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Bounds of Hausdorff measure of the Sierpinski gasket [PDF]

Journal of Mathematical Analysis and Applications, 2007
By a new method, we obtain the lower and upper bounds of the Hausdorff measure of the Sierpinski gasket, which can approach the Hausdorff measure of the Sierpinski gasket ...
Baoguo Jia
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On the Centred Hausdorff Measure

Journal of the London Mathematical Society, 2000
Summary: Let \(\nu\) be a measure on a separable metric space. For \(t,q\in\mathbb{R}\), the centred Hausdorff measure \(\mu^h\) with the gauge function \(h(x,r)= r^t(\nu B(x,r))^q\) is studied. The dimension defined by these measures plays an important role in the study of multifractals. It is shown that if \(\nu\) is a doubling measure, then \(\mu^h\)
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Hausdorff and packing measure for solenoids

Ergodic Theory and Dynamical Systems, 2003
Summary: We prove that the solenoid with two different contraction coefficients has zero Hausdorff and positive packing measure in its own dimension and the SBR measure is equivalent to the packing measure on the attractor. Further, we prove similar statements for Slanting Baker maps with intersecting cylinders (in \(\mathbb{R}^{2}\)).
Rams, Michał, Simon, Károly
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Hausdorff measures on the Wiener space

Potential Analysis, 1992
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Feyel, D., de La Pradelle, A.
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Hausdorff measure and linear forms.

Journal für die reine und angewandte Mathematik (Crelles Journal), 1997
It is shown that given any dimension function \(f\), the Hausdorff measure \({\mathcal H}^f\) of the set of well approximable linear forms \(W(m,n; \psi)\) is zero or infinity depending on whether a certain volume sum converges or diverges. This is a Hausdorff measure analogue of the classical Khintchine-Groshev theorem where the \(mn\)-dimensional ...
Dickinson, Detta, Velani, Sanju L.
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Hausdorff measure of Sierpinski gasket

Science in China Series A: Mathematics, 1997
The author gives a new estimate on the upper bound of the Hausdorff measure of the Sierpiński gasket \(S: H^s(S)\leq{25\over 22}\left({6\over 7}\right)^s\), where \(s= \log_23\) is the Hausdorff dimension of \(S\). The result improves the previous estimates obtained by the author [Proc. Nat. Sci. (English Ed.) 7, No.
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HAUSDORFF DIMENSION AND HAUSDORFF MEASURES OF JULIA SETS OF ELLIPTIC FUNCTIONS

Bulletin of the London Mathematical Society, 2003
Let \(f: \mathbb{C}\to\overline{\mathbb{C}}\) be an elliptic function and \(q\) be the maximal multiplicity of all poles of \(f\). The authors prove that the Hausdorff dimension of the Julia set of \(f\) is greater than \({2q\over q+1}\), and the Hausdorff dimension of the set of points escaping to infinity is less than or equal to \({2q\over q+1 ...
Kotus, Janina, Urbański, Mariusz
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A Characteristic Property of Hausdorff Measure

Journal of the London Mathematical Society, 1950
Man bezeichne mit \(h(x)\) eine für \(x\ge 0\) definierte, stetige, streng wachsende Funktion mit \(h(0) = 0\) und \(\displaystyle\varliminf_{x\to +0} h(\alpha x)/h(x) > 0\) \((0 < \alpha < 1)\). Ist ein separabler, metrischer Raum \(X\) gegeben, so definiert man für \(E\subset X\) \(\text{h. m. }E = \displaystyle\lim_{\delta\to 0} \Lambda_h(E, \delta)\
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Hausdorff Measure and Local Measure

Journal of the London Mathematical Society, 1982
Johnson, Roy A., Rogers, C. A.
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Singular measures and Hausdorff measures

Israel Journal of Mathematics, 1969
An example is given of a family of singular probability measures on the unit interval which are supported on a set of fractional Hausdorff dimension but cannot be represented as Hausdorff measures.
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