A homology theory for Smale spaces: a summary

TL;DR

This paper develops a homology theory for Smale spaces, linking topological dynamics with algebraic invariants, and provides a Lefschetz formula connecting trace data to periodic points.

Contribution

It introduces a novel homology framework for Smale spaces, extending the dimension group concept and establishing a Lefschetz formula for these systems.

Findings

Homology theory for Smale spaces is formulated.

A Lefschetz formula relating trace data and periodic points is derived.

The theory generalizes the dimension group for shifts of finite type.

Abstract

We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the special case of shifts of finite type. This theory provides a Lefschetz formula relating trace data with the number of periodic points of the system.