Topological and Measurable Shadowing Properties in Dynamical Systems

In the realm of dynamical systems, the study of the Shadowing property has emerged as a pivotal endeavor, offering profound insights into the behavior of chaotic systems.

This project delves into the intricacies of the Shadowing property across three distinct frameworks: Shadowable Measures and Points, CW-hyperbolic Homeomorphisms, and Singular Flows, each with their unique characteristics and challenges.

With a focus on these disparate contexts, our investigation aims to unravel the implications of shadowing for the chaotic nature of such systems, while also unraveling the obstructions that arise in continuous time cases.Our exploration into Shadowable Measures and Points harnesses we seeks the existence of invariant shadowable measures and their role in the existence of odometers, shedding light on the fundamental nature of periodicity in chaotic dynamics.

In the domain of CW-hyperbolic Homeomorphisms, we navigate the dynamic interplay of hyperbolicity and chaos.

Our focus extends to investigating periodic shadowing, periodic specification, measures of maximal entropy, and the discovery of new examples that contribute to the broader landscape of chaotic systems theory.

Furthermore, our investigation extends to Singular Flows, where we confront the nuanced challenges presented by singularities in continuous time dynamics. We investigate when the existence of attached singularities forbids a flow to satisfy the shadowing property.

This exploration not only advances our understanding of continuous-time systems but also evidences a remarkable difference between the discrete-time and continuous-time contexts.