL (2 , R ) such that forevery probability measure µ ′ ∈ V whose support is contained in the δ -neighborhoodof the support of µ , we have | χ ( µ, A ) − χ ( µ ′ , A ′ ) | < ε. Now, we can prove Theorem 1.3.
Theorem 4.2.
Suppose that T satisfies the Anosov closing property. Then themaximal Lyapunov exponent β (Φ) can be approximated by Lyapunov exponents ofmeasures supported on periodic orbits.Proof. Let µ be an ergodic maximizing measure, that is β (Φ) = χ ( µ, Φ) .Let x be a generic point for µ . Then there exists µ n,x := n P n − j =0 δ T j ( x ) , where δ x is the Dirac measure at the point x , so that µ x,n → µ . According to Theorem2.6, and (1.1) lim n →∞ n log φ n ( x ) = χ ( µ, Φ) . Let p ∈ X be a periodic point associated to ε, C, δ and { x, T ( x ) , ..., T n − ( x ) } by the Anosov closing property. Denote by µ p := n P n − j =0 δ T j ( p ) the ergodic T − invariant measure supported on the orbit of p . Lemma 4.3. µ p → µ in weak ∗ topology.Proof. We will use the Anosov closing property.Assume that ( f m ) is a sequence of continuous functions. The periodic orbit p has length n and is close to the initial segment of the orbit of x . Since the f m ’s are continuous, the average of f m along the periodic orbit is very close to theaverage of f m along the first n iterates of x , and that is very close to R f m dµ by thegenericity condition. Then, for n large enough, we get longer and longer periodicorbits, approaching x more closely, and we obtain the convergence of the measuresto µ . (cid:3) We now use Lemma 4.3 to finish the proof. By the Anosov closing property, theperiodic point p is close to x , with iterates also close to the iterates of x . Therefore,Theorem 4.1 implies for every ε > (4.1) χ ( p, Φ) = lim n →∞ n log φ n ( p ) = lim n →∞ n log φ n ( x ) + ε. Applying Lemma 4.3, Theorem 2.6 and (4.1), we obtain χ ( p, Φ) = lim n →∞ n log φ n ( p ) = χ ( µ, Φ) + ε = β (Φ) + ε. (cid:3) Remark 3.
Avila, Eskin and Viana [2] announced recently that the Theorem 4.1remains true in arbitrary dimensions. By their result, the proof given for Theorem4.1 works for arbitrary dimensions.
Remark 4.
Morris [26] showed that the speed of convergence of Theorem 4.2 is al-ways superpolynomial for locally constant cocycles. Moreover, Bochi and Garibaldi [3] showed that it is true for general cocycles under certain assumptions.
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