Weakly mixing, proximal topological models for ergodic systems and applications

TL;DR

This paper constructs specific topological models for ergodic systems, demonstrating the existence of weakly mixing and proximal models, and introduces methods for tower constructions with applications in ergodic theory.

Contribution

It provides new topological models for ergodic systems, including weakly mixing and proximal models, and develops tower construction techniques with broad applications.

Findings

Existence of weakly mixing, fully supported models for ergodic systems.

Construction of proximal models with specific properties.

Method for creating taller Kakutani-Rokhlin towers with desired features.

Abstract

In this paper it is shown that every non-periodic ergodic system has two topologically weakly mixing, fully supported models: one is non-minimal but has a dense set of minimal points; and the other one is proximal. Also for independent interests, for a given Kakutani-Rokhlin tower with relatively prime column heights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower with same property, which can be used in Weiss's proof of the Jewett-Krieger's theorem and the proofs of our theorems. Applications of the results are given.