# Transitivity and Mixing Properties of Set-Valued Dynamical Systems

## TL;DR

This paper explores the properties of topological transitivity and mixing in set-valued dynamical systems, generalizing classical results and establishing their equivalence on compact intervals.

## Contribution

It introduces and analyzes these properties within the set-valued context, extending known single-valued dynamical system results.

## Key findings

- Topologically transitive and mixing properties are equivalent for set-valued systems on compact intervals.
- Generalization of classical dynamical properties to set-valued functions.
- Implications of these properties for the behavior of set-valued dynamical systems.

## Abstract

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two topological properties for set-valued functions and generalize some results from single-valued case to set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.