Homology for one-dimensional solenoids

TL;DR

This paper computes a homology theory for one-dimensional generalized solenoids, extending the understanding of Smale spaces and their invariants in dynamical systems.

Contribution

It applies a homology theory for Smale spaces to compute invariants specifically for one-dimensional generalized solenoids, a class of hyperbolic dynamical systems.

Findings

Homology computed explicitly for one-dimensional generalized solenoids.

Extension of homology theory to a new class of dynamical systems.

Deeper understanding of the structure of Smale spaces.

Abstract

Smale space is a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The third author constructed a homology theory for Smale spaces which is based on Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R.F. Williams.