Measures and their dimension spectrums for cookie-cutter sets in ℝd

The characteristics of cookie-cutter sets in ℝ^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly α-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved. This project is supported by the National Natural Science Foundation of China.     

Dimensions of cookie-cutter-like sets

The cookie-cutter-like sets are defined as the limit sets of a sequence of classical cookie-cutter mappings. By introducing Gibbs-like measures, we study the dimensions, Hausdorff and packing measures of the CC-like sets, and then discuss the continuous dependence of the dimensions.     

d维平稳高斯过程极集的充分条件及维数

本文研究了d维平稳高斯过程极集的性质,给出了d维平稳高斯过程广义极性的充分条件,并通过一个特殊的Cantor型集的构造将极集的维数与容度巧妙地结合起来,得到了d维平稳高斯过程非极集的Hausdorff维数的下确界.     

控制系统中的分形

本文将整数维与分形的Hausdorff测度引入并应用于控制系统,同时也介绍了准自相似集这个新概念,证明了这种集合的存在性与唯一性.并将计算自相似集维数的公式推广到准自相似集,在此基础上,说明了控制系统的可达集可以具有分数维.表明在分析非线性系统可控性与可观性时,分形几何学也将是一种有意义的工具.     

An asymptotic analysis in thermodynamic formalism

We consider an asymptotic behaviour of the topological pressure, the Gibbs measure and the measure-theoretic entropy concerning a potential defined on a subshift. Our results are obtained by considering asymptotic perturbation of transfer operators and by using a method that avoids resolvent??s perturbation. In application, we investigate an asymptotic behaviour of the Hausdorff dimension of a perturbed cookie-cutter set.     

关于自相似集的Hausdorff测度

得到了 Ｈａｕｓｄｏｒｆｆ容度与 Ｈａｕｓｄｏｒｆｆ测度相等的集的充分必要条件．对于满足开集条件的自相似集,验证了它的Ｈａｕｓｄｏｒｆｆ容度与Ｈａｕｓｄｏｒｆ测度相等并给出了它的Ｈａｕｓｄｏｒｆｆ测度的一个便于应用的公式．作为例子,给出了均匀康托集的Ｈａｕｓｄｏｒｆｆ测度的一种新的计算方法,对于Ｋｏｃｈ曲线的Ｈａｕｓｄｏｒｆｆ测度的上限也作了讨论．     

Boundedly expressible sets

For a given sequence a boundedly expressible set is introduced. Three criteria concerning the Hausdorff dimension of such sets are proved.     

Hausdorff Dimension of the Set of Generic Points for Gibbs Measures

For a Gibbs measure on the configuration space of a finite spin lattice system, we find (in terms of entropy) the Hausdorff dimension of the set of generic points. Using this result, we evaluate the Hausdorff dimension of level sets for Birkhoff ergodic averages of some continuous functions on the configuration space.     

Conformal iterated function systems with applications to the geometry of continued fractions

In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.

     

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.     

Hausdorff measures for a class of homogeneous Moran sets

This paper provides an explicit formula for the Hausdorff measures of a class of regular homogeneous Moran sets. In particular, this provides, for the first time, an example of an explicit formula for the Hausdorff measure of a fractal set whose Hausdorff dimension is greater than 1.     

GEOMETRY AND DIMENSION OF SELF—SIMILAR SET

The authors show that the self-similar set for a finite family of contractive similitudes (similarities, i.e., |fi(x) - fi(y)| = αi|x - y|, x,y ∈ RN, where 0 < αi < 1) is uniformly perfect except the case that it is a singleton. As a corollary, it is proved that this self-similar set has positive Hausdorff dimension provided that it is not a singleton. And a lower bound of the upper box dimension of the uniformly perfect sets is given. Meanwhile the uniformly perfect set with Hausdorff measure zero in its Hausdorff dimension is given.     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

Dimension sets for infinite IFSs: the Texan Conjecture

We consider the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system and refer to it as the restricted dimension set. The corresponding set for all subsystems will be referred to as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both general results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval.     

Harmonic measure and expansion on the boundary of the connectedness locus

The paper develops a technique for proving properties that are typical in the boundary of the connectedness locus with respect to the harmonic measure. A typical expansion condition along the critical orbit is proved. This condition implies a number of properties, including the Collet-Eckmann condition, Hausdorff dimension less than 2 for the Julia set, and the radial continuity in the parameter space of the Hausdorff dimensions of totally disconnected Julia sets. Oblatum 6-XI-1998 & 12-V-2000?Published online: 11 October 2000     

Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.     

Completeness of Muckenhoupt classes

In this note we prove that the Hausdorff distance between compact sets and the Kantorovich distance between measures, provide an adequate setting for the convergence of Muckenhoupt weights. The results which we prove on compact metric spaces with finite metric dimension can be applied to classical fractals.     

Fractal entropies and dimensions for microstates spaces

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite-dimensional algebra or where the set consists of a single selfadjoint. We show that the Hausdorff dimension becomes additive for such sets in the presence of freeness.     

Intersection of triadic Cantor sets with their translates— I. Fundamental properties

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.     

Bilipschitz embedding of self-similar sets

In this paper, we prove that each self-similar set satisfying the strong separation condition can be bilipschitz embedded into each self-similar set with larger Hausdorff dimension. A bilipschitz embedding between two self-similar sets of the same Hausdorff dimension both satisfying the strong separation condition is only possible if the two sets are bilipschitz equivalent.