The dynamical properties of Penrose tilings

Robinson, E. Arthur

doi:10.1090/s0002-9947-96-01640-6

Cited by 59 publications

(46 citation statements)

References 16 publications

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“…We extend this idea by showing, for substitution tilings, that ( , d, ω) is a Smale space. First we prove that the inflation map is topologically mixing; Robinson proved this for the Penrose tilings in a different way [ERob2]. PROPOSITION 3.1. ω is a topologically mixing homeomorphism of ( , d).…”

Section: Dynamicsmentioning

confidence: 89%

“…Here [Ken] have noted that the inflation map is hyperbolic: with each inflation, tilings that agree around the origin become exponentially closer together, while those that are close translations of each other become exponentially farther apart. Robinson has used an algebraic theory of de Bruijn to prove some measure theoretic results about this for the Penrose tilings [ERob2]. We extend this idea by showing, for substitution tilings, that ( , d, ω) is a Smale space.…”

Section: Dynamicsmentioning

confidence: 90%

“…Then we prove that the resulting dynamical system is a Smale space. Other authors Substitution tilings 513 [S,ERob2] study the different tiling dynamical system consisting of and the action of R d by translation. We shall see that these two dynamical systems are closely related in that the unstable equivalence classes in the Smale space are the orbits in the tiling dynamical system.…”

Section: Dynamicsmentioning

confidence: 99%

“…We study the dynamics of substitution tiling systems, the most famous example being the Penrose tilings of the plane. These have been studied by many other authors [Ga,GS,Ken,Mo,Ra1,ERob2,Rud1,S]. We work in R d , d-dimensional Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

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Topological invariants for substitution tilings and their associated $C^\ast$-algebras

Anderson

Putnam

1998

Ergod. Th. Dynam. Sys.

229

480

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We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

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Section: Dynamicsmentioning

confidence: 89%

Section: Dynamicsmentioning

confidence: 90%

Section: Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological invariants for substitution tilings and their associated $C^\ast$-algebras

Anderson

Putnam

1998

Ergod. Th. Dynam. Sys.

229

480

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“…In the abelian case, Schlottmann has established that there exists a continuous

G

‐equivariant surjective almost everywhere one‐to‐one map

X_{P_{0}} \to Y

known as Schlottmann's generalized torus parametrization . If

P_{0}

is non‐uniform, then no such map can exist, since

X_{P_{0}}

contains a fixpoint and is compact, wheras

Y

does not contain a fixpoint and is non‐compact, so the best one can hope for is to obtain a parametrization map between the punctured hull

X_{P_{0}}^{\times} : = X_{P_{0}} ∖ false{\emptyset false}

and

Y

.…”

Section: Introductionmentioning

confidence: 99%

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Björklund

Hartnick

Pogorzelski

2017

Proc. London Math. Soc.

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ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

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