Cram\'er distance and discretizations of circle expanding maps I: theory

TL;DR

This paper investigates the short-term ergodic behavior of discretized circle expanding maps using Cramér distance, providing asymptotic results and a linearized analysis under generic conditions.

Contribution

It introduces a theoretical framework for analyzing the discretizations of circle expanding maps with Cramér distance and establishes asymptotic behavior under explicit genericity assumptions.

Findings

Asymptotic estimates of measure distances for fixed iterations as grid size increases

Connection between discretization behavior and equirepartition on high-dimensional tori

Theoretical insights supported by numerical studies

Abstract

This paper is aimed to study the ergodic short-term behaviour of discretizations of circle expanding maps. More precisely, we prove some asymptotics of the distance between the $t$-th iterate of Lebesgue measure by the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of order $N$ by the discretization on this grid, when $t$ is fixed and the order $N$ goes to infinity. This is done under some explicit genericity hypotheses on the dynamics, and the distance between measures is measured by the mean of \emph{Cram\'er} distance. The proof is based on a study of the corresponding linearized problem, where the problem is translated into terms of equirepartition on tori of dimension exponential in $t$. A numerical study associated to this work is presented in arXiv:2206.08000 [math.DS].