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

TL;DR

This paper demonstrates the existence of a finitely generated automorphism subgroup with highly transitive action on finite points of a four-symbol full shift, leading to a finitary version of Ryan's theorem and exploring related automorphism group properties.

Contribution

It introduces a finitely generated subgroup with transitive action on finite points of the shift, providing a finitary analogue of Ryan's theorem and analyzing its algebraic structure.

Findings

Existence of a finitely generated subgroup with transitive action on finite points.

A finite set of automorphisms with centralizer equal to the shift group.

Construction of a subgroup within the commutator subgroup with involution-based elements.

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 $\mathbb{Z}$ (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~$V$.