Cramér distance and discretizations of circle expanding maps II: simulations

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 P.-A. Guihéneuf and M. Monge, [Cramér distance and discretizations of circle expanding maps I: theory, (2022). arXiv 2206.07991]. Our aim is to identify the phenomena driving the evolution of the distance between the tth iterate of Lebesgue measure by the dynamics f and the tth 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.

Acknowledgments

This project was partially supported by a PEPS/CNRS project and the ANR CODYS. The authors warmly thank Nina Heloin for his careful reading of a first version of this text, Djalil Chafai for the references about the name of our distance $d_C$, and the anonymous referees for their very careful reading and unseful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Meaning that the densities of these measures converge exponentially fast towards the density of $\mathrm{S}\mathrm{R}\mathrm{B}$ in the $C^{r-1}$ topology

2 Here, ‘discretization’ stands for the projection on the nearest element of $\mathbf{Z}$, i.e. the image under the projection $\mathbf{R}\to \mathbf{Z}$ on the nearest integer.

3 By Perron–Frobenius theorem, the measures obtained from MapToCombination tend (when the time goes to infinity) to some measure depending on N. We do not know if these measures tend to $\mathrm{S}\mathrm{R}\mathrm{B}$ when N goes to infinity. It may be possible to prove it using the ideas of [19], by checking that the distance between both Perron–Frobenius and discretized (associated to MapToCombination) transfer operators are close relative to $B{V}_1$ distance. As this is not in the scope of this article, we do not investigate this question.

4 There is a conflict of notations with the dynamics f, we hope that which one is used is clear from the context.

5 In practical, we will see the asymptotic regime's behaviour as soon as most of points of $E_N$ already have browsed a whole cycle.