This book introduces a new measurable perspective on dynamical systems by connecting concepts from topological dynamics with their measure-theoretic counterparts. A central theme is the translation of topological notions into measurable ones. For example, minimality in topological dynamics suggests a measurable analogue in ergodicity, where every invariant measurable set has either zero or full measure, offering an intuitive parallel between the two settings. Likewise, the notion of expansiveness is reinterpreted through expansive measures, in which almost all orbits separate beyond a fixed radius. These measurable analogues extend naturally to homeomorphisms and flows on compact metric spaces, which are explored in depth in Chapters 3 and 7.

Building on this framework, the book develops measurable versions of several structural results from topological dynamics. Walters' stability theorem-grounded in shadowing, expansiveness, and topological stability-is revisited in Chapters 4 and 8 from a measurable perspective, while Smale's spectral decomposition theorem is reformulated in measurable terms in Chapters 5 and 9. By bridging topological and measurable viewpoints, the book offers a cohesive approach that provides new insights and directions for the study of dynamical systems.

Offer a fresh measurable perspective on topological dynamics, bridging classical theory with modern insights Translate classical topological results into measurable terms to provide new interpretations and applications Guide young researchers toward a unified approach to measure-preserving and topological dynamical systems

Autorentext

Keonhee Lee is a professor emeritus of Chungnam National University.

Carlos Arnoldo Morales Rojas is a researcher at Beihang University.

Ngocthach Nguyen is a lecturer of the University of Danang.

Inhalt

Preliminaries.- Topological Dynamics.- Expansive Measures.- Shadowing and Topological Stability.- Measurable Spectral Decomposition.- Topological Dynamics for Flows.- Measure Expansiveness for Flows.- Shadowing and Topological Stability for Flows.- Measurable Spectral Decomposition for Flows.