Statistical stability and limit laws for Rovella maps

Abstract

We consider the family of one-dimensional maps arising from the contracting Lorenz attractors studied by Rovella. Benedicks-Carleson techniques were used by Rovella to prove that there is a one-parameter family of maps whose derivatives along their critical orbits increase exponentially fast and the critical orbits have slow recurrent to the critical point. Metzger proved that these maps have a unique absolutely continuous ergodic invariant probability measure (SRB measure). Here we use the technique developed by Freitas and show that the tail set (the set of points which at a given time have not achieved either the exponential growth of derivative or the slow recurrence) decays exponentially fast as time passes. As a consequence, we obtain the continuous variation of the densities of the SRB measures and associated metric entropies with the parameter. Our main result also implies some statistical properties for these maps.

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References in corpus (1)

• Strong statistical stability of non-uniformly expanding maps

Cited by in corpus (5)

• On the statistical stability of families of attracting sets and the contracting Lorenz attractor
• Statistical instability for contracting Lorenz flows
• Decay of correlations, quantitative recurrence and logarithm law for contracting Lorenz attractors
• Noise induced order for skew-products over a non-uniformly expanding base
• Quenched decay of correlations for random contracting Lorenz maps