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Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence

Published online by Cambridge University Press:  11 December 2009

S. GALATOLO and
MARIA JOSÉ PACIFICO
S. GALATOLO
Affiliation:
Dipartimento di Matematica Applicata via Buonarroti 1, 56100, Pisa, Italia (email: s.galatolo@docenti.ing.unipi.it)
MARIA JOSÉ PACIFICO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, Brazil (email: pacifico@im.ufrj.br)
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Abstract

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In this paper we prove that the Poincaré map associated to a Lorenz-like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x,x0) is the time needed for the orbit of a point x to enter a ball Br(x0) centered at x0, with small radius r, for the first time. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the physical measure at x0: for each x0 such that the local dimension dμ (x0)exists, \[ \lim _{r\rightarrow 0} \frac {\log \tau _r(x,x_0)}{-{\log }\, r} = d_{\mu }(x_0)-1 \] holds for μ almost each x. In a similar way, it is possible to consider a quantitative recurrence indicator quantifying the speed of an orbit to come back to its starting point. Similar results hold for this recurrence indicator.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010, pp. 1703 - 1737
Copyright
Copyright © Cambridge University Press 2009

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