Robust Exponential Mixing and Convergence to Equilibrium for Singular-Hyperbolic Attracting Sets

Statistical determinism in non-Lipschitz dynamical systems

“…Having in mind the above discussion, assume that the attractor A + has C 2 -robust convergence to equilibrium (for example, a hyperbolic attractor as in [11] or the Lorenz attractor as in [5]). Supposing moreover the hypothesis of Proposition 4.1, we obtain that A + has robust convergence to equilibrium.…”

“…The classical results on the ergodic theory of hyperbolic flows show that a -hyperbolic attractor satisfying the C-dense condition of Bowen–Ruelle (density of the stable manifold of some orbit) has -robust convergence to equilibrium, see [11, Theorem 5.3]. In the last decades, many statistical properties have been studied for the larger class of singular hyperbolic attractors, which includes the hyperbolic and the Lorenz attractors; see for example [3] as a basic reference and [1, 2, 4, 5] for more recent advances. Robust convergence to equilibrium was naturally conjectured for such attractors [10, Problem E.4].…”

“…Robust convergence to equilibrium was naturally conjectured for such attractors [10, Problem E.4]. Although the general proof is not available yet, recently in [5, Corollary B and §4] were given examples of singular hyperbolic attractors having robust convergence to equilibrium, which include perturbations of the Lorenz attractor. In particular, it was shown there exists an arbitrary small -perturbation of the Lorenz attractor so that the resulting system has -robust convergence to equilibrium with respect to -observables.…”