Hyperbolic Potentials for Continuous Non-Uniformly Expanding Maps
aa r X i v : . [ m a t h . D S ] F e b HYPERBOLIC POTENTIALS FOR CONTINUOUSNON-UNIFORMLY EXPANDING MAPS
EDUARDO SANTANA
Abstract.
In this work, we give a class of examples of hyperbolic potentials (includingthe null one) for continuous non-uniformly expanding maps. It implies the existence anduniqueness of equilibrium state (in particular, of maximal entropy measure). Among themaps considered is the important class known as Viana maps. Introduction
The theory of equilibrium states on dynamical systems was firstly developed by Sinai,Ruelle and Bowen in the sixties and seventies. It was based on applications of techniquesof Statistical Mechanics to smooth dynamics. Given a continuous map f : M → M on acompact metric space M and a continuous potential φ : M → R , an equilibrium state isan invariant measure that satisfies a variational principle, that is, a measure µ such that h µ ( f ) + Z φdµ = sup η ∈M f ( M ) (cid:26) h η ( f ) + Z φdη (cid:27) , where M f ( M ) is the set of f -invariant probabilities on M and h η ( f ) is the so-calledmetric entropy of η .In the context of uniform hyperbolicity, which includes uniformly expanding maps,equilibrium states do exist and are unique if the potential is H¨older continuous and themap is transitive. In addition, the theory for finite shifts was developed and used toachieve the results for smooth dynamics.Beyond uniform hyperbolicity, the theory is still far from complete. It was studied byseveral authors, including Bruin, Keller, Demers, Li, Rivera-Letelier, Iommi and Todd[14, 13, 16, 22, 23, 24] for interval maps; Denker and Urbanski [17] for rational maps;Leplaideur, Oliveira and Rios [25] for partially hyperbolic horseshoes; Buzzi, Sarig andYuri [15, 42], for countable Markov shifts and for piecewise expanding maps in one andhigher dimensions. For local diffeomorphisms with some kind of non-uniform expansion,there are results due to Oliveira [26]; Arbieto, Matheus and Oliveira [10]; Varandas andViana [39]. All of whom proved the existence and uniqueness of equilibrium states forpotentials with low oscillation. Also, for this type of maps, Ramos and Viana [32] proved itfor potentials so-called hyperbolic , which includes the previous ones. The hyperbolicityof the potential is characterized by the fact that the pressure emanates from the hyperbolicregion. In all these studies the maps does not have the presence of critical sets and recently,Alves, Oliveira and Santana proved the existence of at most finitely many equilibrium Date : February 9, 2021. states for hyperbolic potentials, possible with the presence of a critical set (see [6]). Morerecently, Santana completed this by showing uniqueness in [33].In this work, we give examples of a class of potentials whose Birkhoff sums are uniformlybounded. In particular, we show that the null potentials ϕ ≡ Preliminaries and Main Result
Non-uniformly Expanding Maps.
We recall the definition of a ( σ, δ )-hyperbolictime for x ∈ M .Let M be a connected compact metric space, f : M → M a continuous map and µ areference Borel measure on M . Fix σ ∈ (0 , , δ > x ∈ M . We say that n ∈ N is a( σ, δ )- hyperbolic time for x if • there exists a neighbourhood V n ( x ) of x such that f n sends V n ( x ) homeomorphi-cally onto the ball B δ ( f n ( x )); • d ( f i ( y ) , f i ( z )) ≤ σ n − i d ( f n ( y ) , f n ( z )) , ∀ y, z ∈ V n ( x ) , ∀ ≤ i ≤ n − . The sets V n ( x ) are called hyperbolic pre-balls and their images f n ( V n ( x )) = B δ ( f n ( x )), hyperbolic balls .We observe that if n is a ( σ, δ )-hyperbolic time for x , then n is a ( σ, δ ′ )-hyperbolic timefor x , for every 0 < δ ′ < δ .We say that x ∈ M has positive frequency of hyperbolic times iflim sup n →∞ n { ≤ j ≤ n − | j is a hyperbolic time for x } > , and define the expanding set H = { x ∈ M | the frequency of hyperbolic times of x is positive } . We say that a Borel probability measure µ on M is expanding if µ ( H ) = 1.Given a measure µ on M , its Jacobian is a function J µ f : M → [0 , + ∞ ) such that µ ( f ( A )) = Z A J µ f dµ for every A domain of injectivity , that is, a measurable set such that f ( A ) is measurableand f A : A → f ( A ) is a bijection. We say that the measure has bounded distortion ifthere exists ρ > (cid:12)(cid:12)(cid:12)(cid:12) log J µ f n ( y ) J µ f n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρd ( f n ( y ) , f n ( z )) , for every y, z ∈ V n ( x ), µ -almost everywhere x ∈ M , for every hyperbolic time n of x .A map with an expanding measure with bounded distortion associated is called non-uniformly expanding . YPERBOLIC POTENTIALS 3
Relative Pressure.
