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Article Dans Une Revue Nonlinearity Année : 2023

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].
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Dates et versions

hal-03883534 , version 1 (03-12-2022)
hal-03883534 , version 2 (01-11-2023)

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Pierre-Antoine Guihéneuf, Maurizio Monge. Cramér distance and discretizations of circle expanding maps I: theory. Nonlinearity, 2023, 36 (9). ⟨hal-03883534v2⟩
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