The homology core of matchbox manifolds and invariant measures

Clark, Alex; Hunton, John

Abstract

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group $ \Gamma $ of a closed orientable manifold on a zero-dimensional compact space $ Z$, this invariant corresponds to the space of finite Borel measures on $ Z$ which are invariant under the action of $ \Gamma $. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.

Citation

Clark, A., & Hunton, J. (2019). The homology core of matchbox manifolds and invariant measures. Transactions of the American Mathematical Society, 371(3), 1771-1793. https://doi.org/10.1090/tran/7398

• Journal Article Type: Article
• Acceptance Date: Dec 16, 2016
• Online Publication Date: Oct 11, 2018
• Publication Date: 2019-02
• Deposit Date: May 5, 2016
• Publicly Available Date: Oct 11, 2018
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 371
• Issue: 3
• Pages: 1771-1793
• DOI: https://doi.org/10.1090/tran/7398
• Public URL: https://durham-repository.worktribe.com/output/1405498
• Related Public URLs: http://arxiv.org/abs/1601.00920