Return-time and pullback-convergence properties of statistical attractors

Abstract Various definitions of an attractor for a nonlinear dynamical system have been proposed. These use various assumptions on the set of initial conditions that should converge (the basin), and various notions of convergence. A weak assumption on the basin is the measure attractor of Milnor, which requires that the basin has positive measure. A weak assumption of the notion of convergence is the statistical attractor due to Ilyashenko, which requires that limiting to the attractor occurs on a set of future times of full density. We point out that many examples of statistical attractors actually satisfy a stronger definition which we call a bounded-return-time attractor, and we investigate such attractors. We also give an improved definition for the notion of pullback measure attraction. This was originally developed to understand attractors in nonautonomous systems, but we note here that it is helpful for understanding convergence towards statistical attractors in the autonomous setting. We investigate implications between all these different notions of attractors. We also investigate which of these notions are fulfilled by a hyperbolic fixed point with a homoclinic loop.