Preprint D58/2009
Lorenz like flows: exponential decay of correlations for the Poincare map. logarithm law, quantitative recurrence

Maria JosÃ© Pacifico | Galatolo, Stefano

**Keywords: **
Lorenz like flows | exponential decay of correlations | local dimension | hitting time | logarithm law | quantitative recurrence

Abstract. In this paper we prove that the Poincar`e map associated to a Lorenz like system has exponential decay of correlations with respect to Lipschitz observables. This
implies that the hitting time associated to the system satisfies a logarithm law. The hitting time T_r(x, x_0) is the time needed for the orbit of a point x to enter for the first time in a ball B_r(x_0) centered at x_0, with small radius r. 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 SRB measure at x_0: for each x_0 such that the local dimension d_μ(x_0) exists,
lim_{r\to\infity}log T(x, x_0)/−log r = dμ(x0) − 1
holds for μ almost each x. In a similar way it is possible to consider a quantitative recurrence indicator quantifying the speed of coming back of an orbit to its starting point.
Similar results holds for this recurrence indicator.