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 τ 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,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.

Decay of Correlations, Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors

“…Given any x we recall that we denote with t(x) the first strictly positive time, such that X t(x) (x) ∈ Σ (the return time of x to Σ). A relation between τ r X (x, x 0 ) and τ Σ r (x, x 0 ) is proved in [8] (Proposition 5.2). Let x ∈ R 3 and π(x) be the projection on Σ given by π(x) = y if x is on the orbit of y ∈ Σ and the orbit from y to x does not cross Σ (if x ∈ Σ then π(x) = x).…”

Extensions with shrinking fibers

“…To prove this result, our main tool is expected: we prove the regularity of the disintegration of with respect to (Theorem 6.3). Such results appeared in the work of Galatolo and Pacifico [GP10] (Appendix A; extra difficulty in the proof there seems to be caused by the way disintegration is set up, making it necessary to deal with non-probability measures) and in the recent works of Butterley and Melbourne [BM17] (Proposition 6, to compare with Theorem 6.3) and of Araujo, Galatolo and Pacifico [AGP14] (Theorem A). Compared to these work, we gain in generality: we can consider very general maps while they tend to restrict to uniformly expanding maps, we consider an arbitrary -invariant measure instead of restricting to the absolutely continuous one.…”

Hitting times distribution and extreme value laws for semi-flows

“…It is possible to study the dynamics of a geometric Lorenz attractor flow through a 1-dimensional map f obtained from the quotient though the leaves of this contracting foliation. We refer to [11] for a didactic exposition of this construction.…”