Sofically presented dynamical systems

TL;DR

This paper introduces the concept of sofically presented systems, generalizes classical theorems to these systems, and explores their properties, automorphisms, and specific case studies like $eta$-shifts and toral automorphisms.

Contribution

It defines sofically presented systems, establishes their properties, and extends Ma ext{n}é's theorem, providing explicit metrics and analyzing automorphisms and kernels.

Findings

Sofically presented systems are a broad class including all finitely presented systems.

Ma ext{n}é's theorem extends to sofically presented systems with finite topological dimension.

Conjugacy of one-dimensional sofically presented systems is undecidable.

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 S_1 by (relative) subshifts in S_2. We show that for all finitely generated infinite monoids M, M-SFT \subsetneq M-sofic \subsetneq M-SFT/SFT = M-sofic/SFT \subsetneq M-SFT/sofic = M-sofic/sofic, and that Ma\~n\'e's theorem about the dimension of expansive systems characterizes the virtually cyclic groups. In the case of one-dimensional actions, Ma\~ne'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 study the automorphism groups and periodic points of sofically presented systems. We also perform two case studies. First, in the context of $\beta$-shifts, we define the $\beta$-kernel -- the least subshift relation that identifies $1$ with its orbit. We give a classification of the $\beta$-shift/$\beta$-kernel pair as a function of $\beta$. Second, we revisit the classical study of toral automorphisms in our framework, and in particular for the classical "golden mean" toral automorphism we explicitly compute the kernel of the standard presentation.