We recall the definition of relative pressure for non-compactsets by dynamical balls.Let M be a compact metric space. Consider f : M → M and φ : M → R . Given δ > n ∈ N and x ∈ M , we define the dynamical ball B δ ( x, n ) as B δ ( x, n ) := { y ∈ M | d ( f i ( x ) , f i ( y )) < δ, for 0 ≤ i ≤ n } . Consider for each N ∈ N , the set F δN = { B δ ( x, n ) | x ∈ M, n ≥ N } . Given Λ ⊂ M , denote by F δN (Λ) the finite or countable families of elements in F δN thatcover Λ. Define for n ∈ N the Birkhoff sum S n φ ( x ) = φ ( x ) + φ ( f ( x )) + · · · + φ ( f n − ( x ))and R n,δ φ ( x ) = sup y ∈ B δ ( x,n ) S n φ ( y ) . Given a f -invariant set Λ ⊂ M , not necessarily compact, define for each γ > m f ( φ, Λ , δ, N, γ ) = inf U∈F δN (Λ) X B δ ( y,n ) ∈U e − γn + R n,δ φ ( y ) . Define also m f ( φ, Λ , δ, γ ) = lim N → + ∞ m f ( φ, Λ , δ, N, γ ) . and P Λ ( φ, δ ) = inf { γ > | m f ( φ, Λ , δ, γ ) = 0 } . Finally, define the relative pressure of φ on Λ as P Λ ( φ ) = lim δ → P Λ ( φ, δ ) . The topological pressure of φ is, by definition, P ( φ ) = P M ( φ ) and satisfies P f ( φ ) = sup { P f ( φ, Λ) , P f ( φ, Λ c ) } (1)where Λ c denotes the complement of Λ on M .2.3. Hyperbolic potentials.
We say that a continuous function φ : M → R is a hyper-bolic potential if the topological pressure P f ( φ ) is located on H , i.e. P f ( φ, H c ) < P f ( φ ) . In [24], H. Li and J. Rivera-Letelier consider other type of hyperbolic potentials forone-dimensinal dynamics that is weaker than ours. In their context, φ is a hyperbolicpotential if sup µ ∈M f ( M ) Z φdµ < P f ( φ ) . E. SANTANA
Viana Maps.
We recall the definition of the open class of maps with critical setsin dimension 2, introduced by M. Viana in [40]. We skip the technical points. It can begeneralized for any dimension (See [1]).Let a ∈ (1 ,
2) be such that the critical point x = 0 is pre-periodic for the quadraticmap Q ( x ) = a − x . Let S = R / Z and b : S → R a Morse function, for instance b ( θ ) = sin(2 πθ ). For fixed small α >
0, consider the map f : S × R −→ S × R ( θ, x ) ( g ( θ ) , q ( θ, x ))where g is the uniformly expanding map of the circle defined by g ( θ ) = dθ ( mod Z ) forsome d ≥
16, and q ( θ, x ) = a ( θ ) − x with a ( θ ) = a + αb ( θ ). It is easy to check that for α > I ⊂ ( − ,
2) for which f ( S × I ) is contained in theinterior of S × I . Thus, any map f sufficiently close to f in the C topology has S × I asa forward invariant region. We consider from here on these maps f close to f restrictedto S × I . Taking into account the expression of f it is not difficult to check that for f (and any map f close to f in the C topology) the critical set is non-degenerate.The main properties of f in a C neighbourhood of f that we will use here are sum-marized below (See [1],[9],[31]):(1) f is non-uniformly expanding , that is, there exist λ > H ⊂ S × I such that for all point p = ( θ, x ) ∈ H , the following holdslim sup n →∞ n n − X i =0 log k Df ( f i ( p )) − k < − λ. (2) Its orbits have slow approximation to the critical set , that is, for every ǫ > δ > p = ( θ, x ) ∈ H ⊂ S × I , the followingholds lim sup n →∞ n n − X i =0 − log dist δ ( p, C ) < ǫ. where dist δ ( p, C ) = (cid:26) dist ( p, C ) , if dist ( p, C ) < δ if dist ( p, C ) ≥ δ (3) f is topologically mixing;(4) f is strongly topologically transitive;(5) it has a unique ergodic absolutely continuous invariant (thus SRB) measure;(6) the density of the SRB measure varies continuously in the L norm with f . Remark 1.
