An elementary approach to rigorous approximation of invariant measures

TL;DR

This paper introduces a general framework for rigorously approximating invariant measures of dynamical systems with explicit error bounds, applicable to various map types, and demonstrates its implementation and effectiveness through experiments.

Contribution

It presents a flexible, error-controlled method for approximating invariant measures and densities in dynamical systems, including systems with indifferent fixed points.

Findings

Effective algorithms for invariant measure approximation with certified error bounds.

Successful application to piecewise expanding maps and maps with indifferent fixed points.

Experimental validation on one-dimensional maps demonstrating practical implementation.

Abstract

We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the $L^{1}$ or $L^\infty $ distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations, and experimental results on some one dimensional maps.