Thermodynamic formalism for expanding measures

Abstract

In this paper we study the thermodynamic formalism of strongly transitive endomorphisms $f$, focusing on the set all expanding measures. In case $f$ is a non-flat $C^{1+}$ map defined on a Riemannian manifold, these are invariant probability measures with all its Lyapunov exponents positive. Given a Hölder continuous potential $\phi$ we prove the uniqueness of the equilibrium state among the space of expanding measures. Moreover, we show that the existence of an expanding measure $\mu$ maximizing the entropy on the the space of expanding measures implies the existence and uniqueness of equilibrium state ${\mu }_{\phi }$ on the space of expanding measures for any Hölder continuous potential $\phi$ with a small oscillation $\text{osc }\phi =\sup \phi -\inf \phi$. As some applications, we prove that Collet-Eckmann quadratic maps does not admit phase transition for Hölder potential, and show that for Viana maps and every Hölder continuous potential of sufficiently small oscillation has a unique equilibrium state.

56 pages