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The homology core of matchbox manifolds and invariant measures

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journal contribution
posted on 12.02.2018, 17:07 by Alex Clark, John Hunton
Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological invariants related to their expansions as an inverse sequence of simplicial complexes. These invariants are related to corresponding inverse sequences of groups arising from applying the top--dimension homology to these sequences. In many cases this leads to a computable invariant based on an inverse sequence of matrices. Significantly, we show that when the space is obtained by suspending a topologically transitive action of the fundamental group $\G$ of a closed orientable on a zero--dimensional compact space this invariant at the same time corresponds to the space of Borel measures on the Cantor set which are invariant under the action of $\G$. This leads to connections between the rank of homology groups we consider 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.

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Citation

Transactions of the American Mathematical Society, 2018

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

Transactions of the American Mathematical Society

Publisher

American Mathematical Society

issn

0002-9947

eissn

1088-6850

Acceptance date

16/09/2017

Copyright date

2018

Available date

30/10/2018

Publisher version

http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2018-07398-6/home.html

Language

en

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