On linear non-uniform cellular automata: duality and dynamics

TL;DR

This paper explores the duality and dynamics of linear non-uniform cellular automata (NUCA), establishing key equivalences between dynamical properties and their duals, and analyzing invertibility and shadowing properties.

Contribution

It generalizes duality results from linear cellular automata to linear NUCA, characterizing invertibility and stability properties in this broader context.

Findings

Duality between injectivity and surjectivity properties in linear NUCA

Characterization of invertibility via pre-injectivity and post-surjectivity

Linear NUCA satisfy the shadowing property

Abstract

For linear non-uniform cellular automata (NUCA) over an arbitrary universe, we introduce and investigate their dual linear NUCA. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Moreover, we show that linear NUCA satisfy the important shadowing property. Applications on the dual surjunctivity are also obtained.