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Asymptotic Randomization of Sofic Shifts by Linear Cellular Automata

Marcus Pivato·Reem Yassawi
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Abstract

Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is then a compact abelian group; a `linear cellular automaton' (LCA) is a topological group endomorphism Φ:A^M --> A^M that commutes with all shift maps. Suppose μis a probability measure on A^M whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φand μ) under which Φ`asymptotically randomizes' μ, meaning that wk*lim_{J\ni j --> oo} Φ^j μ= η, where ηis the Haar measure on A^M, and J has Cesaro density 1. In the case when Φ=1+σ, we provide a condition on μthat is both necessary and sufficient. We then use this to construct an example of a zero-entropy measure which is asymptotically randomized by 1+σ(all previously known examples had positive entropy).

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Marcus Pivato, Reem Yassawi. 2006-04-13. Asymptotic Randomization of Sofic Shifts by Linear Cellular Automata. https://arxiv.org/abs/math/0306136

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