Attractors in Dynamical Systems

Abstract

Mathematical and physical dissipative systems occur naturally; for example in heat diffusion and viscous fluid flow, some of the kinetic energy dissipates into heat. Due to the presence of wandering sets, in general dissipative systems exhibit complicated dynamics on some part of the domain. Loosely speaking, an attractor is “where the wandering sets end up,” and this set of points can take many forms. The subject of attractors in dynamical systems provides an important connection between measure theoretic and topological dynamics. There are several definitions of attractors that have evolved over the years, and we give two here that are closely related under our standing assumption that we always work on standard spaces. When μ(X) = ∞, Example 2.17 shows that ergodicity does not imply conservativity. Indeed, there are many important examples of maps that are ergodic but dissipative. Among these are certain unimodal maps of the interval from well-studied families ([19], and see also Section 3.3.1), and other more exotic unimodal interval maps [28]. Fluid flows offer physical examples of dissipative dynamical systems whose ergodic properties remain mostly open problems.