On the interplay between measurable and topological dynamics

TL;DR

This paper reviews the connections between measurable and topological dynamics, highlighting analogies and embedding ergodic theory into topological systems, and presents new results on Borel cross-sections and the local variational principle.

Contribution

It introduces new results on Borel cross-sections in Polish systems and a converse to the local variational principle for compact systems.

Findings

Equivalence of Borel cross-section existence with recurrence and periodicity in Polish systems

A converse to the local variational principle for compact dynamical systems

Demonstrates that ergodic theory can be embedded into topological dynamics

Abstract

This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological dynamics as shown in the treatment of recurrence phenomena, equicontinuity and weak mixing, distality and entropy. The prototypical result of the second part is the statement that any abstract measure probability preserving system can be represented as a continuous transformation of a compact space, and thus in some sense ergodic theory embeds into topological dynamics. The work also contains several new results. In particular (1) we prove, for a Polish dynamical system, the equivalence of the existence of a Borel cross-section with the coincidence of recurrence and periodicity; and (2) for compact dynamical systems we provide a converse to the local variational principle.