Local dimensions for Poincaré recurrences
Authors:
Valentin Afraimovich, Jean-René Chazottes and Benoît Saussol
Journal:
Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 64-74
MSC (2000):
Primary 37C45, 37B20
DOI:
https://doi.org/10.1090/S1079-6762-00-00082-2
Published electronically:
September 11, 2000
MathSciNet review:
1777857
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Abstract: Pointwise dimensions and spectra for measures associated with Poincaré recurrences are calculated for arbitrary weakly specified subshifts with positive entropy and for the corresponding special flows. It is proved 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. Precise formulas for dimensions of measures associated with Poincaré recurrences are derived, which are comparable to Young’s formula for the Hausdorff dimension of measures and Abramov’s formula for the entropy of special flows.
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walters R. Bowen and P. Walters, Expansive one-parameter flows, J. Diff. Equas. 12 (1972), 180–193.
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young L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dyn. Syst 2 (1982), 109–124.
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Additional Information
Valentin Afraimovich
Affiliation:
IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico
Email:
valentin@cactus.iico.uaslp.mx
Jean-René Chazottes
Affiliation:
IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico
Email:
jeanrene@cpt.univ-mrs.fr
Benoît Saussol
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email:
saussol@math.ist.utl.pt
Received by editor(s):
March 31, 2000
Published electronically:
September 11, 2000
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2000
American Mathematical Society
