Stochastic Stability for Fiber Expanding Maps via a Perturbative Spectral Approach

TL;DR

This paper demonstrates the stability of invariant measures for fiber expanding maps under small perturbations using a spectral approach, providing bounds on correlation decay rates as noise diminishes.

Contribution

It introduces a perturbative spectral method to analyze stability of invariant densities for non-invertible skew-product expanding maps under small perturbations.

Findings

Invariant densities remain stable under perturbations.

Exponential decay rates of fiber correlations are bounded as noise decreases.

The spectral approach effectively analyzes non-invertible base dynamics.

Abstract

We consider small perturbations of expanding maps induced by skew-product mappings whose base dynamics are not invertible necessarily. Adopting a previously developed perturbative spectral approach, we show stability of the densities of the unique absolutely continuous invariant probability measures for expanding maps under these perturbations, and upper bounds on the rate of exponential decay of fiber correlations associated to the measures as the noise level goes to zero.