Thermodynamic formalism for expanding measures

Abstract

We study the Thermodynamic Formalism for strongly transitive endomorphisms f , focusing on the set all expanding measures. When f is a non-flat C 1+ map defined on a Riemannian manifold, being an expanding measure means being an invariant probability with all its Lyapunov exponents positive. Roughly speaking, given a Hölder potential ϕ, we establish the uniqueness of the equilibrium state among the expanding measures. Moreover, the existence of an expanding measure µ maximizing the entropy implies the existence and uniqueness of the equilibrium state µ ϕ among all invariant probabilities, not only the expanding ones, for any given Hölder potential ϕ with a small oscillation osc ϕ = sup ϕ − inf ϕ. As one of the applications, we show that if f is a Viana map [68] and ϕ a Hölder continuous potential with small oscillation, then there exists one and only only one equilibrium state for ϕ.