Continuous variation of SRB measures for the geometric Lorenz attractors
Mohammad Soufi (Faculdade de Ciências da Universidade do Porto )
3 de Dezembro de 2012 / 16h / Sala de Reuniões do DMUBI
In this talk we investigate the stability of the geometric Lorenz attractor in measure theoretical sense . First, we consider the geometric Lorenz flow and its deterministic perturbations. It is well known that each of these flows contains a chaotic attractor which supports a unique SRB measure. We prove that the SRB measures depend continuously on the dynamics in the weak$^{\ast}$ topology. In other words, the geometric Lorenz attractor is statistical stable.
Then, we consider a one parameter family with positive Lebesgue measure of one-dimensional maps arising from the contracting Lorenz attractors studied by Rovella. Benedicks-Carleson technique was used by Rovella to prove the existence of this family and also the exponential grows of their derivatives along their critical orbits whose recurrences to the critical point are slow. Here we use the technique developed by Freitas to 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 existence of an SRB measure for each map in the family, and the continuous variation of the densities of the SRB measures and associated metric entropies with the parameter.