On the Connectedness of Self-Affine Tiles

Let T be a self-affine tile in Rⁿ defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R², for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given.     

Classification of Integral Expanding Matrices and Self-Affine Tiles

   Abstract. Let T be a self-affine tile that is generated by an expanding integral matrix A and a digit set D . It is known that many properties of T are invariant under the Z -similarity of the matrix A . In [LW1] Lagarias and Wang showed that if A is a 2 × 2 expanding matrix with |det(A)| = 2 , then the Z -similar class is uniquely determined by the characteristic polynomial of A . This is not true if |det(A)| > 2. In this paper we give complete classifications of the Z -similar classes for the cases |det(A)| =3, 4, 5 . We then make use of the classification for |det(A)| =3 to consider the digit set D of the tile and show that μ(T) >0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this.     

Disk-Like Self-Affine Tiles in R 2

We give simple necessary and sufficient conditions for self-affine tiles in R ² to be homeomorphic to a disk. Received October 10, 2000, and in revised form February 16, 2001, and April 25, 2001. Online publication July 25, 2001.     

一类平面自仿射tiles的边界

考虑由扩张矩阵A=(?)及数字集D=(?):0≤i≤|p|-1,O≤j≤|q|一1(?)生成的自仿射tiles集T=T(A,D),其中p,q∈Z,|p|≥2,|q|≥2,通过对T中的元素进行分析,得到了计算T的边界的方法.     

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

Integral Self-Affine Tiles in {\Bbb R}^n Part II: Lattice Tilings

Let A be an expanding n × n integer matrix with |det (A)| = m. A standard digit set ${\cal D}Let A be an expanding n×n integer matrix with |det(A)|=m. Astandard digit set D for A is any complete set of coset representatives for? ⁿ /A(? ⁿ ). Associated to a given D is a setT (A, D), which is the attractor of an affine iterated function system, satisfyingT=∪ _d∈D (T+d). It is known thatT (A, D) tiles? ⁿ by some subset of? ⁿ . This paper proves that every standard digit set D gives a setT (A, D) that tiles? ⁿ with a lattice tiling.     

Self-Affine Sets and Graph-Directed Systems

Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ? Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Connectedness and Stability of Julia Sets of the Composition of Polynomials of the Form z2+cn

For a sequence (c_n) of complex numbers, the quadratic polynomialsf_cn(z) := z² + c_n and the sequence (F_n) of iterates F_n := f_cn...f_c1areconsidered. The Fatou set F_(cn) is by definition the set ofall z C^ such that (F_n) is normal in some neighbourhood ofz, while the complement of F_(cn) is called the Julia set J_(cn).The aim of this article is to study the connectedness and stabilityof the Julia set J_(cn) provided that the sequence (c_n) is bounded.     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

DIMENSIONS OF SELF-AFFINE SETS WITH OVERLAPS

The authors develop an algorithm to show that a class of self-affine sets with overlaps canbe viewed as sofic affine-invariant sets without overlaps,thus by using the results of [11]and[10],the Hausdorff and Minkowski dimensions are determined.     

Self-Affine Fractal Functions and Wavelet Series

We consider functions represented by series ∑_g  G c_gψ(g ^− 1(x)) of wavelet-type, where G is a group generated by affine functions L₁,…,L_n and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point.     

Measure of Self-Affine Sets and Associated Densities

Let \(B\) be an \(n\times n\) real expanding matrix and \(\mathcal {D}\) be a finite subset of \(\mathbb {R}^n\) with \(0\in \mathcal {D}\) . The self-affine set \(K=K(B,\mathcal {D})\) is the unique compact set satisfying the set-valued equation \(BK=\bigcup _{d\in \mathcal {D}}(K+d)\) . In the case where \(\#\mathcal D=|\det B|,\) we relate the Lebesgue measure of \(K(B,\mathcal {D})\) to the upper Beurling density of the associated measure \(\mu =\lim _{s\rightarrow \infty }\sum _{\ell _0, \ldots ,\ell _{s-1}\in \mathcal {D}}\delta _{\ell _0+B\ell _1+\cdots +B^{s-1}\ell _{s-1}}.\) If, on the other hand, \(\#\mathcal D<|\det B|\) and \(B\) is a similarity matrix, we relate the Hausdorff measure \(\mathcal {H}^s(K)\) , where \(s\) is the similarity dimension of \(K\) , to a corresponding notion of upper density for the measure \(\mu \) .     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

Properties of Some Overlapping Self-Similar and Some Self-Affine Measures

We generalize theorems of Peres and Solomyak about the abso- lute continuity resp. singularity of Bernoulli convolutions ([19], [16], [17]) to a broader class of self-similar measures on the real line. Using the dimension the- ory of ergodic measures (see [11] and [2]) we find a formula for the dimension of certain self-affine measures in terms of the dimension of the above mentioned self- similar measures. Combining these results we show the identity of Hausdorff and box-counting dimension of a special class of self-affine sets.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Copulae,Self-Affine Functions,and Fractal Dimensions

We use the fact that the functions defined on the unit interval whose graphs support a copula are those that are Lebesgue-measure-preserving in order to characterize self-affine functions whose graphs are the support of a copula. This result allows computation of the Hausdorff, packing, and box-counting dimensions. The discussion is applied to classic examples such as the Peano and Hilbert curves, and the results are extended to discontinuous self-affine functions.     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

Bernoulli多项式的积分多项式

首次研究了 Bernoulli多项式的积分多项式 .首先 ,给出这类多项式的定义和基本性质 ;其次 ,建立两类幂和多项式的相互关系 ;最后 ,介绍上述结果在求解自然数幂和公式方面的应用 .     

Euler Pseudoprime Polynomials and Strong Pseudoprime Polynomials

It is usual to emphasize the analogy between the integers and polynomials with coefficients in a finite field, comparing different notions in the two points of view. We introduce a particular rank one Drinfeld module to get an exponentiation for polynomials and then define the notions of Euler pseudoprimes and strong pseudoprimes for polynomials with coefficients in a finite field. As for the integers, we have Solovay–Strassen and Miller–Rabin tests for polynomials.