Any $ C^d $ conservative map $ f $ of the $ d $-dimensional unit ball $ {\mathbb B}^d $, $ d\geq 2 $, 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 $.
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