Advanced Extensions and Applications of Transitivity and Mixing in Set‐Valued Dynamics With Numerical Simulations and Visual Insights

This article extends Alfredo Peris’s work on chaos in set‐valued dynamics by providing new characterizations and applications of transitivity and mixing properties. Peris demonstrated that the topological transitivity of a set‐valued map is closely related to the weak mixing property of the individual map. In this continuation, we investigate these relationships in a broader context, including general metric spaces and infinite‐dimensional Banach spaces. First, we extend the characterization of transitivity and mixing in set‐valued dynamic systems by exploring additional conditions in general metric spaces. Then, we apply these results to the dynamics of fractal sets, showing how these properties influence the structure and behavior of chaotic attractors and Julia sets. Additionally, we incorporate numerical simulations and visualizations to illustrate the theoretical concepts and demonstrate examples of chaotic behavior in set‐valued systems. These simulations provide a visual and computational tool to better understand the dynamics of set‐valued maps, making abstract theories more accessible and engaging. Finally, we address linear dynamics in infinite‐dimensional Banach spaces, providing new proofs and characterizations that relate the hypercyclicity criterion with the transitivity and mixing properties in linear operators. This work not only expands the existing theoretical results but also offers new perspectives and tools for studying complex dynamic systems with potential applications in mathematics and physics.