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Local dimensions for Poincaré recurrences
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## Local dimensions for Poincaré recurrences

### Valentin Afraimovich, Jean-René Chazottes, and Benoît Saussol

**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.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 64-74
- Publisher Identifier: S 1079-6762(00)00082-2
- 2000
*Mathematics Subject Classification*. Primary 37C45, 37B20
- Received by the editors March 31, 2000
- Posted on September 11, 2000
- Communicated by Svetlana Katok
- Comments (When Available)

**Valentin Afraimovich**

IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico

*E-mail address:* `valentin@cactus.iico.uaslp.mx`

**Jean-René Chazottes**

IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico

*E-mail address:* `jeanrene@cpt.univ-mrs.fr`

**Benoît Saussol**

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

*E-mail address:* `saussol@math.ist.utl.pt`

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