两个齐次Cantor集的算术和

§ 1　IntroductionThe book[1 ] and the references therein show thatthe structure of arithmetic sums ofCantor sets is relevantto natural questions in smooth dynamics.Palis and Takens[1 ] askedabout the structure of the sums of two Cantor sets and conjectured that“typically” theyhave either zero Lebesgue measure or contained intervals. In 1 997,Solomyka[2 ] showedthatfor eachγ∈ 0 ,12 ,the set Kγ+Kλ(where Kλ,Kγis the middle-α Cantorset forα=1 -2λ or 1 -2γ) of two centered Cantor s…     

Sum of Cantor sets: Self-similarity and measure

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).

     

Quasisymmetrically minimal uniform Cantor sets

The uniform Cantor set E(n,c) of Hausdorff dimension 1, defined by a bounded sequence n of positive integers and a gap sequence c, is shown to be minimal for 1-dimensional quasisymmetric maps.     

Intersecting nonhomogeneous Cantor sets with their translations

A scheme is given to compute the Hausdorff dimensions for the intersection of a class of nonhomogeneous Cantor sets with their translations.     

Self-similar structure on intersections of triadic Cantor sets

Let C be the triadic Cantor set. We characterize the all real number α such that the intersection C∩(C+α) is a self-similar set, and investigate the form and structure of the all iterated function systems which generate the self-similar set.     

Optimal quantization for dyadic homogeneous Cantor distributions

For a large class of dyadic homogeneous Cantor distributions in ?, which are not necessarily self‐similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non‐existence of the quantization coefficient. The class contains all self‐similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

Suppose C _r = (r C _r ) ∪ (r C _r + 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C _r ) is the set of all bi-Lipschitz automorphisms on C _r . This paper proves that there exists f* ∈ Aut(C _r ) such that

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.     

On the geometry of random Cantor sets and fractal percolation

Random Cantor sets are constructions which generalize the classical Cantor set, middle third deletion being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]^d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1] ^e . This answers a question of Dekking and Meester,⁽⁹⁾ (b) The special case of fractal percolation arises when the substitution is as follows: The cube [0,1] ^d is divided intoM ^d subcubes of side-lengthM ^–, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep _c(M, d) ofp, marking the existence of crossings of [0,1] ^d contained in the limit set, satisfiesp _c(M, d)p _c(d) asM, wherep _c(d) is the critical probability of site percolation on a latticeL ^d obtained by adding certain edges to the hypercubic lattice ^d . This result generalizes in an unexpected way a finding of Chayes and Chayes,⁽⁴⁾ who studied the special case whend=2.     

Hausdorff dimension of perturbed Cantor sets without some boundedness condition

A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set. This revised version was published online in August 2006 with corrections to the Cover Date.     

Discrete and Cantor Spectrum for Neumann Laplacians of Combs

Extending results of [HSS] on the construction of Neumann Laplacians of combs with given essential spectrum S = S [0,∞], we show that, in addition to the essential spectrum, a bounded sequence of discrete eigenvalues can be prescribed. More generally, we find that certain types of spectra can be constructed precisely, in a bounded interval [0, s].     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.     

若干个齐次对称康托集的交

Γ是齐次对称康托集,对n个实数t_1,…,t_n讨论了交集Γ∩(Γ+t_1)∩…∩(Γ+t_n)≠(?)的条件,以及计算出Γ∩(Γ+t_1)∩…∩(Γ+t_n)的Hausdorff维数的精确表达式.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Liouville numbers, Rajchman measures, and small Cantor sets

We show that the set of Liouville numbers carries a positive measure whose Fourier transform vanishes at infinity. The proof is based on a new construction of a Cantor set of Hausdorff dimension zero supporting such a measure.

     

Hausdorff dimension of homogeneous perfect sets

Summary We introduce the notion of homogeneous perfect sets as a generalization of Cantor type sets and determine their exact dimension based on the length of their fundamental intervals and the gaps between them. Some earlier results regarding the dimension of Cantor type sets are shown to be special cases of our main theorem.     

Hausdorff measures for a class of homogeneous cantor sets

We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.     

Packing dimensions of homogeneous perfect sets

In [10], the notion of homogeneous perfect sets as a generalization of Cantor type sets is introduced and their Hausdorff and lower box-counting dimensions are studied. In this paper, we determine their exact packing and upper box-counting dimensions based on the length of their fundamental intervals and the gaps between them. Some known results concerning the dimensions of Cantor type sets are generalized. This work was supported by NSFC (10571138).     

Cantor集的性质及应用

Cantor集是实函数论中一类重要的集合.本文从定义、性质及应用三方面研究了Cantor集.目的是帮助初学者对Cantor集有一个较全面的认识.     

一类齐次Moran集维数的不连续性

丰德军等人在他们的相关的论文中介绍了齐次均匀康托集和偏齐次均匀康托集,在本文中我们构造介于两者之间的一类齐次Moran集,给出其豪斯多夫维数的精确计算公式,并讨论维数关于参数的不连续性.