Unique equilibrium states for flows and homeomorphisms with non-uniform structure

Abstract

Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and non-uniformly hyperbolic settings.

49 pages, 5 figures. Changes since v1: Improved and expanded exposition, following referee comments. Improvement on a couple of technical points: removed a technical assumption in our definition of the specification property for flows (which was present in the classic work on equilibrium states for flows with specification by Franco); fixed an issue in the proof of Proposition 3.10

References in corpus (2)

• Thermodynamics of Towers of Hyperbolic Type
• Unique equilibrium states for Bonatti-Viana diffeomorphisms

Cited by in corpus (23)

• Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows
• Properties of equilibrium states for geodesic flows over manifolds without focal points
• Ergodic measures of intermediate entropies for dynamical systems with the approximate product property
• Equilibrium measures for certain isometric extensions of Anosov systems
• High pointwise emergence and Katok's conjecture for systems with non-uniform structure
• Geodesic flows of compact higher genus surfaces without conjugate points have expansive factors
• The weak specification property for geodesic flows on CAT(-1) spaces
• Renormalization in Lorenz maps -- completely invariant sets and periodic orbits
• Zero-entropy dynamical systems with the gluing orbit property
• Large deviations of return times and related entropy estimators on shift spaces
• Equilibrium states for certain partially hyperbolic attractors
• On non-uniform specification and uniqueness of the equilibrium state in expansive systems
• Fluctuations of time averages around closed geodesics in non-positive curvature
• Equilibrium states for maps isotopic to Anosov
• The K-Property for Subadditive Equilibrium States
• Topological pressure for discontinuous semiflows and a variational principle for impulsive dynamical systems
• Minimality and Gluing Orbit Property
• Unique equilibrium states for geodesic flows in nonpositive curvature
• Equilibrium states for a class of skew-products
• An Estimation of Phase Transition
• Denseness of intermediate pressures for systems with the Climenhaga-Thompson structures
• Unique equilibrium states for Viana maps with small potentials
• On the Entropy of Flows with Reparametrized Gluing Orbit Property