[Paper Review] One-sided shift spaces over infinite alphabets

William Ott, Mark Tomforde+1 more|arXiv (Cornell University)|Sep 8, 2012

semigroups and automata theory10 citations

TL;DR

This paper introduces a new framework for one-sided shift spaces over infinite alphabets by using the one-point compactification of the alphabet to ensure compactness, enabling the application of classical symbolic dynamics techniques. It defines three classes of shift morphisms generalizing shifts of finite type and proves that row-finite shift spaces admit a sliding block code characterization; as a key application, it establishes that conjugate edge shifts of countable directed graphs imply isomorphism of their associated C*-algebras and groupoids.

ABSTRACT

We define a notion of (one-sided) shift spaces over infinite alphabets. Unlike many previous approaches to shift spaces over countable alphabets, our shift spaces are compact Hausdorff spaces. We examine shift morphisms between these shift spaces, and identify three distinct classes that generalize the shifts of finite type. We show that when our shift spaces satisfy a property that we call "row-finite", then shift morphisms on them may be identified with sliding block codes. As applications, we show that if two (possibly infinite) directed graphs have edge shifts that are conjugate, then the groupoids of the graphs are isomorphic, and the C*-algebras of the graphs are isomorphic.

Motivation & Objective

• To overcome the lack of compactness in classical shift spaces over infinite alphabets, which hinders the application of standard dynamical systems techniques.
• To define a new class of one-sided shift spaces over infinite alphabets that are compact Hausdorff, preserving the topological structure essential for symbolic dynamics.
• To generalize the concept of shifts of finite type to infinite alphabets through three distinct classes of shift morphisms.
• To establish a correspondence between sliding block codes and shift morphisms in the row-finite case, extending classical results to infinite settings.
• To apply the framework to C*-algebras and groupoids of countable directed graphs, proving isomorphism invariants under conjugacy of edge shifts.

Proposed method

• The full shift space is redefined using the one-point compactification $\mathcal{A}_\infty = \mathcal{A} \cup \{\infty\}$, making the product space $X_\mathcal{A} = \mathcal{A}_\infty^\mathbb{N}$ compact.
• Sequences containing $\infty$ are interpreted as finite sequences in $\mathcal{A}$, allowing the construction of a compact topology on the space of infinite and finite sequences.
• Shift spaces are defined as closed, $\sigma$-invariant subsets of $X_\mathcal{A}$, inheriting compactness from the full shift.
• Three classes of shift morphisms are introduced that generalize shifts of finite type, based on finite forbidden patterns and their closure properties.
• For row-finite shift spaces, shift morphisms are shown to correspond exactly to sliding block codes, generalizing the finite-alphabet result.
• The framework is applied to directed graphs: conjugacy of edge shifts implies isomorphism of their associated $C^*$-algebras and groupoids, via explicit isomorphisms constructed using higher block graphs and $1$-block conjugacies.

Experimental results

Research questions

• RQ1 Can a compact topology be defined on one-sided shift spaces over infinite alphabets, overcoming the non-compactness of the standard product topology?
• RQ2 How can the classical theory of shifts of finite type be generalized to infinite alphabets while preserving key structural properties?
• RQ3 Under what conditions do shift morphisms on infinite-alphabet shift spaces correspond to sliding block codes?
• RQ4 What invariants of directed graphs are preserved under conjugacy of their edge shifts?
• RQ5 Can conjugacy of edge shifts of countable directed graphs imply isomorphism of their associated $C^*$-algebras and groupoids?

Key findings

• The full shift space over an infinite alphabet is redefined using the one-point compactification of the alphabet, resulting in a compact Hausdorff space.
• Shift spaces defined as closed, $\sigma$-invariant subsets of this compact space are themselves compact, enabling the use of classical dynamical systems tools.
• Three distinct classes of shift morphisms are identified that generalize the notion of shifts of finite type to infinite alphabets.
• In the row-finite case, shift morphisms are precisely those that can be represented as sliding block codes, extending the finite-alphabet characterization.
• If two countable directed graphs have conjugate edge shifts, then their associated $C^*$-algebras are isomorphic, and their groupoids are isomorphic.
• The isomorphism between $C^*$-algebras is explicitly constructed via higher block graphs and $1$-block conjugacies, generalizing known results to the infinite setting.