On the statistical stability of families of attracting sets and the contracting Lorenz attractor

Abstract

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of C 2 vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions avoid the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincaré sections. Contents 1. Introduction 1 1.1. Statements of the results 3 1.2. Application to singular-hyperbolic attracting sets 4 1.3. Application to the Contracting Lorenz (Rovella) attractor 6 1.4. Organization of the text 8 2. The contracting Lorenz family of attractors 8 2.1. Existence and uniqueness of physical measure 8 2.2. The physical measure is a SRB measure 9 2.3. Robust expansiveness of contracting Lorenz flows 10 3. Proof of Statistical Stability 11 3.1. Entropy expansiveness 12 3.2. Statistical stability 13 References 13