aa r X i v : . [ m a t h . D S ] J un EQUILIBRIUM STATES OF INTERMEDIATE ENTROPIES
Peng Sun
China Economics and Management AcademyCentral University of Finance and EconomicsBeijing 100081, China
Abstract.
We explore an approach to the conjecture of Katok on interme-diate entropies that based on uniqueness of equilibrium states, provided theentropy function is upper semi-continuous. As an application, we prove Ka-tok’s conjecture for Ma˜n´e diffeomorphisms. Introduction
Let (
X, d ) be a compact metric space and f : X → X be a continuous map.Denote by M ( f ) the subspace of all invariant measures for the dynamical system( X, f ) and by M e ( f ) the subset of all ergodic measures. Denote by C ( X ) the spaceof all continuous potentials on X , equipped with the C (supremum) norm k · k .For φ ∈ C ( X ), denote by P ( φ ) = P ( f, φ ) the topological pressure of ( X, f, φ ) andby P µ ( φ ) = P µ ( f, φ ) := h µ ( f ) + R φdµ the pressure of µ ∈ M ( f ). We usually omit f when it is clearly fixed. The Variational Principle states that P ( φ ) = sup { P µ ( φ ) : µ ∈ M ( f ) } . For convenience, we say that (
X, f ) is a
USC system if the entropy function µ h µ ( f ) is upper semi-continuous, where h µ ( f ) denotes the metric entropy of ( X, f )with respect to µ ∈ M ( f ). Denote the set of equilibrium states for ( X, f, φ ) by E ( φ ) := { µ ∈ M ( f ) : P µ ( φ ) = P ( φ ) } . It is well-known that when (
X, f ) is a USC system, E ( φ ) is nonempty for every φ ∈ C ( X ). Study on equilibrium states has a long history. Existence, uniquenessand properties of equilibrium states are important and popular topics in dynamicalsystems. For example, see [1, 2, 3, 7, 8, 16].As indicated in [24, Section 9.4], The topological pressure P ( f, · ) for the system( X, f ), as a functional on continuous potentials, determines all invariant probabilitymeasures and their entropies. In this article, we would like to further explore suchconnections and show its application to the conjecture of Katok on intermediateentropies.Denote P ( f, φ ) := { P µ ( f, φ ) : µ ∈ M e ( f ) } Mathematics Subject Classification.
Primary: 37A35, 37D35.
Key words and phrases. equilibrium state, intermediate entropy, pressure, Ma˜n´e diffeomor-phism, ergodic optimization, thermodynamical formalism. and H ( f ) := P ( f,
0) = { h µ ( f ) : µ ∈ M e ( f ) } . We say that (
X, f ) has the intermediate entropy property if H ( f ) ⊃ [0 , h ( f )), i.e.for every a ∈ [0 , h ( f )), there is an ergodic measure µ ∈ M e ( f ) such that h µ ( f ) = a . Conjecture (Katok) . Let ( X, f ) be a C diffeomorphism on a compact Riemannianmanifold. Then ( X, f ) has the intermediate entropy property. Partial results on Katok’s conjecture have been obtained in [10, 17, 18, 23, 19,15, 6, 12, 13, 21, 20, 22]. They represent two major approaches, which are basedon hyperbolic structures and specification-like properties, respectively. What weillustrate here is a new one, which shows that for USC systems, certain resultson equilibrium states imply the intermediate entropy property. This approach isinspired by some facts in ergodic optimization .Denote U ( f ) := { φ ∈ C ( X ) : ( X, f, φ ) has a unique equilibrium state } . For φ ∈ C ( X ), denote by M max ( f, φ ) the set of maximizing measures for φ , i.e.invariant measures µ ∈ M ( f ) that maximizes R φdµ . Theorem 1.1.
