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 , 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 in the topology
2 Here, ‘discretization’ stands for the projection on the nearest element of , i.e. the image under the projection 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 when N goes to infinity. It may be possible to prove it using the ideas of [Citation19], by checking that the distance between both Perron–Frobenius and discretized (associated to MapToCombination) transfer operators are close relative to 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 already have browsed a whole cycle.