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

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 ${\tau }_r(x,x_0)$ 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 τ_r(x,x_0)}{-\log r} = d_μ(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.

Revision, after some advices

Cited by in corpus (4)

• Extreme value laws in dynamical systems under physical observables
• Thermodynamic formalism for contracting Lorenz flows
• Area expanding C^{1+α} Suspension Semiflows
• Decay of correlations, quantitative recurrence and logarithm law for contracting Lorenz attractors