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

Any \begin{document}$ C^d $\end{document} conservative map \begin{document}$ f $\end{document} of the \begin{document}$ d $\end{document} -dimensional unit ball \begin{document}$ {\mathbb B}^d $\end{document} , \begin{document}$ d\geq 2 $\end{document} , can be realized by renormalized iteration of a \begin{document}$ C^d $\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$ {\mathbb B}^d $\end{document} , arbitrarily close to identity in the \begin{document}$ C^d $\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$ C^d $\end{document} change of coordinates is exactly \begin{document}$ f $\end{document} .