Thermodynamic formalism for expanding measures

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 H\"older continuous potential $\varphi$ 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 $\mu_{\varphi}$ on the space of expanding measures for any H\"older continuous potential $\varphi$ with a small oscillation $\text{osc }\varphi=\sup\varphi-\inf\varphi$. As some applications, we prove that Collet-Eckmann quadratic maps does not admit phase transition for H\"older potential, and show that for Viana maps and every H\"older continuous potential of sufficiently small oscillation has a unique equilibrium state.