Equilibrium stability for non-uniformly hyperbolic systems

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic H\"older continuous potentials. Finally we show that these equilibrium states vary continuously in the weak$^\ast$ topology within such systems.