Topological Factoring of Zero Dimensional Dynamical Systems

TL;DR

This paper characterizes topological factorizations of zero-dimensional dynamical systems using ordered Bratteli diagrams and extends the Curtis-Hedlund-Lyndon theorem to these systems, providing new methods and conditions for such factorizations.

Contribution

It introduces a representation of topological factorizations via morphisms of Bratteli diagrams and generalizes the Curtis-Hedlund-Lyndon theorem for zero-dimensional systems.

Findings

Topological factorizations correspond to morphisms between Bratteli diagrams.

Constructed examples where induced maps are not topological factorizations.

Provided conditions for the existence of topological factorizations.

Abstract

We show that every topological factoring between two zero dimensional dynamical systems can be represented by a sequence of morphisms between the levels of the associated ordered Bratteli diagrams. Conversely, we will prove that given an ordered Bratteli diagram $B$ with a continuous Vershik map on it, every sequence of morphisms between levels of $B$ and $C$, where $C$ is another ordered Bratteli diagram with continuous Vershik map, induces a topological factoring if and only if $B$ has a unique infinite min path. We present a method to construct various examples of ordered premorphisms between two decisive Bratteli diagrams such that the induced maps between the two Vershik systems are not topological factorings. We provide sufficient conditions for the existence of a topological factoring from an ordered premorphism. Expanding on the modelling of factoring, we generalize the Curtis-Hedlund-Lyndon theorem to represent factor maps between two zero dimensional dynamical systems through sequences of sliding block codes.