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

By Stefano Galatolo, Maria José Pacifico. Ergodic Theory and Dynamical Systems, 2009

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Abstract 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.

Subfields: Computer Networks and Communications, Mathematical Physics, Statistical and Nonlinear Physics

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