Accessibility of derived-from-Anosov systems
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- by Andy Hammerlindl and Yi Shi PDF
- Trans. Amer. Math. Soc. 374 (2021), 2949-2966 Request permission
Abstract:
This paper shows any non-Anosov partially hyperbolic diffeomorphism on the 3-torus which is homotopic to Anosov must be accessible.References
- Michael Brin, Dmitri Burago, and Sergey Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, J. Mod. Dyn. 3 (2009), no. 1, 1–11. MR 2481329, DOI 10.3934/jmd.2009.3.1
- Christian Bonatti and Nancy Guelman, Axiom A diffeomorphisms derived from Anosov flows, J. Mod. Dyn. 4 (2010), no. 1, 1–63. MR 2643887, DOI 10.3934/jmd.2010.4.1
- Christian Bonatti, Shaobo Gan, and Dawei Yang, On the hyperbolicity of homoclinic classes, Discrete Contin. Dyn. Syst. 25 (2009), no. 4, 1143–1162. MR 2552132, DOI 10.3934/dcds.2009.25.1143
- Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, and Raúl Ures, Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 75–88. MR 2410948, DOI 10.3934/dcds.2008.22.75
- Dmitri Burago and Sergei Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn. 2 (2008), no. 4, 541–580. MR 2449138, DOI 10.3934/jmd.2008.2.541
- M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 9–19 (Russian). MR 0370660
- David Chillingworth (ed.), Proceedings of the Symposium on Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, Vol. 206, Springer-Verlag, Berlin-New York, 1971. Held at the University of Warwick, Coventry, September 1968-August 1969. Summer School, July 15-25, 1969. MR 0369773
- John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR 0271990
- Matthew Grayson, Charles Pugh, and Michael Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2) 140 (1994), no. 2, 295–329. MR 1298715, DOI 10.2307/2118602
- Shaobo Gan and Yi Shi, Rigidity of center Lyapunov exponents and $su$-integrability, Comment. Math. Helv. 95 (2020), no. 3, 569–592. MR 4152625, DOI 10.4171/CMH/497
- Andy Hammerlindl, Quasi-isometry and plaque expansiveness, Canad. Math. Bull. 54 (2011), no. 4, 676–679. MR 2894517, DOI 10.4153/CMB-2011-024-7
- Andy Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory Dynam. Systems 33 (2013), no. 3, 896–933. MR 3062906, DOI 10.1017/etds.2012.171
- Andy Hammerlindl, Ergodic components of partially hyperbolic systems, Comment. Math. Helv. 92 (2017), no. 1, 131–184. MR 3615038, DOI 10.4171/CMH/409
- A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. Lond. Math. Soc. (2) 89 (2014), no. 3, 853–875. MR 3217653, DOI 10.1112/jlms/jdu013
- Andy Hammerlindl and Rafael Potrie, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, J. Topol. 8 (2015), no. 3, 842–870. MR 3394318, DOI 10.1112/jtopol/jtv009
- Andy Hammerlindl and Rafael Potrie, Partial hyperbolicity and classification: a survey, Ergodic Theory Dynam. Systems 38 (2018), no. 2, 401–443. MR 3774827, DOI 10.1017/etds.2016.50
- Andy Hammerlindl and Raúl Ures, Ergodicity and partial hyperbolicity on the 3-torus, Commun. Contemp. Math. 16 (2014), no. 4, 1350038, 22. MR 3231058, DOI 10.1142/S0219199713500387
- Ricardo Mañé, Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383–396. MR 516217, DOI 10.1016/0040-9383(78)90005-8
- Rafael Potrie, A few remarks on partially hyperbolic diffeomorphisms of $\Bbb {T}^3$ isotopic to Anosov, J. Dynam. Differential Equations 26 (2014), no. 3, 805–815. MR 3274442, DOI 10.1007/s10884-014-9362-5
- Rafael Potrie, Partial hyperbolicity and foliations in $\Bbb {T}^3$, J. Mod. Dyn. 9 (2015), 81–121. MR 3395262, DOI 10.3934/jmd.2015.9.81
- Charles Pugh, Michael Shub, and Amie Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), no. 3, 517–546. MR 1432307, DOI 10.1215/S0012-7094-97-08616-6
- G. Ponce and A. Tahzibi, Central Lyapunov exponent of partially hyperbolic diffeomorphisms of $\Bbb {T}^3$, Proc. Amer. Math. Soc. 142 (2014), no. 9, 3193–3205. MR 3223375, DOI 10.1090/S0002-9939-2014-12063-6
- Gabriel Ponce, Ali Tahzibi, and Régis Varão, Minimal yet measurable foliations, J. Mod. Dyn. 8 (2014), no. 1, 93–107. MR 3296568, DOI 10.3934/jmd.2014.8.93
- G. Ponce, A. Tahzibi, and R. Varão, On the Bernoulli property for certain partially hyperbolic diffeomorphisms, Adv. Math. 329 (2018), 329–360. MR 3783416, DOI 10.1016/j.aim.2018.02.019
- Federico Rodriguez Hertz, Maria Alejandra Rodriguez Hertz, and Raul Ures, A survey of partially hyperbolic dynamics, Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., vol. 51, Amer. Math. Soc., Providence, RI, 2007, pp. 35–87. MR 2388690
- 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
- Federico Rodriguez Hertz, Maria Alejandra Rodriguez Hertz, and Raul Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn. 2 (2008), no. 2, 187–208. MR 2383266, DOI 10.3934/jmd.2008.2.187
- Raúl Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc. 140 (2012), no. 6, 1973–1985. MR 2888185, DOI 10.1090/S0002-9939-2011-11040-2
- Marcelo Viana and Jiagang Yang, Measure-theoretical properties of center foliations, Modern theory of dynamical systems, Contemp. Math., vol. 692, Amer. Math. Soc., Providence, RI, 2017, pp. 291–320. MR 3666078, DOI 10.1090/conm/692
- Amie Wilkinson, Conservative partially hyperbolic dynamics, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1816–1836. MR 2827868
Additional Information
- Andy Hammerlindl
- Affiliation: School of Mathematical Sciences, Monash University, Victoria $3800$ Australia
- Email: andy.hammerlindl@monash.edu
- Yi Shi
- Affiliation: School of Mathematical Sciences, Peking University, Beijing $100871$, People’s Republic of China
- MR Author ID: 1077591
- Email: shiyi@math.pku.edu.cn
- Received by editor(s): March 30, 2020
- Received by editor(s) in revised form: April 15, 2020, August 16, 2020, and August 25, 2020
- Published electronically: December 15, 2020
- Additional Notes: This research was partially funded by the Australian Research Council.
The second author was supported by NSFC 11701015, 11831001. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 2949-2966
- MSC (2020): Primary 37D20, 37D30
- DOI: https://doi.org/10.1090/tran/8292
- MathSciNet review: 4223038