Nonuniform center bunching and the genericity of ergodicity among C 1 partially hyperbolic symplectomorphisms
[Resserrement central non-uniforme et la généricité de l’ergodicité parmi les C 1 -symplectomorphismes partiellement hyperboliques]
Avila, Artur ; Bochi, Jairo ; Wilkinson, Amie
Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 931-979 / Harvested from Numdam
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Nous introduisons une notion non-uniforme de resserrement central pour les difféomorphismes partiellement hyperboliques qui nous permet de généraliser quelques résultats de Burns-Wilkinson et Avila-Santamaria-Viana. Cette nouvelle technique est utilisée, en combinaison avec d’autres constructions, pour démontrer la généricité de l’ergodicité parmi les difféomorphismes symplectiques partiellement hyperboliques de classe C 1 . De plus, nous obtenons de nouveaux exemples de dynamiques stablement ergodiques.

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns-Wilkinson and Avila-Santamaria-Viana. Combining this new technique with other constructions we prove that C 1 -generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

Publié le : 2009-01-01
DOI : https://doi.org/10.24033/asens.2113
Classification:  37D30,  37D25,  37J10
Mots clés: hyperbolicité partielle, resserrement central, ergodicité, difféomorphismes symplectiques 
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     author = {Avila, Artur and Bochi, Jairo and Wilkinson, Amie},
     title = {Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {42},
     year = {2009},
     pages = {931-979},
     doi = {10.24033/asens.2113},
     mrnumber = {2567746},
     zbl = {1191.37017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2009_4_42_6_931_0}
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Avila, Artur; Bochi, Jairo; Wilkinson, Amie. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms. Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009) pp. 931-979. doi : 10.24033/asens.2113. http://gdmltest.u-ga.fr/item/ASENS_2009_4_42_6_931_0/

[1] F. Abdenur & M. Viana, Flavors of partial hyperbolicity, in preparation.

[2] D. V. Anosov & A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow Math. Soc. 23 (1970), 1-35. | MR 370662 | Zbl 0255.58007

[3] A. Arbieto & C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems 27 (2007), 1399-1417. | MR 2358971 | Zbl 1142.37025

[4] M.-C. Arnaud, Le “closing lemma” en topologie C 1 , Mém. Soc. Math. Fr. (N.S.) 74 (1998). | Numdam | MR 1662930 | Zbl 0920.58039

[5] M.-C. Arnaud, C. Bonatti & S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dynam. Systems 25 (2005), 1401-1436. | MR 2173426 | Zbl 1084.37017

[6] L. Arnold, Random dynamical systems, 2nd éd., Springer Mono. Math., Springer, 2002. | MR 1374107

[7] A. Avila, On the regularization of conservative maps, to appear in Acta Math. | MR 2736152 | Zbl 1211.37029

[8] A. Avila, J. Santamaria & M. Viana, Cocycles over partially hyperbolic maps, preprint. | Zbl pre06266975

[9] J. Bochi, C 1 -generic symplectic diffeomorphisms: partial hyperbolicity and zero center Lyapunov exponents, to appear in J. Inst. Math. Jussieu. | MR 2576798 | Zbl 1181.37031

[10] J. Bochi & M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. 161 (2005), 1423-1485. | MR 2180404 | Zbl 1101.37039

[11] C. Bonatti & S. Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), 33-104. | MR 2090361 | Zbl 1071.37015

[12] C. Bonatti, L. J. Díaz & E. R. Pujals, A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003), 355-418. | MR 2018925 | Zbl 1049.37011

[13] C. Bonatti, L. J. Díaz & M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Math. Sciences 102, Springer, 2005. | MR 2105774 | Zbl 1060.37020

[14] C. Bonatti, C. Matheus, M. Viana & A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv. 79 (2004), 753-757. | MR 2099120 | Zbl 1052.37023

[15] M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Functional Anal. Appl. 9 (1975), 8-16. | MR 370660 | Zbl 0357.58011

[16] K. Burns, D. Dolgopyat & Y. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys. 108 (2002), 927-942. | MR 1933439 | Zbl 1124.37308

[17] K. Burns, D. Dolgopyat, Y. Pesin & M. Pollicott, Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn. 2 (2008), 63-81. | MR 2366230 | Zbl 1145.37021

[18] K. Burns & A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. 171 (2010), 429-467. | MR 2630044 | Zbl 1196.37057

[19] D. Dolgopyat & A. Wilkinson, Stable accessibility is C 1 dense, Astérisque 287 (2003), 33-60. | MR 2039999 | Zbl 1213.37053

[20] N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems 27 (2007), 1839-1849. | MR 2371598 | Zbl 1127.37031

[21] M. W. Hirsch, C. Pugh & M. Shub, Invariant manifolds, Lecture Notes in Math. 583, Springer, 1977. | MR 501173 | Zbl 0355.58009

[22] V. Horita & A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 641-661. | Numdam | MR 2259610 | Zbl 1130.37356

[23] R. Mañé, An ergodic closing lemma, Ann. of Math. 116 (1982), 503-540. | MR 678479 | Zbl 0511.58029

[24] J. C. Oxtoby & S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. 42 (1941), 874-920. | MR 5803 | Zbl 0063.06074

[25] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Math. 2, Cambridge Univ. Press, 1989. | MR 1073173 | Zbl 0676.28008

[26] C. Pugh & M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), 1-52. | MR 1750453 | Zbl 0964.37017

[27] C. Pugh, M. Shub & A. Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), 517-546. | MR 1432307 | Zbl 0877.58045

[28] F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi & R. Ures, A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms, Electron. Res. Announc. Math. Sci. 14 (2007), 74-81 (electronic). | MR 2353803 | Zbl 1139.37015

[29] F. Rodriguez Hertz, M. A. Rodriguez Hertz & R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), 353-381. | MR 2390288 | Zbl 1136.37020

[30] R. Saghin & Z. Xia, Partial hyperbolicity or dense elliptic periodic points for C 1 -generic symplectic diffeomorphisms, Trans. Amer. Math. Soc. 358 (2006), 5119-5138 (electronic). | MR 2231887 | Zbl 1210.37014

[31] M. Shub & A. Wilkinson, Stably ergodic approximation: two examples, Ergodic Theory Dynam. Systems 20 (2000), 875-893. | MR 1764933 | Zbl 0970.37022

[32] A. Tahzibi, Robust transitivity and almost robust ergodicity, Ergodic Theory Dynam. Systems 24 (2004), 1261-1269. | MR 2085911 | Zbl 1145.37304

[33] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint.

[34] E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, in Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math. 597, Springer, 1977, 828-854. | MR 467846 | Zbl 0363.58004