Order on the homology groups of Smale spaces

Massoud Amini, Ian F. Putnam and Sarah Saeidi Gholikandi

Vol. 288 (2017), No. 2, 257–288

DOI: 10.2140/pjm.2017.288.257

Abstract

Smale spaces were defined by D. Ruelle to describe the properties of the basic sets of an Axiom A system for topological dynamics. One motivation for this was that the basic sets of an Axiom A system are merely topological spaces and not submanifolds. One of the most important classes of Smale spaces is shifts of finite type. For such systems, W. Krieger introduced a pair of invariants, the past and future dimension groups. These are abelian groups, but are also with an order which is an important part of their structure. The second author showed that Krieger’s invariants could be extended to a homology theory for Smale spaces. In this paper, we show that the homology groups on Smale spaces (in degree zero) have a canonical order structure. This extends that of Krieger’s groups for shifts of finite type.

Keywords

hyperbolic dynamical systems, dimension groups, homology, ordered groups, Smale spaces

Mathematical Subject Classification 2010

Primary: 37D20, 55N35

Secondary: 37B10, 06F15

Milestones

Received: 18 January 2016

Revised: 7 December 2016

Accepted: 7 December 2016

Published: 28 April 2017