May  2006, 15(2): 433-446. doi: 10.3934/dcds.2006.15.433

Hyperbolic sets with nonempty interior

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

Received  October 2004 Revised  November 2005 Published  March 2006

In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.
Citation: Todd Fisher. Hyperbolic sets with nonempty interior. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 433-446. doi: 10.3934/dcds.2006.15.433
[1]

Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130

[2]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085

[3]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002

[4]

Răzvan M. Tudoran. Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 3013-3030. doi: 10.3934/dcds.2020159

[5]

Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295

[6]

Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571

[7]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205

[8]

Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585

[9]

Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187

[10]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253

[11]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004

[12]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012

[13]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377

[14]

Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981

[15]

Aubin Arroyo, Enrique R. Pujals. Dynamical properties of singular-hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 67-87. doi: 10.3934/dcds.2007.19.67

[16]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449

[17]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[18]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[19]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

[20]

John Erik Fornæss. Sustainable dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]