Lorenz like flows: exponential decay of correlations for the Poincar\'e map, logarithm law, quantitative recurrence

Abstract

In this paper we prove that the Poincar\'e 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 $\tau_r(x,x_0)$ 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 $d_{\mu}(x_0)$ exists, \lim_{r\to 0} \frac{\log \tau_r(x,x_0)}{-\log r} = d_{\mu}(x_0)-1 holds for $\mu$ 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.