Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms

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1. [1] F. Abdenur & M. Viana, Flavors of partial hyperbolicity, in preparation. 
2. [2] D. V. Anosov & A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow Math. Soc. 23 (1970), 1–35. Zbl0255.58007MR370662
3. [3] A. Arbieto & C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems27 (2007), 1399–1417. Zbl1142.37025MR2358971
4. [4] M.-C. Arnaud, Le “closing lemma” en topologie $C^1$, Mém. Soc. Math. Fr. (N.S.) 74 (1998). Zbl0920.58039MR1662930
5. [5] M.-C. Arnaud, C. Bonatti & S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dynam. Systems25 (2005), 1401–1436. Zbl1084.37017MR2173426
6. [6] L. Arnold, Random dynamical systems, 2nd éd., Springer Mono. Math., Springer, 2002. MR1374107
7. [7] A. Avila, On the regularization of conservative maps, to appear in Acta Math. Zbl1211.37029MR2736152
8. [8] A. Avila, J. Santamaria & M. Viana, Cocycles over partially hyperbolic maps, preprint. Zbl06266975
9. [9] J. Bochi, $C^1$-generic symplectic diffeomorphisms: partial hyperbolicity and zero center Lyapunov exponents, to appear in J. Inst. Math. Jussieu. Zbl1181.37031MR2576798
10. [10] J. Bochi & M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math.161 (2005), 1423–1485. Zbl1101.37039MR2180404
11. [11] C. Bonatti & S. Crovisier, Récurrence et généricité, Invent. Math.158 (2004), 33–104. Zbl1071.37015MR2090361
12. [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. Zbl1049.37011MR2018925
13. [13] C. Bonatti, L. J. Díaz & M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Math. Sciences 102, Springer, 2005. Zbl1060.37020MR2105774
14. [14] C. Bonatti, C. Matheus, M. Viana & A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv.79 (2004), 753–757. Zbl1052.37023MR2099120
15. [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. Zbl0357.58011MR370660
16. [16] K. Burns, D. Dolgopyat & Y. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys. 108 (2002), 927–942. Zbl1124.37308MR1933439
17. [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. Zbl1145.37021MR2366230
18. [18] K. Burns & A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math.171 (2010), 429–467. Zbl1196.37057MR2630044
19. [19] D. Dolgopyat & A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque287 (2003), 33–60. Zbl1213.37053MR2039999
20. [20] N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems27 (2007), 1839–1849. Zbl1127.37031MR2371598
21. [21] M. W. Hirsch, C. Pugh & M. Shub, Invariant manifolds, Lecture Notes in Math. 583, Springer, 1977. Zbl0355.58009MR501173
22. [22] V. Horita & A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. Inst. H. Poincaré Anal. Non Linéaire23 (2006), 641–661. Zbl1130.37356MR2259610
23. [23] R. Mañé, An ergodic closing lemma, Ann. of Math.116 (1982), 503–540. Zbl0511.58029MR678479
24. [24] J. C. Oxtoby & S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math.42 (1941), 874–920. Zbl0063.06074MR5803
25. [25] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Math. 2, Cambridge Univ. Press, 1989. Zbl0676.28008MR1073173
26. [26] C. Pugh & M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), 1–52. Zbl0964.37017MR1750453
27. [27] C. Pugh, M. Shub & A. Wilkinson, Hölder foliations, Duke Math. J.86 (1997), 517–546. Zbl0877.58045MR1432307
28. [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). Zbl1139.37015MR2353803
29. [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. Zbl1136.37020MR2390288
30. [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). Zbl1210.37014MR2231887
31. [31] M. Shub & A. Wilkinson, Stably ergodic approximation: two examples, Ergodic Theory Dynam. Systems20 (2000), 875–893. Zbl0970.37022MR1764933
32. [32] A. Tahzibi, Robust transitivity and almost robust ergodicity, Ergodic Theory Dynam. Systems24 (2004), 1261–1269. Zbl1145.37304MR2085911
33. [33] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, preprint. Zbl06322495
34. [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. Zbl0363.58004MR467846