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arXiv · 2208.13069

On linear non-uniform cellular automata: duality and dynamics

Xuan Kien Phung
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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.

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Xuan Kien Phung. 2022-08-27. On linear non-uniform cellular automata: duality and dynamics. https://arxiv.org/abs/2208.13069

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