The Moore and the Myhill Property For Strongly Irreducible Subshifts Of Finite Type Over Group Sets

We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular automaton with finite set of states and finite neighbourhood over such a subshift is surjective if and only if it is preinjective. This statement is known as Garden of Eden theorem. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical. The notion of amenability for groups was introduced by John von Neumann in 1929. It generalises the notion of finiteness. A group G is left or right amenable if there is a finitely additive probability measure on P(G) that is invariant under left and right multiplication respectively. Groups are left amenable if and only if they are right amenable. A group is amenable if it is left or right amenable. The definitions of left and right amenability generalise to left and right group sets respectively. A left group set (M,G, .) is left amenable if there is a finitely additive probability measure on P(M) that is invariant under .. There is in general no natural action on the right that is to a left group action what right multiplication is to left group multiplication. Therefore, for a left group set there is no natural notion of right amenability. A transitive left group action . of G on M induces, for each element m0 ∈ M and each family {gm0,m}m∈M of elements in G such that, for each point m ∈ M , we have gm0,m . m0 = m, a right quotient set semi-action P of G/G0 on M with defect G0 given by m P gG0 = gm0,mgg −1 m0,m .m, where G0 is the stabiliser of m0 under .. Each of these right semi-actions is to the left group action what right multiplication is to left group multiplication. They occur in the definition of global transition functions of semi-cellular automata over left homogeneous spaces as defined in [9]. A cell space is a left group set together with choices of m0 and {gm0,m}m∈M . 1 ar X iv :1 70 6. 05 82 7v 1 [ m at h. G R ] 1 9 Ju n 20 17