Mixing properties in coded systems

TL;DR

This paper investigates mixing properties in coded systems, establishing equivalences, providing a unique example with specific properties, and analyzing implications for synchronized systems and the strong property P.

Contribution

It demonstrates the equivalence of mixing, weak mixing, and total transitivity in coded systems and constructs a novel example with unique mixing characteristics.

Findings

Topological mixing, weak mixing, and total transitivity are equivalent in coded systems.

An example of a mixing coded system with only even-period periodic points is constructed.

Such a system cannot be a synchronized system and has the strong property P.

Abstract

We show that topological mixing, weak mixing and total transitivity are equivalent for coded systems. We provide an example of a mixing coded system which cannot be approximated by any increasing sequence of mixing shifts of finite type, has only periodic points of even period and each set of its generators consists of blocks of even length. We prove that such an example cannot be a synchronized system. We also show that a mixing coded systems has the strong property $P$.