Topological transitivity in quasi-continuous dynamical systems

TL;DR

This paper investigates topological transitivity in quasi-continuous dynamical systems, demonstrating the equivalence of various transitivity notions under certain conditions, thus extending classical results from continuous systems.

Contribution

It establishes the equivalence of topological and point transitivity in quasi-continuous systems, extending classical dynamical systems theory.

Findings

Topological and point transitivity are equivalent under certain assumptions.

Extends classical results from continuous to quasi-continuous systems.

Provides a unified framework for understanding transitivity in quasi-continuous dynamics.

Abstract

A quasi-continuous dynamical system is a pair $(X,f)$ consisting of a topological space $X$ and a mapping $f: X\to X$ such that $f^n$ is quasi-continuous for all $n \in \mathbb N$, where $\mathbb N$ is the set of non-negative integers. In this paper, we show that under appropriate assumptions, various definitions of the concept of topological transitivity are equivalent in a quasi-continuous dynamical system. Our main results establish the equivalence of topological and point transitivity in a quasi-continuous dynamical system. These extend some classical results on continuous dynamical systems in [3], [10] and [25], and some results on quasi-continuous dynamical systems in [7] and [8].