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

TL;DR

This paper investigates the spectral properties of the differential of self-consistent transfer operators at fixed points, demonstrating local exponential convergence to equilibrium in mean field coupled systems, including strong coupling scenarios.

Contribution

It introduces a method to analyze local convergence to equilibrium using the differential's spectral properties, applicable to strongly coupled mean field systems.

Findings

Spectral properties of the differential imply exponential convergence.

Results apply to systems with strong coupling.

Examples include deterministic expanding maps with various couplings.

Abstract

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 the differential does also. We present examples of application of the general results to self-consistent transfer operators based on deterministic expanding maps considered with different couplings, outside the weak coupling regime.