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Dimensions of cookie-cutter-like sets   总被引:1,自引:0,他引:1  
The cookie-cutter-like sets are defined as the limit sets of a sequence of classical cookie-cutter mappings. By introducing Gibbs-like measures, we study the dimensions, Hausdorff and packing measures of the CC-like sets, and then discuss the continuous dependence of the dimensions.  相似文献
2.
We consider the convolution transforms of measures on ℝ d defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution function and the local Lipschitz exponent of the measure. In particular, the results can be applied to the Poisson and Gauss-Weierstrass kernels. We can then obtain some singular boundary behavior of positive harmonic or parabolic functions on ℝ + d+1 by multifractal analysis of measures. Research supported by the NNSF of China  相似文献
3.
Let ℳ(¦n k ¦k⩾1,¦c k ¦k⩾1) be the collection of homogeneous Moran sets determined by ¦n k ¦k⩾1 and ¦c k ¦k⩾1, where ¦n k ¦k⩾1 is a sequence of positive integers and ¦c k ¦k⩾1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in ℳ are determined. The result is proved that for any values between the maximal and minimal values, there exists an element in ℳ(¦n k ¦k⩾1,¦c k¦k⩾1) such that its Hausdorff dimension is equal tos. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.  相似文献
4.
The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.  相似文献
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