[Paper Review] On Flow Equivalence of Sofic Shifts

Rune Johansen|arXiv (Cornell University)|Oct 10, 2012

Advanced Operator Algebra Research30 references3 citations

TL;DR

This PhD thesis investigates flow equivalence of sofic shifts using structural properties of their graph covers. It introduces a canonical cover generalizing the left Fischer cover, proves layered structures in Krieger and past set covers, and applies these to classify beta-shifts up to flow equivalence and analyze the range of the Bowen-Franks invariant in renewal systems of finite type.

ABSTRACT

The flow equivalence of sofic shifts is examined using results about the structure of the corresponding covers. A canonical cover generalising the left Fischer cover to arbitrary sofic shifts is introduced and used to prove that the left Krieger cover and the past set cover of a sofic shift can be divided into natural layers. These results are used to find the range of a flow invariant and to investigate the ideal structure of the universal C^*-algebras associated to sofic shifts. The right Fischer covers of sofic beta-shifts are constructed, and it is proved that the covering maps are always 2 to 1. This is used to construct the corresponding fiber product covers and to classify these up to flow equivalence. Additionally, the flow equivalence of renewal systems is studied, and several partial results are obtained in an attempt to find the range of the Bowen-Franks invariant over the set of renewal systems of finite type. In particular, it is shown that the Bowen-Franks group is cyclic for every member of a class of renewal systems known to attain all entropies realised by shifts of finite type.

Motivation & Objective

• To understand the flow equivalence of sofic shifts, a natural generalization of shifts of finite type.
• To develop structural tools—particularly a canonical cover—for analyzing covers of sofic shifts.
• To classify right Fischer covers of sofic beta-shifts and determine their flow equivalence.
• To investigate the range of the Bowen-Franks invariant over renewal systems of finite type.
• To explore the ideal structure of universal C*-algebras associated with sofic shifts.

Proposed method

• Introduces a canonical cover that generalizes the left Fischer cover for arbitrary sofic shifts.
• Uses the canonical cover to decompose the left Krieger cover and past set cover into natural layers.
• Applies layer decomposition to analyze flow invariants and ideal lattices in C*-algebras.
• Constructs right Fischer covers for sofic beta-shifts and proves covering maps are always 2-to-1.
• Uses fiber product covers to classify beta-shifts up to flow equivalence.
• Employs symbol expansion techniques and computational tools (C++ and Maple) to analyze higher block shifts and adjacency matrices of renewal systems.

Experimental results

Research questions

• RQ1 What is the structure of the left Krieger cover and past set cover of a sofic shift, and can they be decomposed into natural layers?
• RQ2 How can the canonical cover be used to determine the range of a flow equivalence invariant?
• RQ3 What is the flow equivalence class of sofic beta-shifts, and how do their right Fischer covers behave?
• RQ4 What is the structure of the Bowen-Franks group for renewal systems of finite type, and can it be non-cyclic?
• RQ5 Can the range of the Bowen-Franks invariant over SFT renewal systems be fully characterized?

Key findings

• The canonical cover generalizes the left Fischer cover and enables a natural layer decomposition of the left Krieger and past set covers.
• Right Fischer covers of sofic beta-shifts are always 2-to-1, enabling construction of fiber product covers.
• Sofic beta-shifts are classified up to flow equivalence using their fiber product covers.
• For a class of renewal systems realizing all entropies of shifts of finite type, the Bowen-Franks group is cyclic.
• Renewal systems constructed as disjoint unions of a base list can yield non-cyclic Bowen-Franks groups, such as $\mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$.
• Computational tools based on Proposition 5.23 detect SFT renewal systems, though with limitations due to sufficiency, not necessity, of conditions.