On the concept of attractor. (English) Zbl 0595.58028

There is no agreement in the literature regarding the definition of the concept of attractor in dynamical systems theory. The purpose of this paper is to propose another definition of this term, based on the concept of probable asymptotic behavior of orbits.
Definitions of attractor range from the concepts of Lyapunov stability and asymptotic stability to the more specialized notion of Axiom A attractors. Different definitions are due to R. Williams [Publ. Math., Inst. Hautes Etud. Sci. 43, 169–203 (1974; Zbl 0279.58013)], D. Ruelle and F. Takens [Commun. Math. Phys. 23, 343–344 (1971; Zbl 0227.76084); see also ibid. 20, 167–192 (1971; Zbl 0223.76041)] and P. Collet and J.-P. Eckmann [Iterated maps on the interval as dynamical systems. Progress in Physics, 1. Basel etc.: Birkhäuser (1980; Zbl 0458.58002)], among many others.
After presenting the basic ingredients of all of these definitions, the author settles on the following definition: A closed subset \(A\subset M\) is an attractor if it satisfies (1) the realm of attraction \(\rho (A)\) consisting of all points whose \(\omega\)-limit set lies in \(A\), has positive measure, and (2) there is no strictly smaller closed subset \(A'\subset A\) for which \(\rho (A')\) coincides with \(\rho (A)\) up to a set of measure zero.
The author applies this definition to a variety of well-known dynamical systems, including iterated maps of the interval and strange attractors.
Reviewer: R.Devaney


37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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