The Differential of Self-Consistent Transfer Operators and the Local Convergence to Equilibrium of Mean Field Strongly Coupled Dynamical Systems

Abstract We consider the differential of a self-consistent transfer operator at a fixed point and show that its spectral properties can be used to establish a form 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 result holds even in the strong coupling case. Additionally, we show that for mean field coupled systems satisfying a uniform Lasota–Yorke inequality the differential also satisfies this condition. Finally, we provide examples applying these general results to self-consistent transfer operators based on deterministic expanding maps with various couplings, extending beyond the weak coupling regime.