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

Bassam Fayad; Maria Saprykina

doi:10.3934/dcds.2021129

Article Contents

2022, Volume 42, Issue 2: 597-604. Doi: 10.3934/dcds.2021129

This issue Previous Article Aubry-Mather theory for contact Hamiltonian systems II Next Article Propagating fronts for a viscous Hamer-type system

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

Bassam Fayad^1, and
Maria Saprykina^2, ,

1.
UP7D, 58-56, avenue de France, Boite Courrier 7012, 75205 Paris Cedex 13, France
2.
Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25,100 44 Stockholm, Sweden

^* Corresponding author: Maria Saprykina
^* Corresponding author: Maria Saprykina

Received: January 31, 2021

Revised: June 30, 2021

Early access: September 2021

Published: February 2022

B. Fayad was supported in part by Knut and Alice Wallenberg foundation, grant KAW 2016.0403, and by the ANR-15-CE40-0001. M.Saprykina was supported in part by the Swedish Research Council, VR 2015-04012

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Abstract

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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Mathematics Subject Classification: Primary: 37C15.

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References

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