September  2003, 9(5): 1185-1192. doi: 10.3934/dcds.2003.9.1185

Periodic probability measures are dense in the set of invariant measures

1. 

Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan

Received  August 2002 Revised  December 2002 Published  June 2003

We show that if a mixing diffeomorphism of a compact manifold preserves an ergodic hyperbolic probability measure, then the measures supported by hyperbolic periodic points are dense in the set of invariant measures. This is a generalization of the result shown by Sigmund.
Citation: Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[1]

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

[2]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307

[3]

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

[4]

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

[5]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[6]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

[7]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[8]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[9]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

[10]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[11]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[12]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[13]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[14]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020181

[15]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115

[16]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201

[17]

Sergei Yu. Pilyugin. Variational shadowing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733

[18]

Fabio Ancona, Andrea Marson. On the Glimm Functional for general hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 44-53. doi: 10.3934/proc.2011.2011.44

[19]

Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147

[20]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]