Results 151 to 160 of about 2,719 (189)

THE MULTIFRACTAL HAUSDORFF AND PACKING MEASURE OF GENERAL SIERPINSKI CARPETS

Acta Mathematica Scientia, 2000
The authors study multifractal Hausdorff and packing measures for a self-affine measure supported by a generalized Sierpiński gasket \(E\) satisfying the disjointness condition.
Jinghu Yu, Lihu Huang
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Self-similar sets of zero Hausdorff measure and positive packing measure

Israel Journal of Mathematics, 2000
The authors prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension. For instance, if \(1/5< r< 1/3\) then the set \({\mathcal K}^r_u\) of all sums \(\sum^\infty_{n=0} a_nr^n\) with \(a_n\in \{0, 1,u\}\) has this property for almost every \(u\) from a certain nonempty interval ...
Károly Simon, Yuval Peres, Yuval Peres, Boris Solomyak   +3 more
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On the equivalence of the multifractal centred Hausdorff measure and the multifractal packing measure [PDF]

Nonlinearity, 2008
In this work, we focus on the multifractal centred Hausdorff measure and the multifractal packing measure in . We find that for 0 < s < t < 2, q < 1, then there exist a set and a probability measure μ on such that and, in addition,
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HAUSDORFF AND PACKING MEASURE OF SETS OF GENERIC POINTS: A ZERO-INFINITY LAW

Journal of the London Mathematical Society, 2004
The main result of the paper consists of a fine analysis of the size of the set \(G_\mu\) of generic points (i.e., satisfying the Birkhoff ergodic theorem for all continuous functions) of an invariant measure \(\mu\) defined on a symbolic space. The symbolic space is endowed with a metric defined via a Gibbs measure \(\mu_\phi\) associated to a ...
Jihua Ma, Zhiying Wen
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Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion

Journal of Theoretical Probability, 1999
Iterated Brownian motion is defined as \(Z(t)=X(Y(t))\) where \(X\) and \(Y\) are independent (two-sided) Brownian motions. The exact Hausdorff dimension gauge of the level sets of iterated Bronwian motion is determined to be \(\varphi(x)=x^{3/4}[\log\log(1/x)]^{3/4}\). This result generalizes earlier work of \textit{K.
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Frostman lemmas for Hausdorff measure and packing measure in a product probability space and their physical application

Chaos, Solitons & Fractals, 2005
The authors establish Frostman-type lemmas for the Hausdorff and pre-packing measures on a product probability measure space \((\Omega, {\mathcal F}, \mu) = (\Omega_1\times \Omega_2, {\mathcal F}_1\times {\mathcal F}_2, \mu_1\times\mu_2)\). Based on these results, they prove a sufficient condition for the Hausdorff and packing dimensions with respect ...
Chaoshou Dai, Yanyan Hou
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Exact Hausdorff and packing measures of Cantor sets with overlaps

Ergodic Theory and Dynamical Systems, 2014
Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [
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Hausdorff and packing measure functions of self-similar sets: continuity and measurability

Ergodic Theory and Dynamical Systems, 2008
AbstractLetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition.
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Density theorems for Hausdorff and packing measures of self-similar sets

Aequationes mathematicae, 2008
We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff dimension and packing dimension equal s, a special case of our main results says that if K satisfies the Open Set Condition, then there exists a number r 0 such ...
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The branching measure, Hausdorff and packing measures on the Galton-Watson tree

2000
We present some recent results concerning the branching measure, the exact Husdorff measure and the exact packing measure, defined on the boundary of the Caalton-Watson tree. The results show that in good cases, these three measures coincide each other up to a constant, that the branching measure is homogeneous (it has the same local dimension at each ...
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