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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.

##### MSC:
 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