Vol. 288, No. 2, 2017

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ISSN: 0030-8730
Order on the homology groups of Smale spaces

Massoud Amini, Ian F. Putnam and Sarah Saeidi Gholikandi

Vol. 288 (2017), No. 2, 257–288
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
Authors
Massoud Amini
Department of Mathematics
Faculty of Mathematical Sciences
Tarbiat Modares University
Tehran 14115-134
Iran
Ian F. Putnam
Department of Mathematics and Statistics
University of Victoria
Victoria BC V8W 3R4
Canada
Sarah Saeidi Gholikandi
Department of Mathematics
Faculty of Mathematical Sciences
Tarbiat Modares University
Tehran 14115-134
Iran