Physical measures of discretizations of generic diffeomorphisms

TL;DR

This paper investigates how numerical discretizations of generic conservative diffeomorphisms reflect ergodic behavior, revealing that computed orbit segments tend to represent all invariant measures rather than just physical ones.

Contribution

It demonstrates that discretized orbit segments from generic points accumulate on all invariant measures, challenging the expectation that they reveal physical measures.

Findings

Discretized orbit segments accumulate on the entire set of invariant measures.

Numerical experiments cannot distinguish physical measures from other invariant measures.

Results hold for generic conservative $C^1$-diffeomorphisms.

Abstract

What is the ergodic behaviour of numerically computed segments of orbits of a diffeomorphism? In this paper, we try to answer this question for a generic conservative $C^1$-diffeomorphism, and segments of orbits of Baire-generic points. The numerical truncation will be modelled by a spatial discretization. Our main result states that the uniform measures on the computed segments of orbits, starting from a generic point, accumulates on the whole set of measures that are invariant under the diffeomorphism. In particular, unlike what could be expected naively, such numerical experiments do not see the physical measures (or more precisely, cannot distinguish physical measures from the other invariant measures).