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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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References

[1]Abadi, M. and Galves, A.. Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov Process. Related Fields 7 (2001), 97–112.Google Scholar
[2]Ambrose, W. and Kakutani, S.. Structure and continuity of measurable flows. Duke Math. J. 9 (1942), 25–42.Google Scholar
[3]Arnold, V.. Équations différentielles ordinaires, Éditions Mir, Moscow, 1974. Champs de vecteurs, groupes à un paramètre, difféomorphismes, flots, systèmes linéaires, stabilités des positions d’équilibre, théorie des oscillations, équations différentielles sur les variétés, traduit du russe par Djilali Embarek.Google Scholar
[4]Barreira, L., Pesin, Y. and Schmeling, J.. Dimension and product structure of hyperbolic measures. Ann. of Math. (2) 149 (1999), 755–783.Google Scholar
[5]Barreira, L. and Saussol, B.. Multifractal analysis of hyperbolic flows. Comm. Math. Phys. 214 (2000), 339–371.Google Scholar
[6]Barreira, L. and Saussol, B.. Hausdorff dimension of measures via Poincaré recurrence. Comm. Math. Phys. 219 (2001), 443–463.Google Scholar
[7]Barreira, L. and Wolf, C.. Pointwise dimension and ergodic decompositions. Ergod. Th. & Dynam. Sys. 26 (2006), 653–671.Google Scholar
[8]Barreira, L. and Wolf, C.. Dimension and ergodic decompositions for hyperbolic flows. Discrete Contin. Dyn. Syst. 17 (2007), 201–212.CrossRefGoogle Scholar
[9]Boshernitzan, M. D.. Quantitative recurrence results. Invent. Math. 113 (1993), 617–631.Google Scholar
[10]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.Google Scholar
[11]Bowen, R. and Walters, P.. Expansive one-parameter flows. J. Differential Equations 12 (1972), 180–193.Google Scholar
[12]Bruin, H., Saussol, B., Troubetzkoy, S. and Vaienti, S.. Return time statistics via inducing. Ergod. Th. & Dynam. Sys. 23 (2003), 991–1013.Google Scholar
[13]Chazottes, J.-R. and Leplaideur, R.. Fluctuations of the Nth return time for Axiom A diffeomorphisms. Discrete Contin. Dyn. Syst. 13 (2005), 399–411.Google Scholar
[14]Galatolo, S. and Pacifico, M. J.. Lorenz like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1703–1737.Google Scholar
[15]Ledrappier, F. and Lindenstrauss, E.. On the projections of measures invariant under the geodesic flow. Internat. Math. Res. Notices 9 (2003), 511–526.CrossRefGoogle Scholar
[16]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122 (1985), 540–574.Google Scholar
[17]Marie, P. and Rousseau, J.. Recurrence for random dynamical systems. Discrete Contin. Dyn. Syst. 30(1) (2011), 1–16.Google Scholar
[18]Ornstein, D. S. and Weiss, B.. Entropy and data compression schemes. IEEE Trans. Inform. Theory 39 (1993), 78–83.Google Scholar
[19]Pène, F. and Saussol, B.. Back to balls in billiards. Comm. Math. Phys. 293 (2010), 837–866.Google Scholar
[20]Pesin, Y. B. and Sadovskaya, V.. Multifractal analysis of conformal Axiom A flows. Comm. Math. Phys. 216 (2001), 277–312.Google Scholar
[21]Rousseau, J. and Saussol, B.. Poincaré recurrence for observations. Trans. Amer. Math. Soc. 362 (2010), 5845–5859.Google Scholar
[22]Ruelle, D.. Flots qui ne mélangent pas exponentiellement. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 191–193.Google Scholar
[23]Saussol, B.. Recurrence rate in rapidly mixing dynamical systems. Discrete Contin. Dyn. Syst. 15 (2006), 259–267.Google Scholar
[24]Saussol, B., Troubetzkoy, S. and Vaienti, S.. Recurrence and Lyapunov exponents. Mosc. Math. J. 3 (2003), 189–203, 260.Google Scholar