scholarly journals Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

2016 ◽  
Vol 38 (3) ◽  
pp. 1086-1117 ◽  
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.

scholarly journals Cohomology of substitution tiling spaces

2009 ◽  
Vol 30 (6) ◽  
pp. 1607-1627 ◽  
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.

scholarly journals Asymptotic structure in substitution tiling spaces

2012 ◽  
Vol 34 (1) ◽  
pp. 55-94 ◽  
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.

scholarly journals Solutions of the nonlinear paraxial equation due to laser plasma–interactions

2004 ◽  
Vol 22 (1) ◽  
pp. 69-74 ◽  
Author(s):  
F. OSMAN ◽  
R. BEECH ◽  
H. HORA
Keyword(s):  
Laser Plasma ◽  
Theoretical Study ◽  
Semiclassical Limit ◽  
General Setting ◽  
One Dimension ◽  
One Dimensional ◽  
Laser Plasma Interactions ◽  
Weak Limits ◽  
The One

This article presents a numerical and theoretical study of the generation and propagation of oscillation in the semiclassical limit ħ → 0 of the nonlinear paraxial equation. In a general setting of both dimension and nonlinearity, the essential differences between the “defocusing” and “focusing” cases are observed. Numerical comparisons of the oscillations are made between the linear (“free”) and the cubic (defocusing and focusing) cases in one dimension. The integrability of the one-dimensional cubic nonlinear paraxial equation is exploited to give a complete global characterization of the weak limits of the oscillations in the defocusing case.

scholarly journals Exact regularity and the cohomology of tiling spaces

2011 ◽  
Vol 31 (6) ◽  
pp. 1819-1834 ◽  
Author(s):  
LORENZO SADUN
Keyword(s):  
Convergence Rate ◽  
One Dimension ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Tiling Spaces ◽  
Ergodic Limit

AbstractExact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate. If T is also one dimensional, we put constraints on the measure of any cylinder set in Ω.

scholarly journals Cohomology of one-dimensional mixed substitution tiling spaces

2013 ◽  
Vol 160 (5) ◽  
pp. 703-719 ◽  
Author(s):  
Franz Gähler ◽  
Gregory R. Maloney
Keyword(s):  
One Dimensional ◽  
Substitution Tiling ◽  
Tiling Spaces

Multidimensional Signal Analysis

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
One Dimension ◽  
One Dimensional ◽  
Multidimensional Signals ◽  
Multidimensional Signal ◽  
The Fourier Transform

N dimensional signals are characterized as values in an N dimensional space. Each point in the space is assigned a value, possibly complex. Each dimension in the space can be discrete, continuous, or on a time scale. A black and white movie can be modelled as a three dimensional signal.Acolor picture can be modelled as three signals in two dimensions, one each, for example, for red, green and blue. This chapter explores Fourier characterization of different types of multidimensional signals and corresponding applications. Some signal characterizations are straightforward extensions of their one dimensional counterparts. Others, even in two dimensions, have properties not found in one dimensional signals. We are fortunate to be able to visualize structures in two, three, and sometimes four dimensions. It assists in the intuitive generalization of properties to higher dimensions. Fourier characterization of multidimensional signals allows straightforward modelling of reconstruction of images from their tomographic projections. Doing so is the foundation of certain medical and industrial imaging, including CAT (for computed axial tomography) scans. Multidimensional Fourier series are based on models found in nature in periodically replicated crystal Bravais lattices [987, 1188]. As is one dimension, the Fourier series components can be found from sampling the Fourier transform of a single period of the periodic signal. The multidimensional cosine transform, a relative of the Fourier transform, is used in image compression such as JPG images. Multidimensional signals can be filtered. The McClellan transform is a powerful method for the design of multidimensional filters, including generalization of the large catalog of zero phase one dimensional FIR filters into higher dimensions. As in one dimension, the multidimensional sampling theorem is the Fourier dual of the Fourier series. Unlike one dimension, sampling can be performed at the Nyquist density with a resulting dependency among sample values. This property can be used to reduce the sampling density of certain images below that of Nyquist, or to restore lost samples from those remaining. Multidimensional signal and image analysis is also the topic of Chapter 9 on time frequency representations, and Chapter 11 where POCS is applied signals in higher dimensions.

scholarly journals Geometric realization for substitution tilings

2012 ◽  
Vol 34 (2) ◽  
pp. 457-482 ◽  
Author(s):  
MARCY BARGE ◽  
JEAN-MARC GAMBAUDO
Keyword(s):  
Geometric Realization ◽  
Toral Automorphism ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Substitution Tilings ◽  
Hyperbolic Toral Automorphism ◽  
Tiling Space ◽  
The One ◽  
Cech Cohomology ◽  
Family Condition

AbstractGiven an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

Characterization of Minimum Route MTM in One-Dimensional Multi-Hop Wireless Networks

2009 ◽  
Vol E92-A (9) ◽  
pp. 2227-2235 ◽  
Author(s):  
Kazuyuki MIYAKITA ◽  
Keisuke NAKANO ◽  
Masakazu SENGOKU ◽  
Shoji SHINODA
Keyword(s):  
Wireless Networks ◽  
One Dimensional
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