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: February 2021

Revised: July 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

[1]	D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, rudy Moskov. Mat. Obsc. 23 (1970), 3–36.
[2]	P. Berger and D. Turaev, On Herman's positive entropy conjecture, Adv. Math., 349 (2019), 1234-1288. doi: 10.1016/j.aim.2019.04.002.
[3]	S. Ferenczi, Systèmes de rang un gauche, Ann. Inst. H. Poincaré Probab. Statist., 21 (1985), 177-186.
[4]	M. Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. 2, Berlin, 1998, Doc. Math., 1998, Extra Vol. II, 797–808.
[5]	J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.
[6]	S. Newhouse, D. Ruelle and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on $ {\mathbb T}^m$, $m\geq 3$, Comm. Math. Phys., 64 (1978/79), 35-40. doi: 10.1007/BF01940759.
[7]	D. Ruelle and F. Takens, On the nature of turbulence,, Comm. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.
[8]	D. Turaev, Maps close to identity and universal maps in the Newhouse domain, Comm. Math. Phys., 335 (2015), 1235-1277. doi: 10.1007/s00220-015-2338-4.