American Institute of Mathematical Sciences

Advanced
March  2004, 11(2&3): 615-619. doi: 10.3934/dcds.2004.11.615

A note on periodic orbits for singular-hyperbolic flows

Carlos Arnoldo Morales 1,

1. 

Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received  May 2003 Revised  February 2004 Published  June 2004

A singular-hyperbolic set for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction [7]. Several properties of hyperbolic systems have been conjectured for the singular-hyperbolic sets [8, p. 335]. Related to these conjectures we shall prove the existence of transitive, isolated, singular-hyperbolic set without periodic orbits on any $3$-manifold. In particular, the periodic orbits are not necessarily dense in the limit set of a isolated singular-hyperbolic set.
Keywords: Flow, partially hyperbolic set., periodic orbit.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37C2.
Citation: Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[1]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[2]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315
[3]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[4]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[5]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
[6]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037
[7]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419
[8]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271
[9]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68
[10]

R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403
[11]

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

Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114
[13]

Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008
[14]

Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195
[15]

Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27
[16]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193
[17]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549
[18]

Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245
[19]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[20]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences