Cramér distance and discretizations of circle expanding maps I: theory
Résumé
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 a distance we call discrepancy. 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 [GM22].
Domaines
Systèmes dynamiques [math.DS]
Origine : Fichiers produits par l'(les) auteur(s)