Stable intersection of affine Cantor sets defined by two expanding maps

It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.     

随机康托集的概率性质

本文讨论了两类随机康托集。关于一类随机康托集，证明它们相交等价于稳定从属过程；关于另一类随机康托集，该文证明了由它们产生的随机过程的密度函数与稳定从属过程的密度函数有很大的相似性。     

Algebraic Topology for Minimal Cantor Sets

It will be shown that every minimal Cantor set can be obtained as a projective limit of directed graphs. This allows to study minimal Cantor sets by algebraic topological means. In particular, homology, homotopy and cohomology are related to the dynamics of minimal Cantor sets. These techniques allow to explicitly illustrate the variety of dynamical behavior possible in minimal Cantor sets. submitted 20/07/05, accepted 18/10/05     

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.     

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.     

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).     

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.     

Rigid cantor sets in with simply connected complement

We prove that there exist uncountably many inequivalent rigid wild Cantor sets in with simply connected complement. Previous constructions of wild Cantor sets in with simply connected complement, in particular the Bing- Whitehead Cantor sets, had strong homogeneity properties. This suggested it might not be possible to construct such sets that were rigid. The examples in this paper are constructed using a generalization of a construction of Skora together with a careful analysis of the local genus of points in the Cantor sets.

     

一类齐次Moran集的维数

本文研究了一类特殊的齐次Moran集的维数.将齐次均匀Cantor集通过一系列平移,获得了一类特殊的齐次Moran集并得到了它们维数的精确值,推广了齐次均匀Cantor集维数的计算公式.     

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.     

Semigroups whose Julia sets are Cantor targets

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2.     

On Cantor sets and doubling measures

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.     

Inferior mean of Cantor boundary measures

We establish various bounds for the inferior mean of positive functions, as defined by M. Heins, with respect to the porosity of Cantor sets that support Cantor boundary measures of harmonic functions on the upper half-space.     

Cantor集并的自相似性(英文)

本文研究了Cantor集和其并的自相似性.利用Cantor展式的方法,得到了关于Cantor集和迭代函数系的一个基本关系:T∪(T+α)为自相似的当且仅当存在一个非负整数n使得α=±(k2-k1)dn.进一步,若T∪(T+α)是自相似的,则它满足开集条件.     

Ultrametric Cantor sets and growth of measure

A class of ultrametric Cantor sets (C, d _u ) introduced recently (S. Raut and D. P. Datta, Fractals 17, 45–52 (2009)) is shown to enjoy some novel properties. The ultrametric d _u is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrization invariant, is identified with the Cantor function associated with a Cantor set $ \tilde C $ \tilde C , where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps of $ \tilde C $ \tilde C . The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrization invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension log₃ 2 and thickness 1 are constructed explicitly.     

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…     

On dimensions and measures of statistically constructed cantor sets

We investigate the Hausdorff dimension and the packing dimension of random Cantor sets. That is, using the Gibbs measures, we can conclude that in our Cantor sets the Hausdorff dimension coincides with the packing dimension and this common value is characterized as the unique zero point of a certain function. A striking difference from deterministic cases appears when we consider measures of these sets. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

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.     

Two point set extensions- a counterexample

We show that there exist Cantor sets in the circle that are not extendable to sets that meet every line in the plane in exactly two points. This result solves a problem that was formulated by R. D. Mauldin.

     

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces.