Results 161 to 170 of about 338 (182)
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Hausdorff and packing dimensions and sections of measures
Mathematika, 1998Summary: Let \(m\) and \(n\) be integers with \(0< m< n\) and let \(\mu\) be a Radon measure on \(\mathbb{R}^n\) with compact support. For the Hausdorff dimension, \(\dim_H\), of sections of measures we have the following equality: for almost all \((n- m)\)-dimensional linear subspaces \(V\) \[ \text{ess inf}\{\dim_H \mu_{V,a}: a\in V^{\perp}\text ...
Järvenpää, Maarit, Mattila, Pertti
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Scaling properties of Hausdorff and packing measures
Mathematische Annalen, 2001Let \(\theta \) be a continuous increasing function defined on the nonnegative number line with some restriction. Among other results, the authors characterize those function \(\theta \) such that the corresponding Hausdorff or packing measure with gauge function \(\theta \) scales with exponent \(\alpha \) by showing it must be a product of a power ...
Csörnyei, Marianna, Mauldin, R. Daniel
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Hausdorff and packing dimensions of Mandelbrot measure
International Journal of Mathematics, 2020We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text].
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Upper and Lower Densities of Measures and Comparison with Hausdorff and Packing Measures
Fractals and Dynamics in Mathematics, Science and the Artsexaly +2 more sources
Dimension and Measures for a Curvilinear Sierpinski Gasket or Apollonian Packing
Advances in Mathematics, 1998 In this paper we apply some results about general conformal iterated function systems to A, the residual set of a standard Apollonian packing or a curvilinear Sierpinski gasket.
Mariusz Urbanski +3 more
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Hausdorff and packing measure functions of self-similar sets: continuity and measurability
Ergodic Theory and Dynamical Systems, 2008AbstractLetNbe 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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Self-similar sets of zero Hausdorff measure and positive packing measure
Israel Journal of Mathematics, 2000The 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 ...
Peres, Yuval +2 more
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Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion
Journal of Theoretical Probability, 1999Iterated 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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HAUSDORFF AND PACKING MEASURE OF SETS OF GENERIC POINTS: A ZERO-INFINITY LAW
Journal of the London Mathematical Society, 2004The 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 ...
Ma, Jihua, Wen, Zhiying
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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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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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