Lorenz like Flows: Exponential Decay of Correlations for the Poincaré Map, Logarithm Law, Quantitative Recurrence

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 for the first time in a ball Br(x0) centered at x0, 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 x0: for each x0 such that the local dimension dμ(x0) exists, lim r→0 log τr(x, x0) − 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.