Stability and approximation of statistical limit laws for multidimensional piecewise expanding maps

TL;DR

This paper proves the stability of statistical limit laws, like the central limit theorem and large deviations, for multidimensional piecewise expanding maps under various perturbations, including numerical schemes.

Contribution

It establishes the stability of eigenvalues and eigenprojections of twisted transfer operators for multidimensional maps, extending previous one-dimensional results.

Findings

Stability of eigenvalues and eigenprojections under broad perturbations.

Demonstration of stable variance in the CLT under perturbations.

New convergence results for Ulam projections on quasi-Hölder spaces.

Abstract

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-H\"older spaces.