We observe that this definition of non-uniformly expansion is included inours by neighbourhoods. Details can be found in [6] or [33] . Other obvious references are [1] , [31] and the original work by M. Viana [40] . YPERBOLIC POTENTIALS 5
Main Result.Theorem 1.
Let f : M → M be a continuous non-uniformly expanding map and H its expanding set and ϕ : M → R a continuous potential with Birkhoff sums uniformlybounded. If H and H c are both dense on M and the topological entropy h ( f ) is positive,then the potential ϕ is hyperbolic. In particular, if ϕ is H¨older there exists a uniqueequilibrium state. Corollary 1.
Let f : M → M be a continuous non-uniformly expanding map and H itsexpanding set. If H and H c are both dense on M and h ( f ) > , the null potential ishyperbolic. In particular, there exists a unique measure of maximal entropy for f . Corollary 2.
Let f : S × I → S × I be a Viana map. There exists a unique measureof maximal entropy for f . Proof of the Main Result
We begin by proving the next Theorem, which is the base to prove our main result.
Theorem 2.
Let f : M → M be a continuous non-uniformly expanding map and H itsexpanding set. If H and H c are both dense on M and h ( f ) > , the null potential ishyperbolic. In particular, there exists a unique measure of maximal entropy for f .Proof. We will show that the null potential is hyperbolic. In order to do that we will usethe power of hyperbolic times. We divide the proof in several lemmas.
Lemma 1. If n ∈ N is a ( σ, δ ) -hyperbolic time for x ∈ M , then B δ ( x, n ) = V n ( x ) .Proof. Firstly, given y ∈ V n ( x ), by definition we have d ( f i ( y ) , f i ( x )) ≤ σ n − i d ( f n ( y ) , f n ( z )) < δ, ∀ ≤ i ≤ n − . and it means that y ∈ B δ ( x, n ). So, V n ( x ) ⊂ B δ ( x, n ).We observe that if y ∈ B δ ( x, n ) then f n ( y ) ∈ B δ ( f n ( x )). If y ∈ V n ( x ) c there exists0 ≤ i ≤ n − d ( f i ( y ) , f i ( x )) > σ n − i d ( f n ( y ) , f n ( x )). Once f n sends V n ( x )homeomorphically to B δ ( f n ( x )), it means that f n ( y ) B δ ( f n ( x )) = ⇒ y ∈ B δ ( x, n ) c = ⇒ V n ( x ) c ⊂ B δ ( x, n ) c . Since V n ( x ) ⊂ B δ ( x, n ) and V n ( x ) c ⊂ B δ ( x, n ) c , we have V n ( x ) = B δ ( x, n ). (cid:3) As as consequence of the previous lemma, if n is a ( σ, δ )-hyperbolic time for x , then B δ ( x, n ) is always an open set. Since H c is dense on M and for every dynamicall ball B δ ( x, n ) where n is a ( σ, δ )-hyperbolic time for x we can always find y ( x ) ∈ B δ ( x, n ) ∩ H c .So, we have the following lemma. Lemma 2.
