On surjectivity and dynamical properties of dill maps

TL;DR

This paper investigates the dynamical properties of dill maps, showing that surjective uniform dill maps are equivalent to surjective cellular automata and providing conditions for equicontinuity.

Contribution

It establishes a characterization of surjective uniform dill maps as cellular automata and offers a new criterion for their equicontinuity.

Findings

Surjective uniform dill maps are exactly surjective cellular automata.

A sufficient condition for dill maps to be equicontinuous is identified.

The work generalizes known properties of cellular automata and substitutions.

Abstract

In this paper, we study certain dynamical properties of dill maps, a class of functions introduced in~\cite{salo2015block} that generalizes both cellular automata and substitutions. In particular, we prove that surjective uniform dill maps are precisely the surjective cellular automata. We also establish a sufficient condition for a dill map to be equicontinuous.