Realizing arbitrary $d$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

TL;DR

This paper demonstrates that any conservative $C^d$ map on the $d$-dimensional unit ball can be approximated by the renormalized iteration of a $C^d$ perturbation of the identity, revealing a universal realization property.

Contribution

It establishes that arbitrary $d$-dimensional dynamics can be realized through renormalization of $C^d$ perturbations of the identity map.

Findings

Any $C^d$ conservative map can be realized by renormalized iteration.

Existence of a conservative diffeomorphism close to identity with prescribed periodic dynamics.

The method applies to arbitrary $d$-dimensional conservative systems.

Abstract

Any $C^d$ conservative map $f$ of the $d$-dimensional unit ball $\mathbb B^d$ can be realized by renormalized iteration of a $C^d$ perturbation of identity: there exists a conservative diffeomorphism of $\mathbb B^d$, arbitrarily close to identity in the $C^d$ topology, that has a periodic disc on which the return dynamics after a $C^d$ change of coordinates is exactly $f$.