Statistical stability and limit laws for Rovella maps

Abstract

We consider a 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 recurrence to the critical point. Metzeger 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 (in the L1-norm) 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.