Let ( X, f ) be a USC system. Suppose that there is φ ∈ C ( X ) suchthat:(1) tφ ∈ U ( f ) for every t ≥ .(2) M max ( f, φ ) = { µ φ } is a singleton and h µ φ ( f ) = 0 .Then ( X, f ) has the intermediate entropy property. In fact, the ergodic measures of intermediate entropies we obtain in Theorem1.1 are just the unique equilibrium states for tφ , whose metric entropy varies con-tinuously with t . When M max ( f, φ ) = { µ φ } is a singleton, µ φ is the unique groundstate and the zero temperature limit of these equilibrium states.We shall prove a more general result than Theorem 1.1. We can relax theuniqueness of equilibrium states and conclude on intermediate pressures. For ψ ∈ C ( X ), let V ( f, ψ ) be the set of all continuous potentials φ such that P µ ( ψ ) is aconstant, denoted by P ψ ( φ ), for all µ ∈ E ( φ ). That is, V ( f, ψ ) := (cid:8) φ ∈ C ( X ) : P µ ( ψ ) = P ψ ( φ ) for every µ ∈ E ( φ ) (cid:9) . In particular, V ( f ) := V ( f,
0) is the set of all potentials whose equilibrium stateshave equal entropies. By definition, we have ψ ∈ V ( f, ψ ) and U ( f ) ⊂ V ( f, ψ ) forevery ψ ∈ C ( X ). Theorem 1.2.
Let ( X, f ) be a USC system. Suppose that there are ψ, φ ∈ C ( X ) and α ∈ R such that the following holds:(1) ψ + tφ ∈ V ( f, ψ ) for every t ≥ .(2) P µ ( ψ ) ≤ α for every µ ∈ M max ( f, φ ) .Then P ( f, ψ ) ⊃ [ α, P ( ψ )] . Theorem 1.1 and 1.2 apply to the Ma˜n´e diffeomorphisms considered in [4] and[22]. The following theorem completely verifies Katok’s conjecture for such systems,which improves [22, Corollary 1.2]. See Section 4 for details.
Theorem 1.3.
Ma˜n´e diffeomorphisms have the intermediate entropy property.
QUILIBRIUM STATES OF INTERMEDIATE ENTROPIES 3 Continuity of Equilibrium States
Readers are also referred to [24] for definitions and basic properties of entropiesand pressures. The following lemma is a simple observation (cf. [16, 6.8]).
Lemma 2.1.
For any system ( X, f ) and any φ, ψ ∈ C ( X ) , we have | P ( φ ) − P ( ψ ) | ≤ k φ − ψ k for any φ, ψ ∈ C ( X ) . So P ( · ) is a (Lipschitz) continuous function on C ( X ) . We try to make our results as more applicable as possible. Proposition 2.2 and2.3 are slight variations of known results (cf. [11, Theorem 4.2.11] and [9, Theo-rem 4.1]). In particular, (1) and (6) (hence Proposition 2.2 and Proposition 2.3)hold when the measures are actually the equilibrium states for the correspondingpotentials, i.e. µ n ∈ E ( φ n ) or µ n ∈ E ( ψ + t n φ n ) for all n . Proposition 2.2.
Let ( X, f ) be a USC system. Let { φ n } ∞ n =1 be a sequence ofcontinuous potentials such that k φ n − φ k → . Let { µ n } ∞ n =1 be a sequence in M ( f ) such that µ n → µ and lim n →∞ | P µ n ( φ n ) − P ( φ n ) | = 0 . (1) Then µ is an equilibrium state for φ , i.e. µ ∈ E ( φ ) .Proof. As P ( · ) is continuous, we have P ( φ n ) → P ( φ ). So (1) is equivalent to thecondition lim n →∞ | P µ n ( φ n ) − P ( φ ) | = 0 . (2)We fix a metric D on M ( f ) that induces the weak- ∗ topology. For every ε > η ε > ν ∈ B ( µ, η ε ) we have h ν ( f ) < h µ ( f ) + ε and (cid:12)(cid:12)(cid:12)(cid:12)Z φdν − Z φdµ (cid:12)(cid:12)(cid:12)(cid:12) < ε. (3)By (2), for every ε >
0, there is N such that k φ N − φ k < ε, D ( µ N , µ ) < η ε and P µ N ( φ N ) > P ( φ ) − ε. (4)Hence by (3), we have h µ ( f ) > h µ N ( f ) − ε and (cid:12)(cid:12)(cid:12)(cid:12)Z φdµ N − Z φdµ (cid:12)(cid:12)(cid:12)(cid:12) < ε. (5)By (4) and (5), we have P µ ( φ ) = h µ ( f ) + Z φdµ> h µ N ( f ) − ε + Z φdµ N − (cid:12)(cid:12)(cid:12)(cid:12)Z φdµ N − Z φdµ (cid:12)(cid:12)(cid:12)(cid:12) > h µ N ( f ) − ε + (cid:18)Z φ N dµ N − (cid:12)(cid:12)(cid:12)(cid:12)Z φ N dµ N − Z φdµ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − ε> P µ N ( φ N ) − k φ N − φ k − ε> P ( φ ) − ε. This implies that P µ ( φ ) ≥ P ( φ ) as it holds for all ε >
0. Hence µ is an equilibriumstate for φ . (cid:3) EQUILIBRIUM STATES OF INTERMEDIATE ENTROPIES
Proposition 2.3.
