Attractors and chain recurrence in generalized semigroup

TL;DR

This paper explores the relationship between attractors and chain recurrence in generalized semigroups, extending classical dynamical concepts to a broader algebraic setting.

Contribution

It introduces an alternative definition of chain recurrent sets using attractors in the context of generalized semigroups, building on previous recurrence notions.

Findings

Established a new link between attractors and chain recurrence in generalized semigroups.

Extended classical dynamical concepts like topological transitivity to this generalized setting.

Provided a framework for analyzing recurrence properties in algebraic structures beyond traditional dynamics.

Abstract

In \cite {kl1}, we extended various notions of recurrence for the generalized semigroup analogous to their counterpart in the classical theory of dynamics. In this paper, we shall address the alternative definition of chain recurrent set in terms of attractors, given by Hurley in \cite {mh} following Conley\rq{}s characterization in \cite {conley}. We shall also discuss the notion of topological transitivity and chain transitivity in this general setting.