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Local dimensions for Poincaré recurrences. (English) Zbl 0955.37013
Summary: 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.

MSC:
37C45 Dimension theory of smooth dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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[1] V. Afraimovich, Pesin’s dimension for Poincaré recurrences, Chaos 7 (1997), no. 1, 12 – 20. · Zbl 0933.37019
[2] V. Afraimovich, J.-R. Chazottes, and B. Saussol, Pointwise dimensions for Poincaré recurrences associated with maps and special flows, in preparation. · Zbl 1029.37007
[3] V. Afraimovich, J. Schmeling, E. Ugalde, and J. Urías, Spectra of dimensions for Poincaré recurrences, to appear in Discrete and Continuous Dynamical Systems (2000). · Zbl 1011.37009
[4] Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180 – 193. · Zbl 0242.54041
[5] J.-R. Chazottes and B. Saussol, Sur les dimensions locales et les dimensions des mesures, preprint (2000).
[6] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. · Zbl 0878.58020
[7] Donald Samuel Ornstein and Benjamin Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory 39 (1993), no. 1, 78 – 83. · Zbl 0764.94003
[8] Vincent Penné, Benoît Saussol, and Sandro Vaienti, Dimensions for recurrence times: topological and dynamical properties, Discrete Contin. Dynam. Systems 5 (1999), no. 4, 783 – 798. · Zbl 0954.37012
[9] Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. · Zbl 0895.58033
[10] B. Saussol, S. Troubetzkoy, and S. Vaienti, In preparation.
[11] Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 109 – 124. · Zbl 0523.58024
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