Hausdorff measure of infinitely generated self-similar sets

We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that ofA.     

Improving dimension estimates for Furstenberg-type sets

In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ?e in the direction of e for which dimH(?e∩F)?α. It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets.The main difficulty we had to overcome, was that if Hh(F)=0, there always exists g?h such that Hg(F)=0 (here ? refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α=0.     

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.     

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.     

On the exact rate of convergence of frequencies of digits in self-similar sets

We study the exact rate of convergence of frequencies of digits of “normal” points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of ?^d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in ?^d—namely, sets with full Hausdorff dimension—for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.     

On the Hausdorff dimension of generalized Besicovitch-Eggleston sets of d-tuples of numbers

We give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.     

Local dimensions of sliced measures and stability of packing dimensions of sections of sets

Let m and n be integers with 0<m<n. We relate the absolutely continuous and singular parts of a measure μ on to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.     

Hausdorff dimension of the limit sets of some planar geometric constructions

We determine the Hausdorff and box dimension of the limit sets for some class of planar non-Moran-like geometric constructions generalizing the Bedford-McMullen general Sierpiński carpets. The class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines, as well as some non-affine examples, e.g. the flexed Sierpiński gasket.     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].     

Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion

In this paper we apply the techniques and results from the theory of multifractal divergence points developed in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] to give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using the method and results from [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] we investigate and compute the Hausdorff dimension of several new sets of d-tuples of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of d-tuples numbers for which the limiting frequencies of the digits in their N-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of higher-dimensional and non-linear generalizations of known results, by Cajar and Volkmann and others, on the Hausdorff dimension of sets of normal and non-normal numbers.     

On the connectedness of limit net sets

In this paper we introduce and study net sets and limit net sets. The construction and geometry of net sets can be described with the help of substitutions with net matrices which we also introduce here. Limit net sets are a special type of Moran fractals. We study connectedness properties of net sets and limit net sets.     

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.     

On the magnification of cantor sets and their limit models

For aC ¹⁺ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC ¹⁺ diffeomorphic withC (for some >0, (0,1)), but not all suchC ¹⁺ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's scaling function. An alternative definition of a fractal is discussed.With 1 Figure     

Integral self-affine sets with positive Lebesgue measures

A self-affine region is an integral self-affine set with positive Lebesgue measure. In this note we give two criteria for integral self-affine sets being self-affine regions. As their applications we study the L ¹-solutions of refinement equations, which play an important role in constructing wavelets, and we give several interesting examples. Received: 8 May 2007     

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.     

Limit sets of automatic sequences

We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (k^pn+j)_n?0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.     

The Lebesgue measure of the algebraic difference of two random Cantor sets

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F₁, F₂ such that the arithmetic difference set F₂ − F₁ does not contain any intervals but ?eb(F₂ − F₁)> 0 almost surely, conditioned on non-extinction.     

Hausdorff dimension of certain sets arising in continued fraction expansions

This paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel-Bernstein's theorem, is completely determined

     

Sliding of self-similar sets

This paper deals with the Lipschitz equivalence of slidings of self-similar sets by graph-directed construction and martingale theory.