Distal systems in topological dynamics and ergodic theory

TL;DR

This paper extends Lindenstrauss's result by showing that every separable ergodic measurably distal system has a unique, canonical minimal distal model, constructed via the Furstenberg--Zimmer tower and a new characterization of isometric extensions.

Contribution

It introduces a canonical construction of minimal distal models for ergodic measurably distal systems using the Furstenberg--Zimmer tower and a novel characterization of isometric extensions.

Findings

Every separable ergodic measurably distal system has a canonical minimal distal model.

The construction uses the Furstenberg--Zimmer tower and a new characterization of isometric extensions.

The model is uniquely determined and can be explicitly constructed.

Abstract

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg--Zimmer tower of a measurably distal system and showing that at each step, there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.