Research Article Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems Koon Sang Wong and Zabidin Salleh Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia Correspondence should be addressed to Zabidin Salleh; zabidin@umt.edu.my Received 9 March 2021; Accepted 6 May 2021; Published 15 May 2021 Academic Editor: Victor Kovtunenko Copyright © 2021 Koon Sang Wong and Zabidin Salleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 properties for set-valued functions and generalize some results from a single- valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals. 1. Introduction In dynamical systems, one of the most important research topics is to determine the chaotic behaviour of the system. Various definitions of chaos have been introduced by math- ematicians in the past (see [1–5]) as there is no universally accepted definition of chaos. Definitions of chaos are con- structed based on some topological properties. One of the commonly used properties is topologically transitive. The concept of topologically transitive was introduced by Birkh- off [6] in 1920. Dynamical systems with a topologically tran- sitive property contain at least one point which moves under iteration from one arbitrary neighborhood to any other neighborhood. This property has been studied intensively by mathematicians since it is a global characteristic in the dynamical system. Some prefer to study topologically mixing of dynamical systems as it is a notion stronger than topolog- ically transitive. Numerous studies related to the transitivity and mixing properties of the dynamical systems especially in a one- dimensional system have been done; see [7–12]. As we all know, in general, dynamical systems are studied in the view of a single point. However, knowing how the points of the system move is not sufficient as there are problems and appli- cations that require one to know how the subsets of the sys- tem move. In recent years, several works and research on the topological dynamics of set-valued dynamical systems can be found (see [13–20]). However, there are many proper- ties for the dynamics of set-valued dynamical systems yet to be discovered. Loranty and Pawlak [21] studied the connec- tion between transitivity and a dense orbit for multifunction in generalized topological spaces. Information about topo- logically transitive for the single-valued dynamical systems can be found in [22–25]. In this paper, we will introduce and study the notion of topologically transitive and topologically mixing for set- valued functions. We prove some elementary results of these two properties. Some of the results are generalization from the single-valued case (e.g., [11, 26–28]). We also prove that the definitions of these two properties for a set-valued func- tion on compact intervals are equivalence. This paper is orga- nized as follows. In Section 2, we give some background settings and define topologically transitive and topologically mixing for set-valued functions. In Section 3, we present some elementary implication results of topologically transi- tive and topologically mixing. In Section 4, we prove the Hindawi Abstract and Applied Analysis Volume 2021, Article ID 5541105, 7 pages https://doi.org/10.1155/2021/5541105