Non-zero Lyapunov exponents for some conservative partially hyperbolic systems
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Abstract:
Let $\text {PH}^{1}_\mu (M,3)$ be the set of $C^{1}$ conservative partially hyperbolic diffeomorphisms with center dimensions three or less. We prove that there is a dense subset $\mathcal {H}\subset \text {PH}^{1}_\mu (M,3)$ such that each $f\in \mathcal {H}$ has non-zero Lyapunov exponents on a set of positive volume.References
- Artur Avila, On the regularization of conservative maps, Acta Math. 205 (2010), no. 1, 5–18. MR 2736152, DOI 10.1007/s11511-010-0050-y
- Artur Avila and Jairo Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Amer. Math. Soc. 364 (2012), no. 6, 2883–2907. MR 2888232, DOI 10.1090/S0002-9947-2012-05423-7
- Alexandre T. Baraviera and Christian Bonatti, Removing zero Lyapunov exponents, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1655–1670. MR 2032482, DOI 10.1017/S0143385702001773
- K. Burns, D. Dolgopyat, and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys. 108 (2002), no. 5-6, 927–942. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933439, DOI 10.1023/A:1019779128351
- Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451–489. MR 2630044, DOI 10.4007/annals.2010.171.451
- Yongluo Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity 16 (2003), no. 4, 1473–1479. MR 1986306, DOI 10.1088/0951-7715/16/4/316
- Dmitry Dolgopyat and Amie Wilkinson, Stable accessibility is $C^1$ dense, Astérisque 287 (2003), xvii, 33–60. Geometric methods in dynamics. II. MR 2039999
- F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J. 160 (2011), no. 3, 599–629. MR 2852370, DOI 10.1215/00127094-1444314
- F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), no. 2, 353–381. MR 2390288, DOI 10.1007/s00222-007-0100-z
- Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
- Yakov Pesin and Vaughn Climenhaga, Open problems in the theory of non-uniform hyperbolicity, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 589–607. MR 2600681, DOI 10.3934/dcds.2010.27.589
- Charles Pugh and Michael Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13 (1997), no. 1, 125–179. MR 1449765, DOI 10.1006/jcom.1997.0437
- Yunhua Zhou, The local $C^1$-density of stable ergodicity, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 2621–2629. MR 3007719, DOI 10.3934/dcds.2013.33.2621
Additional Information
- Yunhua Zhou
- Affiliation: College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China
- Email: zhouyh@cqu.edu.cn
- Received by editor(s): December 29, 2011
- Received by editor(s) in revised form: March 15, 2014
- Published electronically: February 17, 2015
- Additional Notes: The author was supported by NSFC (11471056), Natural Science Foundation Project of CQCSTC (cstcjjA00003) and Fundamental Research Funds for the Central Universities (CQDXWL2012008).
- Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3147-3153
- MSC (2010): Primary 37D25; Secondary 37D30
- DOI: https://doi.org/10.1090/S0002-9939-2015-12498-7
- MathSciNet review: 3336638