Quotient Cohomology of Certain 1- and 2-Dimensional Substitution Tiling Spaces

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

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Cohomology of substitution tiling spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000777 ◽

2009 ◽

Vol 30 (6) ◽

pp. 1607-1627 ◽

Cited By ~ 14

Author(s):

MARCY BARGE ◽

BEVERLY DIAMOND ◽

JOHN HUNTON ◽

LORENZO SADUN

Keyword(s):

Inverse Limit ◽

Direct Limit ◽

Higher Dimensions ◽

Rotation Group ◽

One Dimensional ◽

Substitution Tiling ◽

Tiling Spaces ◽

Tiling Space ◽

Cellular Complex

AbstractAnderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

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Cohomology of one-dimensional mixed substitution tiling spaces

Topology and its Applications ◽

10.1016/j.topol.2013.01.019 ◽

2013 ◽

Vol 160 (5) ◽

pp. 703-719 ◽

Cited By ~ 10

Author(s):

Franz Gähler ◽

Gregory R. Maloney

Keyword(s):

One Dimensional ◽

Substitution Tiling ◽

Tiling Spaces

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A complete invariant for the topology of one-dimensional substitution tiling spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385701001638 ◽

2001 ◽

Vol 21 (05) ◽

Cited By ~ 19

Author(s):

MARCY BARGE ◽

BEVERLY DIAMOND

Keyword(s):

One Dimensional ◽

Substitution Tiling ◽

Tiling Spaces

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Asymptotic structure in substitution tiling spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.118 ◽

2012 ◽

Vol 34 (1) ◽

pp. 55-94 ◽

Cited By ~ 2

Author(s):

MARCY BARGE ◽

CARL OLIMB

Keyword(s):

Branch Locus ◽

One Dimensional ◽

Substitution Tiling ◽

Tiling Spaces ◽

Markov Map ◽

Self Similar ◽

Tiling Space ◽

Definition Of ◽

Periodic Space ◽

Dimensional Simplicial Complex

AbstractEvery 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.

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On the frequency module of the hull of a primitive substitution tiling

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273321012572 ◽

2022 ◽

Vol 78 (1) ◽

Author(s):

April Lynne D. Say-awen ◽

Dirk Frettlöh ◽

Ma. Louise Antonette N. De Las Peñas

Keyword(s):

Statistical Properties ◽

Absolute Frequency ◽

Aperiodic Tilings ◽

Primitive Substitution ◽

Substitution Tiling ◽

Tiling Spaces ◽

The Family ◽

Frequency Module ◽

Substitution Tilings ◽

The Absolute

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

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