The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

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.     

Constructing exact Lagrangian immersions with few double points

We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps, 2013), an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space ${\mathbb{R}^6_{\rm st}}$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding ${S^1 \times S^2 \to \mathbb{R}^6_{\rm st}}$ with vanishing Maslov class.     

The packing measure of a general subordinator

Summary Precise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y ₁ (t), Y ₂ (t)), whereY ₁ andY ₂ are independent copies ofX. A finite and positive packing measure is possible for subordinators close to Cauchy; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).Research supported in part by NSF under contracts (1) DMS 87-01866, and (2) DMS 87-01212     

Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions on a metric space X, multifractal analysis refers to the study of the Hausdorff and/or packing dimension of the level sets(1) of the limit function limnφn. However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit limnφn(x) exists, has emerged and attracted considerable interest. Namely, for a sequence n(xn) in a metric space X, we let A(xn) denote the set of accumulation points of the sequence n(xn). The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence (φnn(x)) equals a given set C, i.e. computing the Hausdorff dimension of the set(2){x∈X|A(φn(x))=C} has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φn, including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.     

Thin points for Brownian motion

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {W_t}_0≤t≤1 in . We prove that for any analytic set E in [0,1], we have

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, where dim_P(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which

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, is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of ‘limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.     

Flat points of two-dimensional Brownian motion

Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt ₀ and >0 such thatZ(t ₀) is an extremal point of the convex hull of {Z(t)|t ₀–tt₀} and also an extremal point of the convex hull of {Z(t)|t ₀tt₀+} and, moreover, the tangent lines to the convex hulls atZ(t ₀) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t ₀ and >0 such that the intersection of the convex hulls of {Z(t)|t ₀–tt₀} and {Z(t)|t ₀tt₀+} contains onlyZ(t ₀)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture     

Plane Brownian motion has strictlyn-multiple points

It is shown that, almost surely, for all naturaln, there are points which the plane Brownian motion visitsexactly n times.     

The packing measure of a class of generalized sierpinski carpet

For 1/4 ＜ a ＜√2/4, let S1(x) = ax, S2(x) = 1 - a ax, x ∈ [0,1]. Ca is the attractor of the iterated function system {S1, S2}, then the packing measure of Ca × Ca is Ps(a)(Ca × Ca) = 4.2s(a)(1 - a)s(a),where s(a) = -loga4.     

The size of irregular points for a measure

For a measure μ on the complex plane μ-regular points play an important role in various polynomial inequalities. In the present work it is shown that every point in the set {μ′>0} (actually of a larger set where μ is strong) with the exception of a set of zero logarithmic capacity is a μ-regular point. Here “set of zero logarithmic capacity” cannot be replaced by “β-logarithmic Hausdorff measure  0” with β=1 (it can be replaced by “β-logarithmic measure 0” with any β>1). On the other hand, for arbitrary μ the set of μ-regular points can be quite small, but never empty.     

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 geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable _boxclose/2{{\mathbb Z}/2} -equivariant map h(F) such that the stable \mathbb Z/2{{\mathbb Z}/2} -equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map F of an immersion f : M^m \looparrowright Nⁿ{f : M^m \looparrowright N^n} in terms of the double point set of f. We interpret the Smale–Hirsch–Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m <  2n – 1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.     

A critical case for Brownian slow points

Summary LetX _t be a Brownian motion and letS(c) be the set of realsr0 such that üX _r+t –X _r üct, 0th, for someh=h(r)>0. It is known thatS(c) is empty ifc<1 and nonempty ifc>1, a.s. In this paper we prove thatS(1) is empty a.s.This research was partially supported by NSF Grant 9322689.     

On hardly visited points of the Brownian motion

LetL(x, t) be the local time process of a standard Wiener process {W(t),t>0}. Denote

Divergence points of self-similar measures and packing dimension

Let μ be a self-similar measure in Rd. A point x∈Rd for which the limit does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.     

The distribution and the exact percentage points for Wilks'L mvc criterion

Summary Wilks'L _mvc is the likelihood ratio criterion for testing the hypothesis that the mean values are equal, the variances are equal and the covariances are equal, in ap-variate normal population. In this article the exact null distribution as well as the exact percentage points are given for the first time. The distribution is obtained for the most general cases and the inverse tables, namely, the values ofu for given values ofF(u) are computed for the values ofF(u)=0.01, 0.02, 0.05 and for the various values ofn andp whereF(u) is the exact distribution function of the test statistic,n=N−1 andN is the sample size. The exact tables are given forp=2, 3, 4, 5, 6, 7, 8, 9.