Asymptotic structure in substitution tiling spaces

Details Asymptotic structure in substitution tiling spaces Asymptotic structure in substitution tiling spaces

2011-01-25

arxiv.org/abs/1101.4902v1

Mathematics

Dynamical Systems

33 pages, 5 figures

Scientific paper

Every sufficiently regular space of tilings of $\R^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity `starts'. This leads to the definition of the {\em branch locus} of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the `asymptotic in at least a half-space' behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a 2-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a 1-dimensional simplicial complex.

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