Binary factors of shifts of finite type

TL;DR

This paper introduces two new classes of topological dynamical systems as factors of shifts of finite type, using graph-based data and binary expansions, and analyzes their algebraic invariants.

Contribution

It constructs novel factor systems of shifts of finite type with explicit algebraic invariants, including Smale space structures and K-theory computations.

Findings

Constructed systems are factors of shifts of finite type.

Established Smale space structure for one class.

Computed algebraic invariants from original graphs.

Abstract

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data is a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$-algebras associated to them, in terms of the pair of original graphs.