Functorial properties of Putnam's homology theory for Smale spaces

TL;DR

This paper explores the functorial aspects of Putnam's homology theory for Smale spaces, emphasizing the importance of conjugacy conditions and extending key lemmas to broader classes of dynamical systems.

Contribution

It introduces a conjugacy condition to ensure functoriality and generalizes Putnam's Pullback Lemma from shifts of finite type to non-wandering Smale spaces.

Findings

Conjugacy condition is necessary for functoriality.

Generalization of Pullback Lemma to non-wandering Smale spaces.

Examples illustrating the need for additional hypotheses.

Abstract

We investigate functorial properties of Putnam's homology theory for Smale spaces. Our analysis shows that the addition of a conjugacy condition is necessary to ensure functoriality. Several examples are discussed that elucidate the need for our additional hypotheses. Our second main result is a natural generalization of Putnam's Pullback Lemma from shifts of finite type to non-wandering Smale spaces.