Topological Dynamics from a Measure-Theoretic Viewpoint

Introduction to Measurable Perspectives in Dynamical Systems

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.

Structural Results and Theorems

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.