Statistical stability of Geometric Lorenz attractors

TL;DR

This paper proves that SRB measures for Geometric Lorenz attractors depend continuously on the dynamics, establishing a form of statistical stability for these chaotic systems.

Contribution

It demonstrates the statistical stability of SRB measures in Geometric Lorenz attractors, a key step in understanding their long-term statistical behavior.

Findings

SRB measures depend continuously on the dynamics in the weak* topology

Geometric Lorenz attractors support SRB measures with full Lebesgue measure basins

The attractors exhibit robust chaotic behavior with sensitive dependence on initial conditions

Abstract

We consider the robust family of Geometric Lorenz attractors. These attractors are chaotic in the sense that they are transitive and have sensitive dependence on the initial conditions. Moreover, they support SRB measures whose ergodic basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB measures depend continuously on the dynamics in the weak$^*$ topology.