Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A chain transitive accessible partially hyperbolic diffeomorphism which is non-transitive


Authors: Shaobo Gan and Yi Shi
Journal: Proc. Amer. Math. Soc. 146 (2018), 223-232
MSC (2010): Primary 37C20, 37D30
DOI: https://doi.org/10.1090/proc/13708
Published electronically: July 28, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we construct a partially hyperbolic skew-product diffeomorphism on $ \mathbb{T}^3$, which is accessible and chain transitive, but not transitive.


References [Enhancements On Off] (What's this?)

  • [1] Flavio Abdenur, Christian Bonatti, and Lorenzo J. Díaz, Non-wandering sets with non-empty interiors, Nonlinearity 17 (2004), no. 1, 175-191. MR 2023438, https://doi.org/10.1088/0951-7715/17/1/011
  • [2] 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, https://doi.org/10.1007/978-3-642-28821-0_1
  • [3] Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33-104 (French, with English and French summaries). MR 2090361, https://doi.org/10.1007/s00222-004-0368-1
  • [4] Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
  • [5] 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
  • [6] 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, https://doi.org/10.3934/dcds.2008.22.75
  • [7] Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451-489. MR 2630044, https://doi.org/10.4007/annals.2010.171.451
  • [8] Dmitry Dolgopyat and Amie Wilkinson, Stable accessibility is $ C^1$ dense, Astérisque 287 (2003), xvii, 33-60. Geometric methods in dynamics. II. MR 2039999
  • [9] Andy Hammerlindl, Ergodic components of partially hyperbolic systems, Comment. Math. Helv. 92 (2017), no. 1, 131-184. MR 3615038, https://doi.org/10.4171/CMH/409
  • [10] 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, https://doi.org/10.1112/jlms/jdu013
  • [11] 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, https://doi.org/10.1007/s00222-007-0100-z
  • [12] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • [13] Rafael Potrie, Partial hyperbolicity and foliations in $ \mathbb{T}^3$, J. Mod. Dyn. 9 (2015), 81-121. MR 3395262, https://doi.org/10.3934/jmd.2015.9.81
  • [14] R. Ures and C. H. Vasquez, On the robustness of intermingled basins, arXiv:1503.07155v2.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37C20, 37D30

Retrieve articles in all journals with MSC (2010): 37C20, 37D30


Additional Information

Shaobo Gan
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: gansb@pku.edu.cn

Yi Shi
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: shiyi@math.pku.edu.cn

DOI: https://doi.org/10.1090/proc/13708
Received by editor(s): December 30, 2016
Received by editor(s) in revised form: February 19, 2017
Published electronically: July 28, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society