The exact Hausdorff measure of the level sets of Brownian motion

Summary We show that if s(t, x) is the local time of a Brownian motion B, and (t)=(2t¦log|logt)^1/2 then –m({s=x})=s(t,x) for all t>=0 and x real a.s., where –m(E) is the Hausdorff -measure of E. This solves a problem of Taylor and Wendel who proved the above equality, up to a multiplicative constant, for x=0.     

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.     

The Hausdorff Centred measure of the symmetry Cantor sets

Let 0＜λ≤1/3,K (λ) be the attractor of an iterated function system { ψ1,ψ2 } on the line, where ψ1 (x ) =λx, ψ2(x)=1-λ+λx, x∈ [0,1]. We call K (λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K (λ).     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

THE HAUSDORFF CENTRED MEASURE OF THE SYMMETRY CANTOR SETS

Let 0<λ≤1/3, K(λ) be the attractor of an iterated function system {ϕ₁ϕ₂} on the line, where ϕ₁(x)=λx, ϕ₁(x)=1-λ+λx,x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the 0123 0132 V 3 exact Hausdorff Centred measure of K(λ).     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide

A set is -straight if has finite Hausdorff -measure equal to its Hausdorff -content, where is continuous and non-decreasing with . Here, if satisfies the standard doubling condition, then every set of finite Hausdorff -measure in is shown to be a countable union of -straight sets. This also settles a conjecture of Foran that when , every set of finite -measure is a countable union of -straight sets.

     

Hausdorff measure of the level sets of multi-parameter wiener processes in one dimension

LetZ _N (t) be anN-parameter Wiener process in one dimension, and $$E(x,T) = \left\{ {t = (t_1 , \cdot \cdot \cdot ,t_N ):Z_N (t) = x,0 \leqslant t_1 , \cdot \cdot \cdot ,t_N \leqslant T} \right\}$$ . Then we obtain that with probability one, the Hausdorff measure function ofE(x,T) is $$\psi _N (r) = r^{N - \tfrac{1}{2}} (\log \log r^{ - 1} )^{\tfrac{1}{2}} ,\forall r \in (0,\frac{1}{4})$$ for anyx∈R ¹ andT>0.     

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.     

Pluripolarity of sets with small Hausdorff measure

We show that any set E⊂C ⁿ , n≥ 2, with finite Hausdorff measure? is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral , , with properties of the pluricomplex relative extremal function for the Bedford–Taylor capacity. Received: 20 May 1999     

A Hausdorff measure classification ofG-polar sets for the superdiffusions

Summary We establish relations betweenG-polar sets of superdiffusions and the restricted Hausdorff dimension. As an application, we give new proofs of Dynkin's criteria for theS-polarity andH-polarity (established earlier by Dawson, Iscoe, Perkins, and Le Gall under more restrictive assumptions.)     

Exact Hausdorff measure and intervals of maximum density for Cantor sets

Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

     

On the increments of the local time of a Wiener sheet

{W(x, y), x≥0, y≥0} be a Wiener process and let η(u, (x, y)) be its local time. The continuity of η in (x, y) is investigated, i.e., an upper estimate of the process η(μ, [x, x + α) × [y, y + β)) is given when αβ is small.     

Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets

A compact set is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition' (OSC), then has positive -dimensional Hausdorff measure, where is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.

     

Mappings of finite distortion: Hausdorff measure of zero sets

We prove that a for a mapping f of finite distortion , the -Hausdorff measure of any point preimage is zero provided is integrable, with , and the multiplicity function of f is essentially bounded. As a consequence for we obtain that the mapping is then open and discrete. Received: 18 June 2001 / Revised version: 31 January 2002 / Published online: 27 June 2002