Cram\'er distance and discretizations of circle expanding maps II: simulations

TL;DR

This paper uses numerical experiments to explore how discretizations of expanding maps affect the Cramér distance between measures, providing insights into the ergodic behavior and effects of numerical truncation.

Contribution

It offers new conjectures on the impact of numerical truncation on the ergodic properties of discretized expanding maps based on simulation results.

Findings

Identification of phenomena influencing Cramér distance evolution

Proposed conjectures on effects of numerical truncation

Insights into short-term ergodic behavior of discretizations

Abstract

This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in arXiv:2206.07991 [math.DS]. Our aim is to identify the phenomena driving the evolution of the Cram\'er 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. Based on numerical simulations we propose some conjectures on the effects of numerical truncation from the ergodic viewpoint.