Given < δ < δ and a covering U of H , such that each B δ ( x, n ) ∈ U with x ∈ H and n ≥ is a ( σ, δ ) -hyperbolic time for x , let y ( x ) ∈ B δ ( x, n ) ∩ H c . We have B δ ( x, n ) ⊂ B δ ( y ( x ) , n ) , ∀ δ < δ and the collection V ( U ) of dynamical balls B δ ( y ( x ) , n ) is a covering of H c with the same cardinality as U . E. SANTANA
Proof.
Let z ∈ B δ ( x, n ) = V n ( x ). We know that d ( f i ( z ) , f i ( x )) ≤ σ n − i d ( f n ( z ) , f n ( x )) < δ, ∀ i ≤ n. Also d ( f i ( y ) , f i ( x )) ≤ σ n − i d ( f n ( y ) , f n ( x )) < δ, ∀ i ≤ n. It implies that d ( f i ( z ) , f i ( y )) < δ, ∀ i ≤ n = ⇒ z ∈ B δ ( y, n ) . So, B δ ( x, n ) ⊂ B δ ( y, n ) and B δ ( x, n ) ⊂ B δ ( x, n ) ⊂ B δ ( y, n ), as we claimed.We know that H and H c are both dense on M . Given a covering U ∈ F δN ( H ), we have H ⊂ [ B δ ( x,n ) ∈U B δ ( x, n ) = ⇒ M = [ B δ ( x,n ) ∈U B δ ( x, n ) ⊂ [ B δ ( y ( x ) ,n ) ∈V ( U ) B δ ( y ( x ) , n ) . and H c ⊂ [ B δ ( y ( x ) ,n ) ∈V ( U ) B δ ( y ( x ) , n ) = ⇒ V ( U ) := { B δ ( y ( x ) , n ) } ∈ F δN ( H c ) . (cid:3) Since M is compact, given X ⊂ M , there exists a countable set X ⊂ X such that X ⊂ M . So, since H and H c are both dense on M , there exist countable sets X ⊂ H and Y ⊂ H c both dense on M . Let X = { x , . . . , x k , . . . } and Y = { y , . . . , y k , . . . } . Lemma 3.
Given θ > and a covering U = { B δ ( x, n ) } ∈ F δN ( H ) , there exists a covering U ′ = { B δ ( x i , n i ) | x i ∈ X } ∈ F δN ( H ) such that n i is a ( σ, δ ) -hyperbolic time for x i and X B δ ( x i ,n i ) ∈U ′ e − γ θ n i ≤ X B δ ( x,n ) ∈U e − γn Proof.
Take γ > U such that τ := X B δ ( x,n ) ∈U e − γn < ∞ . Given a >
0, we have that ∞ X i =1 e − ia = e − a − e − a We take a > N and n i ∈ N a ( σ, δ )-hyperbolic time for x i such that γ θ n i ≥ ia andalso e − a − e − a ≤ τ ⇐⇒ e a − ≤ τ ⇐⇒ τ + 1 ≤ e a ⇐⇒ ln (cid:18) τ + 1 τ (cid:19) ≤ a. We also can take the sequence n i increasing. By considering the collection U ′ = { B δ ( x i , n i ) | x i ∈ X } , we have X ⊂ ∞ [ i =1 B δ ( x i , n i ) = ⇒ M = X = ∞ [ i =1 B δ ( x i , n i ) ⊂ [ B δ ( x,n ) ∈U ′ B δ ( x, n ) . YPERBOLIC POTENTIALS 7 and it means that U ′ = { B δ ( x i , n i ) | x i ∈ X } ∈ F δN ( H ) is such that X B δ ( x i ,n i ) ∈U ′ e − γ θ n i ≤ ∞ X i =1 e − ia = e − a − e − a ≤ τ = X B δ ( x,n ) ∈U e − γn (cid:3) Remark 2.
We have that the collection U ′ is a covering of M by open subsets and since M is compact, there exists a finite subcovering U ′′ . Then X B δ ( x,n ) ∈U ′′ e − γn ≤ X B δ ( x,n ) ∈U ′ e − γn It means that it is enough to consider finite coverings U ′′ = { B δ ( x i , n i ) } of H where n i is a ( σ, δ ) -hyperbolic time for x i because we will consider the infimum of those sums. Lemma 4.
We have that P H c (0) = 0 < P H (0) .Proof. Let δ < δ . We have m f (cid:18) , H c , δ, N, γ θ (cid:19) = inf V∈F δN ( H c ) (cid:26) X B δ ( y,n ) ∈V e − γ θ n (cid:27) ≤≤ inf V ( U ′ ) , U∈F δN ( H ) (cid:26) X B δ ( y ( x ) ,n ) ∈V ( U ′ ) e − γ θ n (cid:27) = inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U ′ e − γ θ n (cid:27) == inf U ′ ∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γ θ n (cid:27) ≤ inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γn (cid:27) = m f (0 , H, δ, N, γ ) . It implies that m f (cid:18) , H c , δ, N, γ θ (cid:19) ≤ m f (0 , H, δ, N, γ ) = ⇒ m f (cid:18) , H c , δ, γ θ (cid:19) ≤ m f (0 , H, δ, γ ) . So, P H c (0 , δ ) = inf { ρ > | m f (0 , H c , δ, ρ ) = 0 } ≤ inf { γ/ (1 + θ ) > | m f (0 , H, δ, γ ) = 0 } == inf { γ > | m f (0 , H, δ, γ ) = 0 } / (1 + θ ) = P H (0 , δ ) / (1 + θ ) . By taking the limits when δ →
0, we have P H c (0) ≤ P H (0) / (1 + θ ) < P H (0) . Since it holds for every θ > P H c (0) = 0 < P H (0) . (cid:3) Since Lemma 4 shows that the null potential is hyperbolic, we obtain the Theorem 2.It is the base for our main result. (cid:3)
With Lemma 4 and the next lemma, we can see that the constant potentials are allhyperbolic.
Lemma 5.
We have that P Λ ( φ + c ) = P Λ ( φ ) + c , for all potential φ and constant c ∈ R . E. SANTANA
Proof.
Let ψ := φ + c . We have R n,δ ψ ( x ) = R n,δ φ ( x ) + nc . So, m f ( ψ, Λ , δ, N, γ ) = inf U∈F N (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn + R n,δ ψ ( y ) (cid:27) =inf U∈F N (Λ) (cid:26) X B δ ( y,n ) ∈U e − ( γ − c ) n + R n,δ φ ( y ) (cid:27) = m f ( φ, Λ , δ, N, γ − c ) . It implies that m f ( ψ, Λ , δ, γ ) = m f ( φ, Λ , δ, N, γ − c ). If γ > m f ( ψ, Λ , δ, γ ) =0, then m f ( φ, Λ , δ, N, γ − c ) = 0 ⇒ γ − c ≥ P Λ ( φ, δ ) or γ ≥ P Λ ( φ, δ ) + c ⇒ P Λ ( ψ, δ ) ≥ P Λ ( φ, δ ) + c .In the other hand, if m f ( φ, Λ , δ, N, β ) = 0, then m f ( ψ, Λ , δ, β + c ) = 0 and P Λ ( ψ, δ ) ≤ β + c ⇒ P Λ ( ψ, δ ) ≤ P ( φ, δ ) + c . It means that P Λ ( ψ, δ ) = P ( φ, δ ) + c or P Λ ( ψ ) = P ( φ ) + c . (cid:3) The previous lemma also shows that if φ is a hyperbolic potential, so is φ + c, ∀ c ∈ R .We can also obtain the following lemma. Lemma 6.
Let ϕ be a potential such that there exist β ∈ R with R n,δ ϕ ( x ) ≤ β, ∀ x ∈ M, ∀ n ≥ . . Then, P Λ ( ϕ ) ≤ P Λ (0) .Proof. m f ( ϕ, Λ , δ, N, γ ) = inf U∈F δN (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn + R n,δ ϕ ( y ) (cid:27) ≤ inf U∈F δN (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn + β (cid:27) ≤ = e β inf U∈F δN (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn (cid:27) = e β m f (0 , Λ , δ, N, γ ) . It implies that m f ( ϕ, Λ , δ, γ ) = lim N → + ∞ m f ( ϕ, Λ , δ, N, γ ) ≤ e β m f (0 , Λ , δ, γ ) . So, m f (0 , Λ , δ, η ) = 0 = ⇒ m f ( ϕ, Λ , δ, η + θ ) = 0 and P Λ ( ϕ, δ ) = inf { γ | m f ( ϕ, Λ , δ, γ ) = 0 } ≤ inf { η | m f (0 , Λ , δ, η ) = 0 } = P Λ (0 , δ ) . Finally, P Λ ( ϕ ) = lim δ → P Λ ( ϕ, δ ) ≤ lim δ → P Λ (0 , δ ) = P Λ (0) . and the lemma is proved. (cid:3) Lemma 7.
