Equilibrium stability for non-uniformly hyperbolic systems

Abstract

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally we show that these equilibrium states vary continuously in the weak * topology within such systems.2010 Mathematics Subject Classification. 37A05, 37A35.

Hyperbolic Potentials for Continuous Non-Uniformly Expanding Maps

“…Remark 4. We observe that the set of continuous hyperbolic potentials is open with respect to the C 0 topology, as can be see in [7], Proposition 3.1.…”

Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

“…Namely, we prove that the equilibrium state, as well as others thermodynamical quantities, such as the topological pressure, vary analytically. In [2], it was proved that the equilibrium state is jointly continuous with respect to the map and the potential while here we establish analyticity but only as a function of the potential.…”

“…Note that, δ > 0 depends only on f and σ. Moreover, δ can be taken uniformly in a neighborhood of f , see [Remark 3.5, [2]]. From now on we fix δ > 0 as above.…”

“…We consider the family G σ of pairs (F, Φ) ∈ S × C α (M × N ) such that ( * ) holds for f and Φ is hyperbolic for F satisfying condition ( * * ). The uniqueness of the equilibrium state for (F, Φ) ∈ G σ was proven in [2].…”

“…The meaning of analyticity in this context is the one given in Section 6. We point out that according [2] the family of hyperbolic potentials is an open class in the C 0 -topology. Since ( * * ) is an open condition on the potential we derive that P σ is an open subset of C α (M ).…”