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

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 $f$.