Linear response formula for equilibrium states in non-uniformly expanding dynamics

In this paper we give further contributions to the ergodic theory of
a robust class of local diffeomorphisms with non-uniform expansion and where
no Markov assumption is required. We prove that the topological pressure is
differentiable as a function of the dynamics and the potential and provide a
formula to the differentiable dependence of equilibrium states. Moreover we
prove differentiability of the maximal entropy measure and continuity of extremal
Lyapunov exponents and metric entropy with respect to the dynamics.
Finally we obtain a local large deviation principle for the equilibrium states
and show that the rate function is continuous with respect to the dynamics
and the potential.