American Institute of Mathematical Sciences

Advanced
January  2005, 12(1): 175-183. doi: 10.3934/dcds.2005.12.175

Local rates of Poincaré recurrence for rotations and weak mixing

Jean René Chazottes 1, and  F. Durand 2,

1. 

CNRS-CPHT, École Polytechnique, 91128 Palaiseau Cedex, France
2. 

Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex 1, France

Received  March 2003 Revised  June 2004 Published  December 2004

We study the lower and upper local rates of Poincaré recurrence of rotations on the circle by means of symbolic dynamics. As a consequence, we show that if the lower rate of Poincaré recurrence of an ergodic dynamical system $(X,\mathcal F, \mu, T)$ is greater or equal to 1 $\mu$-almost everywhere, then it is weakly mixing.
Keywords: rotation, Sturmian subshift, linearly recurrent subshift, weak mixing., Poincaré recurrence.
Mathematics Subject Classification: Primary: 37B20; Secondary: 37B1.
Citation: Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[1]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[2]

Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083
[3]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[4]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[5]

Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
[6]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[7]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[8]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[9]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483
[10]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[11]

Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[12]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287
[13]

Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005
[14]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[15]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[16]

Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[17]

Serge Troubetzkoy. Recurrence in generic staircases. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047
[18]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[19]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[20]

Milton Ko. Rényi entropy and recurrence. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences