DIMENSIONS FOR RANDOM SELF-CONFORMAL SETS

1　IntroductionTherehasbeenconsiderableinterestinfractals,bothintheiroccurrenceinthesciences,andintheirmathematicaltheory .Awideclassoffractalsetsaregeneratedbyiteratedfunc tionsystem .Aself similarsetinRdisacompactsetKfulfillingtheinvarianceK =∪Ni=1 SiK ,whereS1,S2 ,… ,SNarecontractivesimilarities.IfS1,S2 ,… ,SNarecontractiveconfor malmappings,weobtainself conformalset.Itiswell known(seeHutchinson [1 2 ] )that,givenafamilyofsuchmappings,thereisauniquecompactsetwiththisproperty .Ifth…     

THE REGULARITY OF RANDOM GRAPH DIRECTED SELF-SIMILAR SETS

A set in R^d is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..     

Self-similar sets in complete metric spaces

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.

     

低复杂度序列的维数

本文研究符号空间中由零拓扑熵序列组成的集合.通过构造适当的自相似集,证明了该集合的盒维数为1,而Hausdorff维数为0.     

DIMENSIONS FORRANDOM SELF——CONFORMAL SETS

A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper, we prove that the random self-con formal set is regular almost surely. Also we determine the dimen-sions for a class of random self-con formal sets.     

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.     

Hausdorff dimension in a family of self-similar groups

For every prime p and every monic polynomial f, invertible over p, we define a group G _{p, f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group . We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further “finiteness” properties are also discussed (amenability, torsion and intermediate growth). Partially supported by NSF grant DMS-0600975.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

Self-similar random measures are locally scale invariant

Summary In an earlier paper Patzschke and U. Zähle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of . Moreover, we approximate the tangential distribution by means of and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of we obtain an earlier result of Mauldin and Williams [9] as a corollary.     

A central set of dimension

The central set of a domain is the set of centers of maximal discs in . Fremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than . We construct a planar domain with central set of Hausdorff dimension .

     

Sets of minimal Hausdorff dimension for quasiconformal maps

For any , there is a compact set of (Hausdorff) dimension whose dimension cannot be lowered by any quasiconformal map . We conjecture that no such set exists in the case . More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.

     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Rd中一类齐次Moran集的盒维数

给定Rd 中的Moran集类 ,本文证明了对介于该集类中元素的上盒维数的最大值和最小值之间的任何一个数值s,总存在该集类中的一个元素 ,其上盒维数等于s,对下盒维数、修正的下盒维数也有类似的性质成立 ,从而给文 [1 ]中的猜想 1一个肯定的回答 .此外 ,还讨论了齐次Cantor集和偏次Cantor集盒维数存在性之间的关系 .     

Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere. Received: March 2007, Revision: October 2007, Accepted: October 2007     

Dimensions for random self–conformal sets

A set is called regular if its Hausdorff dimension and upper box–counting dimension coincide. In this paper, we prove that the random self–conformal set is regular almost surely. Also we determine the dimensions for a class of random self–conformal sets.     

The dimensions of the divergence points of self-similar measures with weak separation condition

Let \(\mu \) be the self-similar measure supported on the self-similar set K with the weak separation condition, which is weaker than the open set condition. This article uses Hausdorff dimension and packing dimension to investigate the multifractal structure of several sets of divergence points of \(\mu \) in the iterated function system.     

A dimension result arising from the -spectrum of a measure

We give a rigorous proof of the following heuristic result: Let be a Borel probability measure and let be the -spectrum of . If is differentiable at , then the Hausdorff dimension and the entropy dimension of equal . Our result improves significantly some recent results of a similar nature; it is also of particular interest for computing the Hausdorff and entropy dimensions of the class of self-similar measures defined by maps which do not satisfy the open set condition.

     

Singularity of self-similar measures with respect to Hausdorff measures

Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base- expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.

     

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).