Degree of recurrence of generic diffeomorphisms
arXiv:1510.00723 · doi:10.19086/da.7545
Abstract
We study spatial discretizations of dynamical systems: is it possible to recover some dynamical features of a system from numerical simulations? Here, we tackle this issue for the simplest algorithm possible: we compute long segments of orbits with a fixed number of digits. We show that for every , the dynamics of the discretizations of a generic conservative diffeomorphism of the torus is very different from that observed in the regularity. The proof of our results involves in particular a local-global formula for discretizations, as well as a study of the corresponding linear case, which uses ideas from the theory of quasicrystals.
43 pages. A part of this article is repeated in the appendices of arXiv:1510.00720. Part of the proof of Theorem 20 of the previous version was wrong, this new version presents a corrected statement. This article is included in the memoir ArXiv:1603.06480