  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 $f$.
