Cramer distance and discretisations of circle expanding maps I: theory

This paper is aimed to study the ergodic short-term behaviour of discretisations of circle expanding maps. More precisely, we prove some asymptotics of the distance between the tth iterate of Lebesgue measure by the dynamicsf and the tth iterate of the uniform measure on the grid of order N by the discretisation 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 Cramer distance. The proof is based on a study of the corresponding linearised 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 Guiheneuf and Monge (2022 arXiv:2206.08000).