Transitive action on finite points of a full shift and a finitary Ryan’s theorem

Abstract

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is Z (the shift group), giving a finitary version of Ryan's theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing SFTs. We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property which is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson's V .

Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type

“…In Sect. 7.3 of Salo (2017) the question was raised whether for a mixing SFT X and for every G Aut ðXÞ such that CðGÞ ¼ r h i there is a finite subset S G such that also CðSÞ ¼ r h i. In the same section it was noted that to construct a counterexample it would be sufficient to find a locally finite group G Aut ðXÞ whose centralizer is generated by r. We use a different strategy based on Lemma 1 to construct a counterexample in the case when X is the binary full shift.…”

“…We give some examples. The paper (Salo 2017) contains a construction of a finitely generated group G of automorphisms of A Z (when jAj ¼ 4) whose elements can implement any permutation on any finite collection of 0-finite non-constant configurations that belong to different shift orbits. An essential part of the construction is that one of the generators of G is a diffusive glider automorphism on A Z .…”

“…Denote the minimal cardinality of such a finite set S by k(X). In Salo (2017) it was proved that kðXÞ 10 when X is the full shift over the four-letter alphabet. In the same paper it is noted that k(X) is an isomorphism invariant of Aut ðXÞ and therefore computing it could theoretically separate Aut ðXÞ and Aut ðYÞ for some mixing SFTs X and Y.…”

No Tits alternative for cellular automata

“…The first item generalizes to sofic H × Z-shifts where H is finite by [49], and could presumably be generalized to virtually-cyclic groups with the same idea. By symmetry-breaking arguments, we believe the third item generalizes to SFTs with a nontrivial point of finite support, on any group which is not locally finite, and also to cellular automata acting on sets of colorings of undirected graphs, but this is beyond the scope of the present paper.…”

Glider Automorphisms on Some Shifts of Finite Type and a Finitary Ryan’s Theorem

“…There may also be subsets S ⊆ Aut(X) whose centralizers are generated by σ. Denote the minimal cardinality of such a finite set S by k(X). In [3] it was proved that k(X) ≤ 10 when X is the full shift over the four-letter alphabet. In the same paper it is noted that k(X) is an isomorphism invariant of Aut(X) and therefore computing it could theoretically separate Aut(X) and Aut(Y ) for some mixing SFTs X and Y .…”