Hausdorff measures on Julia sets of subexpanding rational maps

Leth be the Hausdorff dimension of the Julia setJ(R) of a Misiurewicz’s rational mapR : (subexpanding case). We prove that theh-dimensional Hausdorff measure H _h onJ(R) is finite, positive and the onlyh-conformal measure forR : up to a multiplicative constant. Moreover, we show that there exists a uniqueR-invariant measure onJ(R) equivalent to H _h .     

Hausdorff dimension and conformal measures of Feigenbaum Julia sets

We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon', there exist many Feigenbaum Julia sets whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent is equal to the hyperbolic dimension . Moreover, if , then . In the stationary case, the last statement can be reversed: if , then . We also give a new construction of conformal measures on that implies that they exist for any , and analyze their scaling and dissipativity/conservativity properties.

     

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.     

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.     

Straight line arrangements in the real projective plane

Let A be an arrangement of n pseudolines in the real projective plane and let p ₃(A) be the number of triangles of A. Grünbaum has proposed the following question. Are there infinitely many simple arrangements of straight lines with p ₃(A)=1/3n(n?1)? In this paper we answer this question affirmatively.     

An algorithm for computing the centered Hausdorff measures of self-similar sets

We provide an algorithm for computing the centered Hausdorff measures of self-similar sets satisfying the strong separation condition. We prove the convergence of the algorithm and test its utility on some examples.     

Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions

ABSTRACT

We consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.     

The exact measures of a class of self-similar sets on the plane

Let S (C) R2 be the attractor of the iterated function system{f1,f2,f3}iterating on the unit equilateral triangle So,where fi(x)=λix bi,I=1,2,3,x=(x1,x2),b1=(0,0),62=(1-λ2,0),63=(1-λ3/2,√3/2(1-λ3)).This paper determines the exact Hausdorff measure,centred covering measure and packing measure of S under some conditions relating to the ontraction parameter.     

Ideals of porous sets in the real line and in metrizable topological spaces

Porous sets and spherically porous sets of a metric space are studied. In particular, porous, superporous, and equiporous sets of a metric space (X, π) are characterized from the topological point of view. Metrizable spaces that are spherically porous in themselves with respect to some metric generating the topology are characterized. Some relations between the ideals of the classes of porous sets in the real line are established. Bibliography: 13 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 221–242.     

Lebesgue measure and Hausdorff dimension of special sets of real numbers from (0,1)

We estimate the Hausdorff dimension and the Lebesgue measure of sets of continued fractions of the type a=[a ₁,a ₂,…] where a _n belongs to a set S _n ⊂ℕ for every n∈ℕ. An upper bound for the Hausdorff dimension of the set of numbers with continued fraction expansions which fulfill some properties of asymptotic densities is also included.     

Hausdorff measures on the Wiener space

We construct a Hausdorff measure of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Krée-Watanabe theorem concerning the regularity of the images of the Wiener measure.     

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on.     

Spaces of piecewise-continuous almost-periodic functions and almost-periodic sets on the real line. II

The authors consider a series of spaces of piecewise-continuous almost periodic functions and study the properties of the elements of these spaces. The theory developed in the paper is then applied to investigate almost periodic linear pulse systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 389–400, March, 1992.     

Hausdorff measures, dimensions and mutual singularity

Let be a metric space. For a probability measure on a subset of and a Vitali cover of , we introduce the notion of a -Vitali subcover , and compare the Hausdorff measures of with respect to these two collections. As an application, we consider graph directed self-similar measures and in satisfying the open set condition. Using the notion of pointwise local dimension of with respect to , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.

     

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Máté, P. Nevai and the author. The present paper solves the general problem. Work was supported by the National Science Foundation, DMS 9801435 and by the Hungarian National Science Foundation for Research, T/022983.