Universality in Multidimensional Symbolic Dynamics

TL;DR

This paper establishes the existence of universal effective $Z$ dynamical systems and multidimensional shifts of finite type, revealing fundamental limitations and capabilities in symbolic dynamics and cellular automata.

Contribution

It proves the existence of universal systems in effective $Z$ dynamics and multidimensional SFTs for certain subactions, and shows limitations for higher dimensions.

Findings

Existence of a universal effective $Z$ system factoring onto all others.

d-dimensional SFTs are universal for 1-dimensional subactions when d ≥ 3.

No universal effective $Z^d$-system exists for d > 1.

Abstract

We show that in the category of effective $Z$ dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for d $\geq 3$ there exist d-dimensional shifts of finite type which are universal for 1-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective $Z^d$-system for $d>1$, and in particular SFTs cannot be universal for subactions of rank $d>1$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.