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Recurrence rates for observations of flows

Published online by Cambridge University Press:  06 September 2011

JÉRÔME ROUSSEAU
JÉRÔME ROUSSEAU*
Affiliation:
Instituto de Matemática, Universidade Federal da Bahia, Avenida Ademar de Barros s/n, Ondina, 40170-110 Salvador, BA, Brazil (email: jerome.rousseau@univ-brest.fr)
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Abstract

We study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observations an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observations. We apply these results to the geodesic flow and we compute the recurrence rates for a particular observation of the geodesic flow, i.e. the projection on the manifold.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1727 - 1751
Copyright
Copyright © Cambridge University Press 2011

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