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

Abstract

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 τ r ( x , x 0 ) is the time needed for the orbit of a point x to enter a ball B r ( x 0 ) centered at x 0 , 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 x 0 : for each x 0 such that the local dimension d μ ( x 0 ) exists, 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.