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The homology core of matchbox manifolds and invariant measures. (English) Zbl 1412.37040

The eponymous matchbox manifolds are topological spaces that locally have the structure of \({\mathbb{R}}^d\times C\) for some Cantor set \(C\). These arise naturally in many settings, and particularly in dynamical systems, where they appear as hyperbolic attractors of smooth diffeomorphisms of manifolds and as tiling spaces associated to aperiodic tilings of Euclidean space with finite translational local complexity, in harmonic analysis as solenoids, and in many other places. The solenoid case gives some insight into the complexity of these objects for the following reason: solenoids of topological dimension \(d\) are (via Čech homology, which behaves well with respect to projective limits of tori), up to homeomorphism, in one-to-one correspondence with subgroups of \(\mathbb{Q}^d\) up to isomorphism, and for \(d\geqslant1\) these are notoriously difficult to classify in a precise set-theoretic sense by work of S. Thomas [J. Am. Math. Soc. 16, No. 1, 233–258 (2003; Zbl 1021.03043)]. Here the authors restrict to matchbox manifolds that admit an expansion as a projective limit of finite simplicial complexes with some hypotheses on the projection and bonding maps. Here a homeomorphism invariant of oriented matchbox manifolds, the so-called homology core, is introduced. The core may be considered as a monoid equipped with a representation in a linear space, and the representation is used to make some fine distinctions between examples. This work generalizes results on weak equivalence of matrices associated to one-dimensional substitution tiling systems by M. Barge and B. Diamond [Fundam. Math. 146, No. 2, 171–187 (1995; Zbl 0851.54037)] and R. Swanson and H. Volkmer [Ergodic Theory Dyn. Syst. 20, No. 2, 611–626 (2000; Zbl 0984.37019)].

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
28D15 General groups of measure-preserving transformations
52C23 Quasicrystals and aperiodic tilings in discrete geometry
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37A15 General groups of measure-preserving transformations and dynamical systems
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