Let ( X, f ) be any system and ψ, φ ∈ C ( X ) . Let { t n } ∞ n =1 be asequence of real numbers such that t n → ∞ . Let { µ n } ∞ n =1 be a sequence in M ( f ) such that µ n → µ and lim n →∞ t n | P µ n ( ψ + t n φ ) − P ( ψ + t n φ ) | = 0 . (6) Then µ is a maximizing measure for φ , i.e. µ ∈ M max ( φ ) .Proof. For every ε >
0, there is η ε > ν ∈ B ( µ, η ε ) we have (cid:12)(cid:12)(cid:12)(cid:12)Z ψdν − Z ψdµ (cid:12)(cid:12)(cid:12)(cid:12) < ε and (cid:12)(cid:12)(cid:12)(cid:12)Z φdν − Z φdµ (cid:12)(cid:12)(cid:12)(cid:12) < ε. (7)As t n → ∞ , µ n → µ , by (6), for every ε >
0, there is N such that for all n > N we have t n ε > h ( f ) , D ( µ n , µ ) < η ε and P ( ψ + t n φ ) − P µ n ( ψ + t n φ ) < t n ε (8)By (7) and (8), for all n > N , we have | P µ ( ψ ) − P µ n ( ψ ) | ≤ | h µ ( f ) − h µ n ( f ) | + (cid:12)(cid:12)(cid:12)(cid:12)Z ψdµ − Z ψdµ n (cid:12)(cid:12)(cid:12)(cid:12)
0. Hence µ is a maximizingmeasure for φ . (cid:3) Intermediate Pressures
Proposition 3.1.
Let ( X, f ) be a USC system and ψ ∈ C ( X ) . Then the function φ P ψ ( φ ) is continuous on V ( f, ψ ) . QUILIBRIUM STATES OF INTERMEDIATE ENTROPIES 5
Proof.
Let { φ n } ∞ n =1 be a sequence in V ( f, ψ ) such that φ n → ˜ φ ∈ V ( f, ψ ). Foreach n , take any µ n ∈ E ( φ n ). Let µ be the limit of a convergent subsequence { µ n k } ∞ k =1 . As φ n k → ˜ φ , by Proposition 2.2, we have µ ∈ E ( ˜ φ ).By Lemma 2.1, we havelim k →∞ (cid:12)(cid:12)(cid:12) P ψ ( φ n k ) − P ψ ( ˜ φ ) (cid:12)(cid:12)(cid:12) = lim k →∞ (cid:12)(cid:12)(cid:12) P µ nk ( ψ ) − P µ ( ψ ) (cid:12)(cid:12)(cid:12) ≤ lim k →∞ (cid:18)(cid:12)(cid:12)(cid:12) P µ nk ( φ n k ) − P µ ( ˜ φ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ( φ n k − ψ ) d µ nk − Z ( ˜ φ − ψ ) d µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ lim k →∞ (cid:12)(cid:12)(cid:12) P ( φ n k ) − P ( ˜ φ ) (cid:12)(cid:12)(cid:12) + lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z ( φ n k − ˜ φ ) d µ nk (cid:12)(cid:12)(cid:12)(cid:12) + lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z ( ˜ φ − ψ ) d µ nk − Z ( ˜ φ − ψ ) d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim k →∞ k φ n k − ˜ φ k + lim k →∞ k φ n k − ˜ φ k + 0=0 . This implies that P ψ ( · ) is a continuous function on V ( f, ψ ). (cid:3) Corollary 3.2.
Let ( X, f ) be a USC system. Denote by µ φ the unique equilibriumstate for each φ ∈ U ( f ) . Then the map φ h µ φ ( f ) is continuous on U ( f ) .Proof. Note that U ( f ) ∈ V ( f ) and P ( φ ) = P µ φ (0) = h µ φ ( f ) for every φ ∈ U ( f ).Apply Proposition 3.1 for ψ = 0. (cid:3) Proposition 3.3.
Let ( X, f ) be a USC system and ψ ∈ C ( X ) . Suppose that thereis a continuous path Φ : [0 , T ] → V ( f, ψ ) and P ψ (Φ(0)) < P ψ (Φ( T )) . Then P ( f, ψ ) ⊃ [ P ψ (Φ(0)) , P ψ (Φ( T ))] (13) Proof.
By Proposition 3.1, P ψ ( · ) is continuous on V ( f, ψ ). Hence P ψ ◦ Φ is con-tinuous. Then (13) follows from the Intermediate Value Theorem. (cid:3)
Analogous to Corollary 3.2, we have:
Corollary 3.4.
Let ( X, f ) be a USC system. Suppose that there is a continuouspath Φ : [0 , T ] → U ( f ) , i.e. Φ( t ) has a unique equilibrium state µ t for each t ∈ [0 , T ] . Assume that h µ ( f ) ≤ h µ T ( f ) . Then we have H ( f ) ⊃ [ h µ ( f ) , h µ T ( f )] Proof of Theorem 1.2.
Take µ n ∈ E ( ψ + tφ ) for each n ∈ N . As M ( f ) is compact,there is a subsequence { t k } ∞ k =1 such that t k → ∞ and { µ t k } ∞ k =1 converges to µ ∈M ( f ). By Proposition 2.3, µ is a maximizing measure for φ and hence P µ ( ψ ) ≤ a .For any b ∈ ( a, P ( ψ )), as µ t k → µ and the entropy map is upper semi-continuous,there is N such that P µ tN ( ψ ) < b . Note that Φ( t ) := ψ + tφ is a continuous pathfrom [0 , t N ] to V ( f, ψ ). By Proposition 3.3, we have P ( ψ ) ⊃ ( b, P ( ψ )] . (14)Then P ( ψ ) ⊃ [ a, P ( ψ )] because P µ ( ψ ) ≤ a and (14) holds for any b ∈ ( a, P ( ψ )). (cid:3) Corollary 3.5.
Let ( X, f ) be a USC system with positive topological entropy h ( f ) > . Suppose that there is a continuous potential φ such that the following holds: EQUILIBRIUM STATES OF INTERMEDIATE ENTROPIES (1) ( X, f, tφ ) has a unique equilibrium state for every t ≥ .(2) ( X, f, φ ) has a unique maximizing measure µ with h µ ( f ) = 0 .Then ( X, f ) has the intermediate entropy property.Proof. Note that for t = 0, µ is just the measure of maximal entropy. Hence h µ ( f ) = h ( f ). Apply Theorem 1.2 for ψ = 0. (cid:3) It is well known that every Axiom A system has unique equilibrium states forH¨older potentials [1]. One can just pick a fixed point p and take φ ( x ) := − d ( x, p )for every x ∈ X , which is a H¨older function with the unique maximizing measuresupported on p . In this case Corollary 3.5 provides another proof that every AxiomA system has the intermediate entropy property.4. Application to Ma˜n´e Diffeomorphisms
Following [4], we consider the Ma˜n´e family M ρ,r , which is a class of DA (derivedfrom Anosov) maps first introduced by Ma˜n´e [14]. They are C perturbations of ahyperbolic toral automorphism f A : T d → T d , which are partially hyperbolic with1-dimensional centers. Let q be a fixed point of f A . As described in [4], for each g ∈ M ρ,r we assume that:(1) ρ > B ( q, ρ ) is the support of the perturbation,i.e. g = f on T \ B ( q, ρ ).(2) r ∈ [0 ,
