The $C^1$ density of nonuniform hyperbolicity in $C^{r}$ conservative diffeomorphisms
HTML articles powered by AMS MathViewer
- by Chao Liang and Yun Yang PDF
- Proc. Amer. Math. Soc. 145 (2017), 1539-1552 Request permission
Abstract:
Let $\mathrm {Diff}^{r}_m(M)$ be the set of $C^{r}$ volume-preserving diffeomorphisms on a compact Riemannian manifold $M$ ($\dim M\geq 2$). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\mathrm {Diff}^{r}_m(M), r\geq 1$. We also prove a weaker result for symplectic diffeomorphisms $\mathrm {Sym}^{r}_{\omega }(M), r\geq 1$ saying that either the symplectic diffeomorphism $f$ is partially hyperbolic or $C^1$ arbitrarily close to $f$ in $\mathrm {Sym}^{r}_{\omega }(M)$, there is a diffeomorphism $g\in \mathrm {Sym}^{r}_{\omega }(M)$ without zero Lyapunov exponents on a set with 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, Density of positive Lyapunov exponents for $\textrm {SL}(2,\Bbb {R})$-cocycles, J. Amer. Math. Soc. 24 (2011), no. 4, 999–1014. MR 2813336, DOI 10.1090/S0894-0347-2011-00702-9
- 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
- A. Avila, S. Crovisier, and A. Wilkinson, Diffeomorphisms with positive metric entropy, arXiv:1408.4252.
- Flavio Abdenur and Sylvain Crovisier, Transitivity and topological mixing for $C^1$ diffeomorphisms, Essays in mathematics and its applications, Springer, Heidelberg, 2012, pp. 1–16. MR 2975581, DOI 10.1007/978-3-642-28821-0_{1}
- Alexander Arbieto and Carlos Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1399–1417. With an appendix by David Diica and Yakov Simpson-Weller. MR 2358971, DOI 10.1017/S014338570700017X
- Jairo Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1667–1696. MR 1944399, DOI 10.1017/S0143385702001165
- 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
- M. Brin, Bernoulli diffeomorphisms with $n-1$ nonzero exponents, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 1–7. MR 627783, DOI 10.1017/s0143385700001127
- M. Brin, D. Burago, and S. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 307–312. MR 2090777
- Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33–104 (French, with English and French summaries). MR 2090361, DOI 10.1007/s00222-004-0368-1
- Keith Burns, Charles Pugh, Michael Shub, and Amie Wilkinson, Recent results about stable ergodicity, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 327–366. MR 1858538, DOI 10.1090/pspum/069/1858538
- C. Bonatti, L. J. Díaz, and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), no. 2, 355–418 (English, with English and French summaries). MR 2018925, DOI 10.4007/annals.2003.158.355
- Chong Qing Cheng and Yi Sui Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom. 47 (1989/90), no. 3, 275–292. MR 1056793, DOI 10.1007/BF00053456
- 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
- Bernard Dacorogna and Jürgen Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 1, 1–26 (English, with French summary). MR 1046081, DOI 10.1016/S0294-1449(16)30307-9
- Dmitry Dolgopyat and Yakov Pesin, Every compact manifold carries a completely hyperbolic diffeomorphism, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 409–435. MR 1898798, DOI 10.1017/S0143385702000202
- Michael Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 797–808. MR 1648127
- Federico Rodriguez Hertz and Zhiren Wang, Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math. 198 (2014), no. 1, 165–209. MR 3260859, DOI 10.1007/s00222-014-0499-y
- Huyi Hu, Yakov Pesin, and Anna Talitskaya, Every compact manifold carries a hyperbolic Bernoulli flow, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 347–358. MR 2093309
- Huyi Hu, Yakov Pesin, and Anna Talitskaya, A volume preserving diffeomorphism with essential coexistence of zero and nonzero Lyapunov exponents, Comm. Math. Phys. 319 (2013), no. 2, 331–378. MR 3037580, DOI 10.1007/s00220-012-1602-0
- A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2) 110 (1979), no. 3, 529–547. MR 554383, DOI 10.2307/1971237
- Boris Kalinin and Ralf Spatzier, On the classification of Cartan actions, Geom. Funct. Anal. 17 (2007), no. 2, 468–490. MR 2322492, DOI 10.1007/s00039-007-0602-2
- Ricardo Mañé, Oseledec’s theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1269–1276. MR 804776
- Sheldon E. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Amer. J. Math. 99 (1977), no. 5, 1061–1087. MR 455049, DOI 10.2307/2374000
- Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regul. Chaotic Dyn. 12 (2007), no. 5, 476–489. MR 2350334, DOI 10.1134/S1560354707050024
- Radu Saghin and Zhihong Xia, Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5119–5138. MR 2231887, DOI 10.1090/S0002-9947-06-04171-7
- Stephen Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR 112149, DOI 10.1090/S0002-9939-1959-0112149-8
- Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057, DOI 10.1007/s002229900035
- Zhihong Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 621–631. MR 1182665, DOI 10.1017/S0143385700006969
- Eduard Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 828–854. MR 0467846
Additional Information
- Chao Liang
- Affiliation: Applied Mathematical Department, Central University of Finance and Economics, Beijing, 100081, People’s Republic of China
- Email: chaol@cufe.edu.cn
- Yun Yang
- Affiliation: Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016
- Email: yyang@gc.cuny.edu
- Received by editor(s): November 21, 2015
- Published electronically: December 15, 2016
- Additional Notes: The first author was supported by NNSFC(# 11471344) and 2016 NSFC/ICTP Grants(# 11681240278) and Beijing Higher Education Young Elite Teacher Project(YETP0986)
- Communicated by: Yingfei Yi
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1539-1552
- MSC (2010): Primary 37D30, 37D25, 37C25
- DOI: https://doi.org/10.1090/proc/13527
- MathSciNet review: 3601546