Attractors of Local Semiflows on Topological Spaces

TL;DR

This paper introduces a new concept of attractors for local semiflows on topological spaces, developing a foundational theory including stability, existence, Lyapunov functions, and Morse decompositions.

Contribution

It proposes a novel attractor notion tailored for topological spaces and establishes a comprehensive theoretical framework for their analysis.

Findings

Defined a new attractor concept suitable for topological spaces

Proved fundamental properties like maximality and stability of attractors

Established a converse Lyapunov theorem and addressed Morse decompositions

Abstract

In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.