On the Induction Operation for Shift Subspaces and Cellular Automata as Presentations of Dynamical Systems

TL;DR

This paper explores how shift subspaces and cellular automata can be represented as dynamical systems, focusing on induction operations, group actions, and the simulation of structures, with particular attention to sofic shifts.

Contribution

It introduces methods to relate properties of induced structures to original ones and demonstrates how to simulate smaller structures within larger ones for dynamical system presentations.

Findings

Properties of induced entities can be derived from original structures

Simulation techniques for embedding smaller structures into larger ones

Characterization of systems via group actions, especially for sofic shifts

Abstract

We consider continuous, translation-commuting transformations of compact, translation-invariant families of mappingsfrom finitely generated groups into finite alphabets. It is well-known that such transformations and spaces can be described "locally" via families of patterns and finitary functions; such descriptions can be re-used on groups larger than the original, usually defining non-isomorphic structures. We show how some of the properties of the "induced" entities can be deduced from those of the original ones, and vice versa; then, we show how to "simulate" the smaller structure into the larger one, and obtain a characterization in terms of group actions for the dynamical systems admitting of presentations via structures as such. Special attention is given to the class of sofic shifts.