On the Concept of Attractor

Abstract

There is no agreement in the literature regarding the definition ofthe concept of attractor in dynamical systems theory. The purposeof this paper is to propose another definition of this term, basedon the concept of probable asymptotic behavior of orbits. Definitionsof attractor range from the concepts of Lyapunov stability and asymptoticstability to the more specialized notion of Axiom A attractors. Differentdefinitions are due to {\it R. Williams} [Publ. Math., Inst. HautesEtud. Sci. 43, 169--203 (1974; Zbl 0279.58013)], {\it D. Ruelle}and {\it F. Takens} [Commun. Math. Phys. 23, 343--344 (1971; Zbl0227.76084); see also ibid. 20, 167--192 (1971; Zbl 0223.76041)]and {\it P. Collet} and {\it J.-P. Eckmann} [Iterated maps on theinterval as dynamical systems. Progress in Physics, 1. Basel etc.:Birkhäuser (1980; Zbl 0458.58002)], among many others. After presentingthe basic ingredients of all of these definitions, the author settleson the following definition: A closed subset A\subset M is an attractorif it satisfies (1) the realm of attraction ρ(A) consistingof all points whose ω-limit set lies in A, has positivemeasure, and (2) there is no strictly smaller closed subset A'\subsetA for which ρ(A') coincides with ρ(A) up to a set ofmeasure zero. The author applies this definition to a variety ofwell-known dynamical systems, including iterated maps of the intervaland strange attractors.