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.     

Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

The BV-capacity in metric spaces

We study basic properties of the BV-capacity and Sobolev capacity of order one in a complete metric space equipped with a doubling measure and supporting a weak Poincaré inequality. In particular, we show that the BV-capacity is a Choquet capacity and the Sobolev 1-capacity is not. However, these quantities are equivalent by two sided estimates and they have the same null sets as the Hausdorff measure of codimension one. The theory of functions of bounded variation plays an essential role in our arguments. The main tool is a modified version of the boxing inequality.     

关于自相似集的Hausdorff测度的一个判据及其应用

讨论了满足开集条件的自相似集。对于此类分形，用自然覆盖类估计它的Hausdorff测度只能得到一个上限，因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用，统一处理了一类自相似集，得到了平面上的一个Cantor集－Cantor尘的Hausdorff测度的准确值，并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

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．     

Homogeneous sets from several ultrafilters

We consider Ramsey-style partition theorems in which homogeneity is asserted not for subsets of a single infinite homogeneous set but for subsets whose elements are chosen, in a specified pattern, from several sets in prescribed ultrafilters. We completely characterize the sequences of ultrafilters satisfying such partition theorems. (Non-isomorphic selective ultrafilters always work, but, depending on the specified pattern, weaker hypotheses on the ultrafilters may suffice.) We also obtain similar results for analytic partitions of the infinite sets of natural numbers. Finally, we show that the two P-points obtained by applying the maximum and minimum functions to a union ultrafilter are never nearly coherent.     

On spectrums of certain harmonic functions attached to the Riemann zeta-function

A. Beurling introduced harmonic functions attached to measurable functions satisfying suitable conditions and defined their spectral sets. The concept of spectral sets is closely related to approximations by trigonometric polynomials. In this paper we consider spectral sets of the harmonic functions attached to the Riemann zeta-function and its modification.     

A decomposition theorem for maxitive measures

A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. In contrast to charges, maxitive measures often have a density. We show that maxitive measures can be decomposed as the supremum of a maxitive measure with density, and a residual maxitive measure that is null on compact sets under specific conditions.     

A criterion on a repeller being a null set of any limit measure for stochastic differential equations

This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure. Using this criterion, we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures, which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits,and saddles. Interesting open questions on exact supports of limit measures are proposed.     

Isomorphism and Embedding of Borel Systems on Full Sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

The Hausdorff measure of the self-similar sets

The self-similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important. A negative criterion has been given. As a consequence, the Marion’s conjecture on the Hausdorff measure of the Koch curve has been proved invalid. Project partially supported by the State Scientific Commission and the State Education Commission.     

Eine Bemerkung zum Begriff der zufÄlligen Folge

Summary Martin-Löf has defined random sequences to be those sequences which withstand a certain universal stochasticity test. On the other hand one can define a sequence to be random if it is not contained in any species of measure zero in the sense of Brouwer. Both definitions imply that these random sequences possess all statistical properties which can be checked by algorithms. We draw a comparison between the two concepts of constructive null sets and prove that they induce concepts of randomness which are not equivalent. The union of all species of measure zero in the sense of Brouwer is a proper subset of the universal constructive null set defined by Martin-Löf.     

Sublevel representations of epi-Lipschitz sets and other properties

Mathematical Programming - Epi-Lipschitz sets in normed spaces are represented as sublevel sets of Lipschitz functions satisfying a so-called qualification condition. Canonical representations...     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz     

The Dynkin-Lamperti arc-sine laws for measure preserving transformations

Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.

     

Metrics defined via discrepancy functions

We introduce the notion of a discrepancy function, as an extended real-valued function that assigns to a pair (A,U) of sets a nonnegative extended real number ω(A,U), satisfying specific properties. The pairs (A,U) are certain pairs of sets such that A⊆U, and for fixed A, the function ω takes on arbitrarily small nonnegative values as U varies. We present natural examples of discrepancy functions and show how they can be used to define traditional pseudo-metrics, quasimetrics and metrics on hyperspaces of topological spaces and measure spaces.     

On the measure of intersecting families,uniqueness and stability

Let t≥1 be an integer and let A be a family of subsets of {1,2,…,n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0,1} ⁿ we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the t-intersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable: any t-intersecting family that is close to attaining the maximal measure must in fact be close in structure to a genuine maximum family. This is stated precisely in Theorem 1.6. We deduce some similar results for the more classical case of Erdős-Ko-Rado type theorems where all the sets in the family are restricted to be of a fixed size. See Corollary 1.7. The main technique that we apply is spectral analysis of intersection matrices that encode the relevant combinatorial information concerning intersecting families. An interesting twist is that part of the linear algebra involved is done over certain polynomial rings and not in the traditional setting over the reals. A crucial tool that we use is a recent result of Kindler and Safra [22] concerning Boolean functions whose Fourier transforms are concentrated on small sets. Research supported in part by the Israel Science Foundation, grant no. 0329745.