Self-Consistent Transfer Operators: Invariant Measures, Convergence to Equilibrium, Linear Response and Control of the Statistical Properties

Abstract

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We apply the general statements to examples of different nature: coupled continuous maps, coupled expanding maps, coupled systems with additive noise, systems made of different maps coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

Globally Coupled Anosov Diffeomorphisms: Statistical Properties

“…Note that the above strategy is natural when the transfer operator associated with the site dynamics admits a spectral gap on a Banach space. See [15] for a general strategy similar to the one we implement in this work.…”

“…Remark 2.11. If one wants to follow [15] in the Anosov setting, one has to choose the regularity in our spaces carefully. It seems to us that such a choice may then require more regularity on the map.…”

“…This was taken one step further in [31] where linear response was shown in a rather general, smooth setting. Recent advances on globally coupled maps can be found in [15]. The work of [15] includes an abstract framework and applications to study statistical aspects of globally coupled circle maps.…”

“…Recent advances on globally coupled maps can be found in [15]. The work of [15] includes an abstract framework and applications to study statistical aspects of globally coupled circle maps. However, up to date there are no ergodic theoretic results on globally coupled higher dimensional hyperbolic systems in an infinite limit.…”

Globally coupled Anosov diffeomorphisms: Statistical properties

“…This was taken one step further in [28] where linear response was shown in a rather general, smooth setting. Recent advances on globally coupled circle maps can be found in [14]. However, up to date there are no ergodic theoretic results on globally coupled higher dimensional hyperbolic systems in an infinite limit.…”