Cohomology in one-dimensional substitution tiling spaces

Barge, Marcy; Diamond, Beverly

doi:10.1090/s0002-9939-08-09225-3

Cited by 31 publications

(48 citation statements)

References 7 publications

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“…However, the advantages afforded to the Barge-Diamond method are less apparent when there exist bounded letters in the alphabet. When all letters are expanding and the substitution is aperiodic, a very similar argument to the original proof presented by Barge and Diamond [3] will carry through, and one can then apply the usual method of replacing the induced substitution on the BD-complex with a homotopic map which is simplicial on the vertex-edges.…”

Section: 3mentioning

confidence: 94%

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Maloney¹,

Rust²

2016

Ergod. Th. Dynam. Sys.

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ABSTRACT. We study the topology and dynamics of subshifts and tiling spaces associated to nonprimitive 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 characterisation 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 CW complex 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 that theČech cohomology of the tiling space alone.1. PRELIMINARIES 1.1. Outline. The goal of this work is to study one-dimensional tiling spaces arising from nonprimitive substitution rules, in terms of the topology, dynamics, and cohomology. This study naturally divides into two cases: the case where the tiling space is minimal, and the case where it is non-minimal. The minimal case is treated in Section 2, where 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. In particular, all aperiodic substitutions will be seen to be tame. By aperiodic, we mean that the subshift of the substitution has no periodic orbits. The first main result is Theorem 2.9, which gives a characterisation of tameness. This theorem is used to prove the following result.Theorem 2.1. Let ϕ be a minimal substitution with non-empty minimal subshift X ϕ . There exists an alphabet Z and a primitive substitution θ on Z such that X θ is topologically conjugate to X ϕ . This is similar to, but slightly stronger than, a result from the section on Open problems and perspectives (Section 6.2) of [10].Examples and applications based on this section are then given in Section 3.

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Section: 3mentioning

confidence: 94%

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Maloney¹,

Rust²

2016

Ergod. Th. Dynam. Sys.

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“…Consider a Z d action on a Cantor set C. For n ∈ Z d , we denote the action of n on x by φ n (x). A 1-cocycle with values in Z is a continuous map θ : C × Z d → Z such that, for all n, m ∈ Z d and all χ ∈ C, 1 (1) θ(χ, n + m) = θ(χ, n) + θ(φ n (χ), m).…”

Section: Z D Actions Cochains and Cohomologymentioning

confidence: 99%

Small Cocycles, Fine Torus Fibrations, and a $$\varvec{\mathbb {Z}^{2}}$$ Z 2 Subshift with Neither

Clark

Sadun

2017

Ann. Henri Poincaré

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Abstract.Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free Z d actions on Cantor sets admit "small cocycles." These represent classes in H 1 that are mapped to small vectors in R d by the Ruelle-Sullivan (RS) map. We show that there exist Z 2 actions where no such small cocycles exist, and where the image of H 1 under RS is Z 2 . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of "virtual eigenvalues," i.e., elements of R d that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.

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“…Consider the irreducible substitution σ on three letters, with σ(a) = abca, σ(b) = abb, and σ(c) = ac. Since each substituted letter begins with a, the first cohomology of the resulting tiling space is the direct limit of the transpose of the substitution matrix M = 2 1 1 1 2 0 1 0 1 (see [5]). The eigenvalues of this matrix are λ 1 ≈ 3.247, λ 2 ≈ 1.555, and λ 3 ≈ 0.1981.…”

Section: Scrambled Fibonacci Tilingsmentioning

confidence: 99%

Meyer sets, topological eigenvalues, and Cantor fiber bundles

Kellendonk

Sadun

2013

Journal of the London Mathematical Society

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The diffraction spectrum of a point pattern is very closely related to the dynamical spectrum of an associated dynamical system. This dynamical spectrum is invariant under topological conjugacy and measurable conjugacy, and in particular under a large class of shape deformations. Using measure theory and topology, we construct a pure-point diffractive set, with finite local complexity, that is not a Meyer set. This provides a counterexample to a famous conjecture of Lagarias.

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Cohomology in one-dimensional substitution tiling spaces

Cited by 31 publications

References 7 publications

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Small Cocycles, Fine Torus Fibrations, and a $$\varvec{\mathbb {Z}^{2}}$$ Z 2 Subshift with Neither

Meyer sets, topological eigenvalues, and Cantor fiber bundles

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