-
Previous Article
Propagating fronts for a viscous Hamer-type system
- DCDS Home
- This Issue
-
Next Article
Aubry-Mather theory for contact Hamiltonian systems II
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 |
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 $.
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. |
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. 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. |
[1] |
Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018 |
[2] |
I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 |
[3] |
Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241 |
[4] |
Xuzhou Chen, Xinghua Shi, Yimin Wei. The stationary iterations revisited. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 261-270. doi: 10.3934/naco.2013.3.261 |
[5] |
Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214 |
[6] |
João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837 |
[7] |
Yunping Jiang. On a question of Katok in one-dimensional case. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1209-1213. doi: 10.3934/dcds.2009.24.1209 |
[8] |
Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195 |
[9] |
Mostapha Benhenda. Nonstandard smooth realization of translations on the torus. Journal of Modern Dynamics, 2013, 7 (3) : 329-367. doi: 10.3934/jmd.2013.7.329 |
[10] |
Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 |
[11] |
L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183 |
[12] |
Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 |
[13] |
Jianhong Wu, Weiguang Yao, Huaiping Zhu. Immune system memory realization in a population model. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 241-259. doi: 10.3934/dcdsb.2007.8.241 |
[14] |
Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531 |
[15] |
Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 |
[16] |
João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641 |
[17] |
Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061 |
[18] |
Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121 |
[19] |
Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 |
[20] |
Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]