# 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.

### Keywords

Cellular automata Group actions Homogeneous spaces Properness Curtis–Hedlund–Lyndon theorem Invertibility### References

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