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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.

37C45 Dimension theory of smooth dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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