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Cellular automata on group sets and the uniform Curtis–Hedlund–Lyndon theorem

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Abstract

We introduce cellular automata whose cell spaces are left-homogeneous spaces, show that their global transition functions are closed under composition, prove a uniform as well as a topological variant of the Curtis–Hedlund–Lyndon theorem, and deduce that such an automaton is invertible if and only if its global transition function is bijective. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations.

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  1. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Simon Wacker

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  1. Simon Wacker
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Correspondence to Simon Wacker.

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Wacker, S. Cellular automata on group sets and the uniform Curtis–Hedlund–Lyndon theorem. Nat Comput 18, 459–487 (2019). https://doi.org/10.1007/s11047-017-9645-y

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