Subshifts on groups and computable analysis

TL;DR

The thesis develops a computable-analysis framework for subshifts on finitely generated groups, connecting dynamics with recursion theory. It proves that any effective dynamical system on a general metric space is a topological factor of a zero-dimensional computable system, enabling broad simulations of dynamics by SFTs through an effective zero-dimensional extension. It then introduces Medvedev degrees as a dynamical invariant for subshifts, classifies their possible values for several group families, and analyzes how these degrees transfer under group-theoretic constructions and quasi-isometries, linking to the domino problem and isolated points in spaces of subshifts. A parallel computable-analysis program develops computable structures for the space of G-subshifts S(G), studying isolated points, entropy, and the embedding into computable hyperspaces. Collectively, the work extends simulation results to non-zero-dimensional settings, clarifies the algorithmic content of subshifts across groups, and lays a computable foundation for understanding the space of group actions in topological dynamics.

Abstract

The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and recursion theory. Different questions about the dynamics of these systems have been answered in recursion-theoretical terms. In this work we further explore this connection. We use the framework of computable analysis to explore the class of effective dynamical systems on metric spaces, and relate these systems to subshifts of finite type (SFTs) on groups. We prove that every effective dynamical system on a general metric space is the topological factor of an effective dynamical system with topological dimension zero. We combine this result with existing simulation results to obtain new examples of systems that are factors of SFTsWe also study a conjugacy invariant for subshifts on groups called Medvedev degree. This invariant is a complexity measure of algorithmic nature. We develop the basic theory of these degrees for subshifts on arbitrary finitely generated groups. Using these tools we are able to classify the values that this invariant attains for SFTs and other classes of subshifts on several groups. Furthermore, we establish a connection between these degrees and the distribution of isolated points in the space of all subshifts. Motivated by the study of Medvedev degrees of subshifts, we also consider translation-like actions of groups on graphs. We prove that every connected, locally finite, and infinite graph admits a translation by $\mathbb{Z}$, and that this action can be chosen transitive exactly when the graph has one or two ends. This generalizes a result of Seward about translation-like actions of $\mathbb{Z}$ on finitely generated groups.

Figures (3)

• Figure 1: Many decorative tilings can be seen as $\Z^2$-SFTs. Given a finite set of decorated square tiles of the same size, we can define an SFT whose alphabet is the set of tiles, and with the local rule that adjacent tiles preserve the intended decoration. Conversely, every SFT on $\Z^2$ is topologically conjugate to one defined by square decorated tiles. This identification between SFTs and tilings is also valid on finitely generated groups. One just needs to fix a generating set for the group, and then use it to express the adjacency relations. For this purpose it is also convenient to consider Wang tiles, a standard model for decorated tiles proposed by Wang wang_proving_1961.
• Figure 2: A representation of the actions $\ZZ\curvearrowright X$ and $\ZZ\curvearrowright Y$.
• Figure :

Theorems & Definitions (338)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Remark 2.5
• Definition 2.6: rabin_computable_1960
• Proposition 2.7
• proof : Proof sketch
• Remark 2.8
• Definition 2.9