Statistical stability and limit laws for Rovella maps

TL;DR

This paper studies the statistical stability and limit laws of Rovella maps, a family of one-dimensional maps related to Lorenz attractors, showing exponential decay of tail sets and continuous variation of SRB measures.

Contribution

It applies Freitas' technique to prove exponential tail decay and establishes statistical stability and limit laws for Rovella maps, extending previous results.

Findings

Exponential decay of tail sets for Rovella maps

Continuous variation of SRB measures with parameters

Statistical properties derived from tail decay

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.