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.     

Gauges for the self‐similar sets

For a self‐similar set E with the open set condition we completely determine the class of its Hausdorff gauges and the class of its prepacking gauges. Moreover, its Hausdorff measures and its packing premeasures with respect to the corresponding gauges are estimated. Without the open set condition we prove that a doubling gauge function is a packing gauge of E if and only if it is a prepacking gauge of E. Also, we give some extensions and applications of these results. Here a gauge function is called a Hausdorff, a prepacking, and a packing gauge of a set, if with respect to the function the set has positive and finite Hausdorff measure, packing premeasure, and packing measure, respectively. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundedly expressible sets

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

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

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.     

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).

     

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.

     

A certain family of self-similar sets

In this paper we consider an one-parameter family of iterated function systems. For every value of the parameter we find the set of top addresses. We prove that this set is a countable disjoint union of self-similar sets and calculate its Hausdorff dimension.     

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.     

A dimensional result for random self-similar sets

A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal

     

Hausdorff dimension and dendritic limit sets

Let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3-space, H³, such that H³/ has positive injectivity radius. It is known that the limit set is a dendrite of Hausdorff dimension 2. We show that the cut-point set of the limit set has Hausdorff dimension strictly less than 2.     

On the exact Hausdorff measure of a class of self-similar sets satisfying open set condition

In this paper,we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition(OSC).As applications,we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.     

Baire category and zero sets

Quasi-all continuous functions have a zero set which is a perfect Kronecker set with Hausdorff dimension zero. To cite this article: T. Körner, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Dimension and measures on sub-self-affine sets

We show that in a typical sub-self-affine set, the Hausdorff and the Minkowski dimensions coincide and equal the zero of an appropriate topological pressure. This gives a partial positive answer to the question of Falconer. We also study the properties of the topological pressure and the existence and the uniqueness of natural measures supported on a sub-self-affine set.     

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.     

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 Hausdorff dimension of the level sets of Takagi’s function

Recently, Maddock (2006) [12] has conjectured that the Hausdorff dimension of each level set of Takagi’s function is at most 1/2. We prove this conjecture using the self-affinity of the function of Takagi and the existing relationship between the Hausdorff and box-counting dimensions.     

Fixed point theorems for paracompact convex sets

In the present paper a few fixed point theorems are given for upper hemi-continuous mappings from a paracompact convex set to its embracing space, a real, locally convex, Hausdorff topological vector space.     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.