1] such that if an orbit of g spends a proportion at least r of its timeoutside B ( q, ρ ), then it contracts the vectors in the central direction.(3) The C distance between g and f A is sufficiently small, i.e. there is aconstant η = η ( f A ) depending only on f A such that d C ( g, f A ) < η . Inparticular, this holds when ρ is sufficiently small.For φ ∈ C ( T d ), ρ, L > r ∈ (0 , ρ, r, φ, L ) := (1 − r ) sup B ( q,ρ ) φ + r (sup T d φ + h ( f A ) + L ) + H (2 r ) , where H ( r ) := − r ln r − (1 − r ) ln(1 − r ) . Denote C ρ,r,g,L := { φ ∈ C ( T d ) : P ( g, φ ) > Ξ( ρ, r, φ, L ) } Let φ be an α -H¨older potential on T d and | φ | α := sup (cid:26) | φ ( x ) − φ ( y ) | d ( x, y ) α : x, y ∈ X, x = y (cid:27) be its H¨older semi-norm. Denote C αM := { φ ∈ C ( T d ) : | φ | α < M } . Theorem 4.1 ([4, Theorem A and B]) . Let g ∈ M ρ,r and φ be an α -H¨oldercontinuous function on T .(1) There is a constant L = L ( f A ) depending only on f A such that ( T d , g, φ ) has a unique equilibrium state as long as φ ∈ C ρ,r,g,L .(2) There is a function M ( ρ, r ) such that M ( ρ, r ) → ∞ as ρ, r → and ( T d , g, φ ) has a unique equilibrium sate as long as φ ∈ C αM ( ρ,r ) . Let p be another fixed point of f A such that p / ∈ B ( q, ρ ). We can fix Ψ ∈ C ( T d )such that QUILIBRIUM STATES OF INTERMEDIATE ENTROPIES 7 (1) Ψ( p ) = 0.(2) Ψ( x ) = − x ∈ B ( q, ρ ).(3) Ψ( x ) < x ∈ T d \{ p } .(4) Ψ is α -H¨older.Denote by µ p the Dirac measure on p . Then µ p is the unique maximizing measurefor Ψ and h µ p ( g ) = 0. Lemma 4.2.
There are ρ ∗ > and r ∗ ∈ (0 , such that for every g ∈ M ρ ∗ ,r ∗ , wehave t Ψ ∈ U ( g ) for all t ≥ .Proof. Denote β r := r ( h ( f A ) + L ) + H (2 r )1 − r . Note that β r → r →
0. By Theorem 4.1(2), there are ρ ∗ > r ∗ ∈ (0 , β r ∗ | Ψ | α < M ( ρ ∗ , r ∗ )This implies that for every g ∈ M ρ ∗ ,r ∗ , we have t Ψ ∈ C αM ( ρ ∗ ,r ∗ ) ⊂ U ( g ) for every t ∈ [0 , β r ∗ ] . For t > β r ∗ , as h µ p ( g ) = 0, by the Variational Principle, we haveΞ( ρ ∗ , r ∗ , t Ψ , L ) = (1 − r )( − t ) + r ( h ( f A ) + L ) + H (2 r ) < h µ p ( g ) + Z ( t Ψ) dµ p ≤ P ( g, t Ψ) . By Theorem 4.1(1), for every g ∈ M ρ ∗ ,r ∗ , we have t Ψ ∈ C ρ ∗ ,r ∗ ,g,L ⊂ U ( g ) for every t > β r ∗ . (cid:3) Remark.
We may choose Ψ such that | Ψ | α is as small as possible to achieve largervalues of ρ ∗ and r ∗ as in Lemma 4.2. Theorem 4.3.
For every g ∈ M ρ ∗ ,r ∗ , the system ( T d , g ) has the intermediateentropy property.Proof. By [5, Proposition 6], every Ma˜n´e diffeomorphism is entropy expansive. So( T d , g ) is a USC system. As µ p is the unique maximizing measure for Ψ and h µ p ( g ) = 0, the conclusion follows from Lemma 4.2 and Corollary 3.5. (cid:3) Acknowledgments
The author is supported by National Natural Science Foundation of China (No.11571387) and CUFE Young Elite Teacher Project (No. QYP1902). The authorlearned about ergodic optimization from Yiwei Zhang and Yun Yang, and wouldlike to thank them for fruitful discussions.
EQUILIBRIUM STATES OF INTERMEDIATE ENTROPIES
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