Orbit sets, transitivity, and sensitivity with upper semicontinuous maps

TL;DR

This paper investigates the dynamics of upper semicontinuous maps on compact metric spaces, introducing new concepts and results related to orbit sets, transitivity, and sensitivity, with a focus on their properties and interrelations.

Contribution

It introduces new concepts for upper semicontinuous maps and establishes results on orbit set compactness, connectedness, and the relationship between transitivity and orbit density.

Findings

Orbit set $\\mathcal{O}_F(p)$ is compact.

Conditions for connectedness of orbit sets.

Relationships between transitivity and orbit density.

Abstract

Given a compact metric space $X$ and an upper semicontinuous function $F\colon X \to 2^X$, we explore the dynamic system $(X,F)$. In this study, we introduce new concepts, demonstrate various results, and provide numerous examples. In particular, we define the orbit set $\mathcal{O}_F(p)$ and prove that it is compact. We also establish conditions for connectedness of the orbit sets and pose several questions related to the system. We also investigate transitivity and its relation to the density of orbits. In addition, we present strong and weak notions of sensitivity and examine the relationships between these concepts.