The differential of self-consistent transfer operators and the local convergence to equilibrium of mean field strongly coupled dynamical systems

We consider the differential of a self-consistent transfer operator at a fixed point of the operator itself and show that its spectral properties can be used to establish a kind of local exponential convergence to equilibrium: probability measures near the fixed point converge exponentially fast to the fixed point by the iteration of the transfer operator. This holds also in the strong coupling case. We also show that for mean field coupled systems satisfying uniformly a Lasota-Yorke inequality also the differential does. We present examples of application of the general results to deterministic expanding maps outside the weak coupling regime.