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Weak attractors and Lyapunov-like functions. (English) Zbl 0964.37015
It is well-known that the qualitative behaviour of dynamical systems can be described in terms of invariant sets called attractors. In this note, the authors adapt the definition of attractor in a compact space in the sense of Conley, and then extend this concept to the dynamical system generated by a continuous map \(f\) on a noncompact space, which will be called the weak attractor of \(f\). Recently M. Hurley [Proc. Am. Math. Soc. 115, 1139-1148 (1992; Zbl 0759.58031)] proved that if \({\mathcal A}\) is a weak attractor of a discrete dynamical system \(f\) then there exists a Lyapunov-like function for \(A\). The purpose of this note is to investigate whether the converse of the above theorem does hold or not.

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37B25 Stability of topological dynamical systems
37C05 Dynamical systems involving smooth mappings and diffeomorphisms