  In this paper we study the ergodic theory of a robust non-uniformly expanding
maps where no Markov assumption is required. We prove that the topological
pressure is differentiable as a function of the dynamics and analytic with
respect to the potential. Moreover we not only prove the continuity of the
equilibrium states and their metric entropy as well as the differentiability of
the maximal entropy measure and extremal Lyapunov exponents with respect to the
dynamics. We also prove a local large deviations principle and central limit
theorem and show that the rate function, mean and variance vary continuously
with respect to observables, potentials and dynamics. Finally, we show that the
correlation function associated to the maximal entropy measure is
differentiable with respect to the dynamics and it is $C^1$-convergent to zero.
In addition, precise formulas for the derivatives of thermodynamical quantities
are given.
