Topology of the Random Fibonacci Tiling Space

AbstractA weighted substitution is a substitution that has weights associated with each occurrence of the substituted symbols. It defines a tiling space that admits the translation and scaling operators; the translation is the additive ℝ-action and the scaling is the multiplicative G-action, where G is a closed multiplicative subgroup of ℝ+. We obtained necessary and sufficient conditions for the additive action to be strongly mixing and for it to be weakly mixing.

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Tiling space with the aid of the holomorph

Journal of Combinatorial Theory Series A ◽

10.1016/0097-3165(73)90018-6 ◽

1973 ◽

Vol 14 (2) ◽

pp. 167-172 ◽

Cited By ~ 2

Author(s):

James P Conlan

Keyword(s):

Tiling Space

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Cuntz-Krieger Algebras Associated with Hilbert $C^*$-Quad Modules of Commuting Matrices

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-22239 ◽

2015 ◽

Vol 117 (1) ◽

pp. 126 ◽

Cited By ~ 1

Author(s):

Kengo Matsumoto

Keyword(s):

Euclidean Algorithm ◽

C Algebra ◽

Commuting Matrices ◽

Tiling Space ◽

Positive Integers

Let $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^\kappa$ is transitive, the $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ is simple and purely infinite. In particular, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{[N],[M]}_{\kappa}}$ are computed by using the Euclidean algorithm.

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Complexity and cohomology for cut-and-projection tilings

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000194 ◽

2009 ◽

Vol 30 (2) ◽

pp. 489-523 ◽

Cited By ~ 12

Author(s):

ANTOINE JULIEN

Keyword(s):

Polynomial Growth ◽

Finitely Generated ◽

Complexity Function ◽

Tiling Space ◽

Dimension One ◽

Natural Complexity

AbstractWe consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponentαin terms of the ranks of certain groups which appear in the construction. We give bounds forα. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

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Factoring groups and tiling space

Aequationes Mathematicae ◽

10.1007/bf01832620 ◽

1973 ◽

Vol 9 (2-3) ◽

pp. 145-149 ◽

Cited By ~ 11

Author(s):

William Hamaker

Keyword(s):

Tiling Space

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Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.58 ◽

2016 ◽

Vol 38 (3) ◽

pp. 1086-1117 ◽

Cited By ~ 2

Author(s):

GREGORY R. MALONEY ◽

DAN RUST

Keyword(s):

Invariant Subspaces ◽

Finite Graph ◽

One Dimension ◽

One Dimensional ◽

Substitution Tiling ◽

Cohomological Invariants ◽

Tiling Spaces ◽

Tiling Space ◽

Cech Cohomology

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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