WAP Systems and Labeled Subshifts

TL;DR

This paper introduces a novel method for constructing weakly almost periodic (WAP) systems using labeled subshifts, enabling the creation of diverse examples with specific properties such as nullity, height, and scrambling.

Contribution

It presents a powerful construction technique for subshifts, leading to new examples of WAP systems with various dynamical properties not previously demonstrated.

Findings

Constructed null and non-null WAP subshifts.

Created WAP subshifts of arbitrary countable height.

Developed examples of recurrent non-tame subshifts.

Abstract

The main object of this work is to present a powerful method of construction of subshifts which we use chiefly to construct WAP systems with various properties. Among many other applications of this so called labeled subshifts, we obtain examples of null as well as non-null WAP subshifts, WAP subshifts of arbitrary countable (Birkhoff) height, and completely scrambled WAP systems of arbitrary countable height. We also construct LE but not HAE subshifts, and recurrent non-tame subshifts.