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Asymptotic structure in substitution tiling spaces

Published online by Cambridge University Press:  27 September 2012

MARCY BARGE
Affiliation:
Department of Mathematics, Montana State University, Bozeman, MT 59717, USA (email: [email protected])
CARL OLIMB
Affiliation:
Department of Mathematics, Southwest Minnesota State University, Marshall, MN 56258, USA (email: [email protected])

Abstract

Every sufficiently regular non-periodic space of tilings of $\mathbb {R}^d$ 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 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 two-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a zero- or one-dimensional simplicial complex.

Type
Research Article
Copyright
Copyright © 2012 Cambridge University Press 

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References

[AP]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated $C^*$-algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509537.CrossRefGoogle Scholar
[BD]Barge, M. and Diamond, B.. A complete invariant for the topology of one-dimensional substitution tiling spaces. Ergod. Th. & Dynam. Sys. 21 (2001), 13331358.Google Scholar
[BD2]Barge, M. and Diamond, B.. Proximality in Pisot tiling spaces. Fund. Math. 194 (2007), 191238.CrossRefGoogle Scholar
[BDH]Barge, M., Diamond, B. and Holton, C.. Asymptotic orbits of primitive substitutions. Theoret. Comput. Sci. 301 (2003), 439450.Google Scholar
[BDHS]Barge, M., Diamond, B., Hunton, J. and Sadun, L.. Cohomology of substitution tiling spaces. Ergod. Th. & Dynam. Sys. 30 (2010), 16071627.Google Scholar
[BDS]Barge, M., Diamond, B. and Swanson, R.. The branch locus in one-dimensional substitution tiling spaces. Fund. Math. 204 (2009), 215240.Google Scholar
[BKe]Barge, M. and Kellendonk, J.. Proximality and pure point spectrum for tiling dynamical systems, 2011, Preprint, arXiv:1108.4065.Google Scholar
[BL]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349(1) (1997), 55102.Google Scholar
[BS]Barge, M. and Swanson, R.. Rigidity in one-dimensional tiling spaces. Top. Appl. 154(17) (2007), 30953099.Google Scholar
[BSa]Barge, M. and Sadun, L.. Quotient cohomology for tiling space. New York J. Math. 17 (2011), 579599.Google Scholar
[BSW]Barge, M., Štimac, S. and Williams, R. F.. Pure discrete spectrum in substitution tiling spaces, 2011, Preprint, arXiv:1107.3598.Google Scholar
[FHK]Forrest, A., Hunton, J. and Kellendonk, J.. Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(758) (2002).Google Scholar
[G-S]Goodman-Strauss, C.. Matching rules and substitution tilings. Ann. of Math. (2) 147(1) (1998), 181223.Google Scholar
[H]Hochman, M.. Non-expansive directions for $\mathbb {Z}^2$ actions. Ergod. Th. & Dynam. Sys. 31(1) (2011), 91112.Google Scholar
[K]Kwapisz, J.. Rigidity and mapping class group for abstract tiling spaces. Ergod. Th. & Dynam. Sys. 31(6) (2011), 17451783.Google Scholar
[M]Mozes, S.. Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. 53 (1989), 139186.Google Scholar
[O]Olimb, C.. The branch locus for two dimensional substitution tiling spaces. PhD Thesis, Montana State University, 2010.Google Scholar
[S]Sadun, L.. Topology of Tiling Spaces. American Mathematical Society, Providence, RI, 2008.Google Scholar
[So1]Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695738.Google Scholar
[So2]Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265279.Google Scholar
[So3]Solomyak, B.. Eigenfunctions for substitution tiling systems. Adv. Stud. Pure Math. 49 (2007), 433454.Google Scholar
[W]Williams, R. F.. Classification of one-dimensional attractors. Proc. Symp. Pure Math. 14 (1970), 341361.Google Scholar