American Institute of Mathematical Sciences

Advanced
March  2003, 9(2): 263-280. doi: 10.3934/dcds.2003.9.263

Pointwise dimensions for Poincaré recurrences associated with maps and special flows

V. Afraimovich 1, , Jean-René Chazottes 2, and  Benoît Saussol 3,

1. 

IICO-UASLP, A. Obregón 64, 78000 San Luis Postosí, SLP, Mexico
2. 

Centre de Physique Théorique, CNRS-Ecole polytechnique, UMR 7644, F-91128 Palaiseau Cedex, France
3. 

CNRS-LAMFA, Université de Picardie Jules Verne, 80039 Amiens, France

Received  November 2001 Revised  May 2002 Published  December 2002

We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
Keywords: special flows., pointwise dimensions, spectra for measures, Poincaré recurrences.
Mathematics Subject Classification: 37C45, 37B20, 37A1.
Citation: V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[4]

João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641
[5]

Aleksander Ćwiszewski, Wojciech Kryszewski. On a generalized Poincaré-Hopf formula in infinite dimensions. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953
[6]

Oliver Butterley, Carlangelo Liverani. Smooth Anosov flows: Correlation spectra and stability. Journal of Modern Dynamics, 2007, 1 (2) : 301-322. doi: 10.3934/jmd.2007.1.301
[7]

Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013
[8]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
[9]

Luis Barreira, Yakov Pesin and Jorg Schmeling. On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture. Electronic Research Announcements, 1996, 2: 69-72.

[10]

Przemysław Berk, Krzysztof Frączek. On special flows over IETs that are not isomorphic to their inverses. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 829-855. doi: 10.3934/dcds.2015.35.829
[11]

Boris Khesin, Fedor Soloviev. The pentagram map in higher dimensions and KdV flows. Electronic Research Announcements, 2012, 19: 86-96. doi: 10.3934/era.2012.19.86
[12]

Wenyu Pan. Joining measures for horocycle flows on abelian covers. Journal of Modern Dynamics, 2018, 12: 17-54. doi: 10.3934/jmd.2018003
[13]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501
[14]

Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313
[15]

Moisey Guysinsky, Serge Yaskolko. Coincidence of various dimensions associated with metrics and measures on metric spaces. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 591-603. doi: 10.3934/dcds.1997.3.591
[16]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[17]

Niklas Behringer. Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions. Mathematical Control and Related Fields, 2021, 11 (2) : 313-328. doi: 10.3934/mcrf.2020038
[18]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101
[19]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[20]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy