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doi: 10.3934/dcds.2021129
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## 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 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 & Continuous Dynamical Systems, 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.  Google Scholar [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.  Google Scholar [3] S. Ferenczi, Systèmes de rang un gauche, Ann. Inst. H. Poincaré Probab. Statist., 21 (1985), 177-186.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] D. Ruelle and F. Takens, On the nature of turbulence,, Comm. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] S. Ferenczi, Systèmes de rang un gauche, Ann. Inst. H. Poincaré Probab. Statist., 21 (1985), 177-186.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] D. Ruelle and F. Takens, On the nature of turbulence,, Comm. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar [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.  Google Scholar
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