Dynamics of self-maps in their primal topologies

TL;DR

This paper explores various dynamical properties of self-maps within their primal topologies, including stability, chaos, and mixing, extending some results to continuous maps on Alexandroff spaces.

Contribution

It introduces a comprehensive analysis of dynamical concepts for self-maps in primal topologies and proves Lyapunov stability for continuous self-maps on Alexandroff spaces.

Findings

Lyapunov stability of continuous self-maps on Alexandroff spaces

Characterization of chaos and mixing in primal topologies

Extension of dynamical concepts to Alexandroff spaces

Abstract

We study a series of dynamical concepts for self-maps in the primal topology induced by them. Among the concepts studied are non-wandering points, limit points, recurrent points, minimal sets, transitive points and self-maps, topologically ergodic self-maps, weakly mixing self-maps, strongly mixing self-maps, Lyapunov stable self-maps, chaotic self-maps in the sense of Auslander-Yorke, chaotic self-maps in the sense of Devaney, asymptotic pairs, proximal pairs, and syndetically proximal pairs. Some results are given in the more general context of continuous self-maps in an Alexandroff topological space. We prove that a continuous self-map of an Alexandroff space is always Lyapunov stable.