共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well
known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution
is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures
are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some
mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of
non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings,
the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling. 相似文献
2.
Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which
such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point
set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects
are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with
a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone
multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely
between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions),
being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary.
This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model
sets for lattice substitution systems begun by the first two authors of this paper. 相似文献
3.
Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which
such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point
set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects
are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with
a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone
multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely
between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions),
being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary.
This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model
sets for lattice substitution systems begun by the first two authors of this paper. 相似文献
4.
Marcy Barge Henk Bruin Leslie Jones Lorenzo Sadun 《Israel Journal of Mathematics》2012,188(1):281-300
We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and the first rational Čech cohomology is d-dimensional. We construct examples of such “homological Pisot” substitutions whose tiling flows do not have pure discrete
spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the
norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity
Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The
ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space. 相似文献
5.
Mark J. Nielsen 《Israel Journal of Mathematics》1993,81(1-2):129-143
We prove that a tiling of the plane by topological disks is locally finite at most boundary points of tiles, confirming a
conjecture by Valette. This comes by way of a much more general theorem on tilings of topological vector spaces. We also investigate
a question raised by Klee as to whether or not there is a tiling of separable Hilbert space by bounded convex tiles. We present
evidence to support the conjecture that the answer is negative. 相似文献
6.
Dirk Frettlh 《European Journal of Combinatorics》2008,29(8):1881
Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all primitive substitution tilings showing tiles in infinitely many orientations. 相似文献
7.
A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a nonexact substitution rule: the substitution
for a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with rhombi. We prove that a nonperiodic
repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Vorono\"{\i} tilings
up to similarity. To establish this characterization, we settle (in the planar case) a conjecture of E. A. Robinson by providing
an algorithm which converts any pseudo-self-similar tiling of R
2
into a self-similar tiling of R
2
in such a way that the translation dynamics associated to the two tilings are topologically conjugate.
Received June 20, 2000, and in revised form January 25, 2001. Online publication July 25, 2001. 相似文献
8.
We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value. 相似文献
9.
We study the spaces of rhombus tilings, i.e. the
graphs whose vertices are tilings of a fixed zonotope. Two
tilings are linked if one can pass from one to the other
by a local
transformation, called a flip.
We first use a decomposition method to encode rhombus tilings and
give a useful characterization for a sequence of bits to encode a
tiling.
We use the previous coding to get a canonical
representation of tilings, and two order structures on the space of
tilings.
In codimension 2 we prove that the two order structures are equal.
In larger codimensions we study the lexicographic case, and get an order
regularity result. 相似文献
10.
Matthew Foreman 《Topology and its Applications》2010,157(8):1404-1414
Many recent results about the classification problem for ergodic measure preserving transformations involve global considerations about spaces of measure preserving transformations. This paper surveys recent joint work with Dan Rudolph and Benjamin Weiss in determining when various spaces of measure preserving transformations are equivalent in the sense of conjugacy preserving Borel isomorphism and in having the same generic dynamical properties. 相似文献
11.
Shahar Mozes 《Journal d'Analyse Mathématique》1989,53(1):139-186
The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical
systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence
of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for
one-dimensional tilings the dynamical system always contains periodic points. 相似文献
12.
Eric Rémila 《Discrete and Computational Geometry》2005,34(2):313-330
We fix two rectangles with integer dimensions. We give a quadratic
time algorithm which, given a polygon F as input, produces a tiling
of F with translated copies of our rectangles (or indicates that there is no
tiling). Moreover, we prove that any pair of tilings can be linked by a sequence of
local transformations of tilings, called flips. This study is based on the
use of Conway’s tiling groups and extends the results of Kenyon and Kenyon (limited to the
case when each rectangle has a side of length 1). 相似文献
13.
Daniel Gonçalves 《Journal of Functional Analysis》2011,260(4):998-1019
We describe a method to compute the K-theory of the C?-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group K0 is always torsion free and give an example of a two-dimensional tiling such that K0 has torsion. 相似文献
14.
Juan García Escudero 《Mathematical Methods in the Applied Sciences》2011,34(5):587-594
A class of non‐periodic tilings in n‐dimensions is considered. They are based on one‐dimensional substitution tilings that force the border, a property preserved in the construction for higher dimensions. This fact allows to compute the integer?ech cohomology of the tiling spaces in an efficient way. Several examples are analyzed, some of them with PV numbers as inflation factors, and they have finitely or infinitely generated torsion‐free cohomologies. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
15.
Valérie Berthé Anne Siegel Wolfgang Steiner Paul Surer Jörg M. Thuswaldner 《Advances in Mathematics》2011,226(1):139
Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine). 相似文献
16.
Anatoly M. Vershik Pavel B. Zatitskiy Fedor V. Petrov 《Central European Journal of Mathematics》2013,11(3):379-400
We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ?-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ?-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded. 相似文献
17.
We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue spectrum in a suitable subspace. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemanzyk, D. Rudolph and the second author in the one-dimensional setting. Our proofs rely on the use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional torus. 相似文献
18.
Natalie Priebe Frank 《Expositiones Mathematicae》2008,26(4):295-326
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions. 相似文献
19.
We produce an algorithm that is optimal with respect to both space and execution time to generate all the lozenge (or domino) tilings of a hole-free, general-shape domain given as input.We first recall some useful results, namely the distributive lattice structure of the space of tilings and Thurston's algorithm for constructing a particular tiling. We then describe our algorithm and study its complexity. 相似文献
20.
Anderson 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. For one-dimensional substitution tiling spaces, we describe a modification of the Anderson-Putnam complex on collared tiles that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology.