Sofically presented dynamical systems

Abstract

Systems obtained by quotienting a subshift of finite type (SFT) by another SFT are called finitely presented in the literature. Analogously, if a sofic shift is quotiented by a sofic equivalence relation, we call the resulting system sofically presented. Generalizing an observation of Fried, for all discrete countable monoids M , we show that M -subshift SFT systems are precisely the expansive dynamical M -systems, where S 1 S 2 denotes the class of systems obtained by quotienting subshifts in S1 by (relative) subshifts in S2. We show that for all finitely generated infinite monoids M ,and that Mañé's theorem about the dimension of expansive systems characterizes the virtually cyclic groups.In the case of one-dimensional actions, Mañé's theorem generalizes to sofically presented systems, which also have finite topological dimension. The basis of this is the construction of an explicit metric for a sofically presented system. We show that any finite connected simplicial complex is a connected component of a finitely presented system, and prove that conjugacy of one-dimensional sofically presented dynamical systems is undecidable. A key idea is the introduction of so-called automatic spaces. We also briefly study the automorphism groups and periodic points of these systems.We also perform two case studies. First, in the context of β-shifts, we define the β-kernelthe least subshift relation that identifies 1 with its orbit. We give a classification of the β-shift/βkernel pair as a function of β. Second, we revisit the classical study of toral automorphisms in our framework, and in particular for the toral automorphism ( 1 1 1 0 ), we explicitly compute the kernel of the standard presentation.

Distortion element in the automorphism group of a full shift

“…Second, in the setting of general expansive homeomorphisms, finding distortion elements is very easy. Namely, if S is the invertible natural extension of the ×2-map on the circle, Aut(S n ) contains a natural copy of GL(n, Z) by simply summing tracks to each other [34]. For n = 3, the group GL(n, Z) contains the Heisenberg group, thus has distortion elements.…”