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.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

Restricted orbit equivalence for ergodic ${\Bbb Z}^{d}$ actions I

1997 ◽  
Vol 17 (5) ◽  
pp. 1083-1129 ◽  
Author(s):  
JANET WHALEN KAMMEYER ◽  
DANIEL J. RUDOLPH
Keyword(s):  
Equivalence Relation ◽  
Dimensional Case ◽  
Orbit Equivalence ◽  
One Dimensional ◽  
Ergodic Transformations ◽  
Higher Dimensional ◽  
The One ◽  
Equivalence Invariant ◽  
Classical Entropy

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

One-dimensional Hardy-type inequalities in many dimensions

1998 ◽  
Vol 128 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Gord Sinnamon
Keyword(s):  
Integral Operators ◽  
Weighted Norm ◽  
Weighted Inequalities ◽  
Weighted Norm Inequalities ◽  
Averaging Operators ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Higher Dimensional ◽  
The One

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.

CORRELATION AND SPECTRAL PROPERTIES OF MULTIDIMENSIONAL THUE–MORSE SEQUENCES

2007 ◽  
Vol 17 (04) ◽  
pp. 1265-1303 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Spectral Properties ◽  
Three Dimensional ◽  
Number System ◽  
Dimensional Case ◽  
Binary Number ◽  
Singular Continuous Spectrum ◽  
One Dimensional ◽  
Number Systems ◽  
Higher Dimensional ◽  
The One

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤm instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

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.

A Plessner Decomposition Along Transverse Curves

1988 ◽  
Vol 40 (5) ◽  
pp. 1243-1255
Author(s):  
Frank Beatrous ◽  
Songying Li
Keyword(s):  
Harmonic Functions ◽  
Entire Plane ◽  
Cartesian Products ◽  
One Dimensional ◽  
Cluster Set ◽  
Higher Dimensional ◽  
Distinguished Boundary ◽  
The One ◽  
By Products ◽  
Null Set

A classical theorem of Plessner [6] asserts that any holomorphic function f on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, f has a non-tangential limit, and at each point of the second piece, the cluster set of f in any Stolz angle is the entire plane. Higher dimensional versions of this result were first obtained by Calderon [2], who considered holomorphic functions on Cartesian products of half-planes. In this setting, an exact analogue of the one-dimensional result is obtained, in which the circle is replaced by the distinguished boundary, and the Stolz angles are replaced by products of cones in the coordinate half-planes. The ideas of Calderon were further developed by Rudin [8, pp. 79-83], who considered holomorphic and invariant harmonic functions in the ball of Cn. In this case, the circle is replaced by the unit sphere, and the Stolz angles are replaced by the approach regions of Korányi [4].

On the Ordering Conditions for Self-Organizing Maps

1995 ◽  
Vol 7 (2) ◽  
pp. 284-289 ◽  
Author(s):  
Marco Budinich ◽  
John G. Taylor
Keyword(s):  
Geometric Interpretation ◽  
Dimensional Case ◽  
Feature Maps ◽  
Self Organizing Maps ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One ◽  
Ordered State ◽  
Self Organizing

We present a geometric interpretation of ordering in self-organizing feature maps. This view provides simpler proofs of Kohonen ordering theorem and of convergence to an ordered state in the one-dimensional case. At the same time it explains intuitively the origin of the problems in higher dimensional cases. Furthermore it provides a geometric view of the known characteristics of learning in self-organizing nets.

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

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