Cellular automata and substitutions in topological spaces defined via edit distances

TL;DR

This paper investigates the dynamics of cellular automata and substitutions within topological spaces defined by edit distances, introducing new metrics and analyzing their properties and implications for system behavior.

Contribution

It characterizes when substitutions form well-defined dynamical systems and extends analysis to dill maps, unifying cellular automata and substitutions under new pseudo-metrics.

Findings

Substitutions are Lipschitz in the Feldman pseudo-metric.

Cellular automata are Lipschitz in both Besicovitch and Feldman spaces.

Conditions for equicontinuity of these systems are established.

Abstract

The Besicovitch pseudo-metric is a shift-invariant pseudo-metric on the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. They correspond to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudo-metric, the Feldman pseudo-metric, where the Hamming distance is replaced by the Levenshtein distance. Like in the Besicovitch space, cellular automata are Lipschitz in this space, but here also all substitutions are Lipschitz. In both spaces, we discuss equicontinuity of these systems, and give a number of examples, and generalize our results to the class of the dill maps, that embed both cellular automata and substitutions.