February  2022, 42(2): 597-604. doi: 10.3934/dcds.2021129

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

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

Received  February 2021 Revised  July 2021 Published  February 2022 Early access  September 2021

Fund Project: 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

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 $.

Citation: Bassam Fayad, Maria Saprykina. Realizing arbitrary $d$-dimensional dynamics by renormalization of $C^d$-perturbations of identity. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 597-604. doi: 10.3934/dcds.2021129
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. NewhouseD. 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.

show all references

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. NewhouseD. 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.

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