Let ϕ be a potential such that there exist α, β > with α ≤ R n,δ ϕ ( x ) ≤ β, ∀ x ∈ M, ∀ n ≥ . . Then, P H c ( ϕ ) ≤ P H c (0) = 0 < P H ( ϕ ) ≤ P H (0) . YPERBOLIC POTENTIALS 9
Proof.
It is enough to show that 0 < P H ( ϕ ) because Lemma 6 gives the other inequalities.We follow along the same lines as in Lemma 4Let δ < δ . We have m f (cid:18) , H c , δ, N, γ θ (cid:19) = inf V∈F δN ( H c ) (cid:26) X B δ ( y,n ) ∈V e − γ θ n (cid:27) ≤≤ inf V ( U ′ ) , U∈F δN ( H ) (cid:26) X B δ ( y ( x ) ,n ) ∈V ( U ′ ) e − γ θ n (cid:27) = inf U ′ ∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U ′ e − γ θ n (cid:27) == inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γ θ n (cid:27) ≤ inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γn (cid:27) ≤≤ e α inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γn (cid:27) = inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γn + α (cid:27) ≤≤ inf U∈F δN ( H ) (cid:26) X B δ ( x,n ) ∈U e − γn + R n,δ ϕ ( x ) (cid:27) = m f ( ϕ, H, δ, N, γ ) . It implies that m f (cid:18) , H c , δ, N, γ θ (cid:19) ≤ m f ( ϕ, H, δ, N, γ ) = ⇒ m f (cid:18) , H c , δ, γ θ (cid:19) ≤ m f ( ϕ, H, δ, γ ) . So, P H c (0 , δ ) = inf { ρ > | m f (0 , H c , δ, ρ ) = 0 } ≤ inf { γ/ (1 + θ ) > | m f ( ϕ, H, δ, γ ) = 0 } == inf { γ > | m f ( ϕ, H, δ, γ ) = 0 } / (1 + θ ) = P H ( ϕ, δ ) / (1 + θ ) . By taking the limits when δ →
0, we have P H c (0) ≤ P H ( ϕ ) / (1 + θ ) < P H ( ϕ ) . Since it holds for every θ > P H c (0) = 0 < P H ( ϕ ) . and the lemma is proved. (cid:3) In particular, we have P H c ( ϕ ) ≤ P H c (0) and P H ( ϕ ) ≤ P H (0) . We recall that a hyperbolic potential ϕ satisfies P H c ( ϕ ) < P H ( ϕ ) . It means that the potential ϕ is hyperbolic: P H c ( ϕ ) ≤ P H c (0) = 0 < P H ( ϕ ) ≤ P H (0) . Lemma 8.
Let φ : M → N with its Birkhoff sums uniformly bounded, that is, there exists r > such that | S n φ ( x ) | < r, ∀ n ∈ N , ∀ x ∈ M. Then, φ is a hyperbolic potential. Proof.
In fact, by following the proof of Lemma 6 we can see that P H c ( φ ) ≤ P H c (0) = 0.Also, Birkhoff’s Ergodic Theorem implies that R φdη = 0 for every f -invariant probability η . It means that P ( φ ) = sup η ∈M f ( M ) (cid:26) h η ( f ) + Z φdη (cid:27) = sup η ∈M f ( M ) (cid:26) h η ( f ) (cid:27) = h ( f ) = P (0) = P H (0) > . So, φ is a hyperbolic potential. (cid:3) Remark 3.
