American Institute of Mathematical Sciences

Advanced
September  2010, 26(3): 949-966. doi: 10.3934/dcds.2010.26.949

On the distribution of periodic orbits

Katrin Gelfert 1, and  Christian Wolf 2,

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil
2. 

Department of Mathematics, Wichita State University, Wichita, Kansas, 67260

Received  January 2009 Revised  September 2009 Published  December 2009

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.
Keywords: equilibrium states, large deviation., topological pressure, non-uniformly hyperbolic dynamics, Pesin theory.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37D50, 37C3.
Citation: Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[1]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045
[2]

Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013
[3]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74
[4]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[5]

Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224
[6]

José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363
[7]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[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 & Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585
[9]

Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane, Azam Ehsani. Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1777-1807. doi: 10.3934/dcds.2018073
[10]

Mark Pollicott. Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1247-1256. doi: 10.3934/dcds.2005.13.1247
[11]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699
[12]

Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106
[13]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
[14]

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
[15]

Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401
[16]

José F. Alves. A survey of recent results on some statistical features of non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 1-20. doi: 10.3934/dcds.2006.15.1
[17]

Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206
[18]

Jose F. Alves; Stefano Luzzatto and Vilton Pinheiro. Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension. Electronic Research Announcements, 2003, 9: 26-31.

[19]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
[20]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences