Equilibrium states for Mañé diffeomorphisms

Abstract

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the SRB measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

Equilibrium states for a class of skew products

“…Although powerful, to apply the main theorem of [12] to particular cases beyond hyperbolicity, one has to cope with these additional demands on the dynamics and the potentials. Climenhaga, Fisher and Thompson succeeded in using this machinery to settle a thermodynamic formalism for both Bonatti-Viana diffeomorphisms [4,10] and the family of partially hyperbolic, robustly transitive, derived from Anosov systems introduced by Mañé [11,26]. One of the key ideas of these two articles was the reformulation of those supplementary requests on the dynamics in terms of the C 0 -closeness to an Anosov system and the denseness of each center-stable/centerunstable foliation, both of which are essential to ensure specification at a small scale.…”

Equilibrium stability for non-uniformly hyperbolic systems

“…In [9], Buzzi, Fisher, Sambarino and Vásquez obtained uniqueness of the maximal entropy measure for partially hyperbolic maps derived from Anosov. Climenhaga, Fisher and Thompson in [14,15] address the question of existence and uniqueness of equilibrium states for Bonatti-Viana diffeomorphisms and Mañé diffeomorphisms for suitable classes of potentials. Castro and Nascimento in [12] showed uniqueness of the maximal entropy measure for partially hyperbolic attractors semiconjugated to nonuniformly expanding maps.…”

Specification and Towers in Shift Spaces

“…This will be particularly valuable if it can be extended to the smooth setting. The non-uniform specification properties from [CT12,CT13] have been extended and applied to various smooth systems [CT16,CFT18,CFT,BCFT], such as geodesic flows over rank 1 manifolds of nonpositive curvature. It is expected that the results given here will admit a similar generalisation.…”