The previous lemma proved that the potential φ is hyperbolic if we have R φdη = 0 for every invariant measure η . In particular, if we have the Birkhoff sumsuniformly bounded. Lemma 9. If φ ≤ ψ , then P Λ ( φ ) ≤ P Λ ( ψ ) .Proof. If φ ≤ ψ , we obtain R n,δ φ ( x ) ≤ R n,δ ψ ( x ). So, m f ( φ, Λ , δ, N, γ ) = inf U∈F N (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn + R n,δ φ ( y ) (cid:27) ≤≤ inf U∈F N (Λ) (cid:26) X B δ ( y,n ) ∈U e − γn + R n,δ ψ ( y ) (cid:27) = m f ( ψ, Λ , δ, N, γ ) ⇒ m f ( φ, Λ , δ, γ ) = lim N → + ∞ m f ( φ, Λ , δ, N, γ ) ≤ lim N → + ∞ m f ( ψ, Λ , δ, N, γ ) = m f ( ψ, Λ , δ, γ ) . It implies that m f ( φ, Λ , δ, γ ) = 0 if m f ( ψ, Λ , δ, γ ) = 0, that is, P Λ ( φ, δ ) = inf { γ | m f ( φ, Λ , δ, γ ) = 0 } ≤ inf { γ | m f ( ψ, Λ , δ, γ ) = 0 } = P Λ ( ψ, δ ) . Finally, P Λ ( φ ) = lim δ → P Λ ( φ, δ ) ≤ lim δ → P Λ ( ψ, δ ) = P Λ ( ψ ) . (cid:3) With the previous lemma we can obtain the following example.
Example 1.
Let ϕ : M → R be a hyperbolic potential and φ : M → R such that max φ − min φ < P H ( ϕ ) − P H c ( ϕ ) . It implies that P H c ( ϕ + φ ) ≤ P H c ( ϕ + max φ ) = P H c ( ϕ ) + max φ << P H ( ϕ ) + min φ = P H ( ϕ + min φ ) ≤ P H ( ϕ + φ ) . So, ϕ + φ is a hyperbolic potential.If | t | ≤ , we also have max tφ − min tφ < P H ( ϕ ) − P H c ( ϕ ) . and ϕ + tφ is also a hyperbolic potential.In particular, since the null potential is hyperbolic, if we have max φ − min φ < P H (0) = P (0) = h ( f ) , then φ is also a hyperbolic potential. YPERBOLIC POTENTIALS 11
Remark 4.
We observe that the set of continuous hyperbolic potentials is open with respectto the C topology, as can be see in [7] , Proposition . . Example 2.
Now, for Viana maps, we construct a potential with Birkhoff sums uniformlybounded.Let B be an open set and V = f − ( B ) such that V ∩ B = ∅ and V ∩ C = ∅ , where C is the critical set. Let φ : B → R be a C ∞ function such that φ | ∂B ≡ and we define apotential ϕ : X → R as ϕ ( x ) = φ ( x ) , if x ∈ B − φ ( f ( x )) , if x ∈ V , if x ∈ ( V ∪ B ) c Claim 1.
The Birkhoff sums S n ϕ are uniformly bounded.Proof. For x ∈ V , we have that S n ϕ ( x ) = ϕ ( x ) + ϕ ( f ( x )) + · · · + ϕ ( f n − ( x )) = − φ ( f ( x )) + φ ( f ( x )) + 0 + · · · + φ ( f n − ( x )) ≤ sup φ, For x ∈ B , we have S n ϕ ( x ) = ϕ ( x ) + ϕ ( f ( x )) + · · · + ϕ ( f n − ( x )) ≤ sup φ − inf φ, if f n − ( x ) ∈ V and it is equal to φ ( x ), otherwise.For x ∈ ( V ∪ B ) c , we have at most the same estimate for S n ϕ ( x ) because the orbit of x may intersect V ∪ B . (cid:3) So, the Birkhoff sums are uniformly bounded and Lemma 8 guarantees that ϕ is hy-perbolic. Moreover, ϕ is H¨older, which means that we have existence and uniqueness ofequilibrium state. Acknowledgements:
The author would like to thank K. Oliveira for pointing out thisproblem and R. Bilbao for fruitful conversations.
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Eduardo Santana, Universidade Federal de Alagoas, 57200-000 Penedo, Brazil
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