Unique equilibrium states for geodesic flows over surfaces without focal points

Abstract

We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.Theorem A. Let M be a closed rank 1 Riemannian manifold without focal points, F be the geodesic flow over M , and ϕ : T 1 M → R be a Hölder potential. If ϕ satisfies the pressure gap condition, then ϕ has a unique equilibrium state µ ϕ .The origin of Theorem A traces back to the work of Bowen [Bow74] where he showed that the bounded distortion property (also known as the Bowen property) on the potential and the expansivity and the specification property on the dynamical system guarantee the existence of a unique equilibrium state. Natural examples of such systems are uniformly hyperbolic systems. Since the work of Bowen, his result has been extended in various directions, and one such direction aims at relaxing the assumptions on the base dynamical systems. Recently, Climenhaga and Thompson [CT16] developed a set of criteria that guarantees the existence of a unique equilibrium state which applies to many non-uniformly hyperbolic systems; see [CFT18], [BCFT18], [CKP18], and [CFT19].Among non-uniformly hyperbolic dynamical systems arising from geometry, the first result in the same flavor as Theorem A was the work of Knieper [Kni98]. He employed Patterson-Sullivan theory to establish the uniqueness of the measure of maximal entropy for geodesic flows over nonpositively curved manifolds. Twenty years later, Burns, Climenhaga, Fisher, and Thompson [BCFT18] extended Knieper's result to equilibrium states for potentials with the pressure gap. Their approach is inspired by the previously mentioned work of Bowen [Bow74]. Previous work by the authors [CKP18] used the same approach to further generalize this result to surfaces without focal points. Finally, in this work, Theorem A shows that the same philosophy holds for higher dimensional manifolds without focal points.From Theorem A, it automatically follows that these equilibrium states are ergodic. It is natural to ask whether such equilibrium states possess stronger ergodic properties. Indeed, for uniformly hyperbolic systems, the unique equilibrium states for Hölder potentials have many stronger ergodic properties; these include being Bernoulli, and having equidistribution property by weighted periodic orbits as well as statistical properties such as the central limit theorem and the large deviation property; see [PP90] and [KH97].The following theorem partially answers the question above, and establishes several ergodic properties of the unique equilibrium state µ ϕ from Theorem A.Theorem B. In the setting of Theorem A, the unique equilibrium state µ ϕ has the following properties: µ ϕ has the K-property and is fully supported. Moreover, µ ϕ is equal to the weak- * limit of the weig...

Symbolic dynamics for non-uniformly hyperbolic systems

“…Gelfert and Ruggiero proved the uniqueness for geodesic flows on surfaces without focal points and genus greater than one [GR19]. Burns et al proved the uniqueness of many equilibrium states (including some multiples of the geometric potential and the zero potential) of geodesic flows on rank-one manifolds [BCFT18], and there is a recent preprint that obtains similar results for geodesic flows on surfaces without focal points [CKP20]. There is also a recent preprint that proves the uniqueness of the measure of maximal entropy for geodesic flows on surfaces without conjugate points [CKW19].…”

TheK-property for subadditive equilibrium states

“…Proof: First, as η is a weak mixing equilibrium state, η × η is an ergodic equilibrium state. Therefore, if we can show Lemma 3.1 holds for the system (X × X, f × f ) with potential defined as (12), then it follows that η × η is the unique equilibrium state. Therefore, by the subadditive version of Ledrappier's criterion, it immediately follows that η is K.…”

“…Since then, the theory has been extended in mainly two different directions. One direction aims to relax the uniform hyperbolicity of the base dynamics (see, for instance [8,12,15,24]). The other aims to relax and generalize the assumptions on the potential.…”

Multifractal analysis of geodesic flows on surfaces without focal points

“…Setting α 2 := D − P(1), the first case concerns with the domain β ∈ (−α 2 , −α 1 ) corresponding to the time before the phase transition. In this case, we compute the entropy h(L(β)) using the fact that P is C 1 when t < 1, which is based on the uniqueness of the equilibrium state for tϕ geo obtained in [CKP20]; see § 2 for further discussions.…”

“…As the entropy spectrum is well-understood on the basic sets, and the increasingly nested basic sets are constructed so that they eventually intersect L(β) non-trivially, we use such information to establish an effective lower bound for the entropy of L(β). In our case, the construction of such a sequence of basic sets relies on the hyperbolic index function λ T introduced in [CKP20]; see § 4.…”