共查询到20条相似文献,搜索用时 62 毫秒
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本文研究了Cantor集和其并的自相似性.利用Cantor展式的方法,得到了关于Cantor集和迭代函数系的一个基本关系:T∪(T+α)为自相似的当且仅当存在一个非负整数n使得α=±(k2-k1)dn.进一步,若T∪(T+α)是自相似的,则它满足开集条件. 相似文献
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设K■R为由满足强分离条件的相似压缩映射族{h_k(x)=a_kx+b_k,k=1,…,N}所生成的自相似集,此处N≥2.对一个概率向量p=(p_1,…p_N),设γ_p为对应的支撑在K上的自相似测度.在单位线段上定义广义Cantor函数f(x)=γ_p([0,x]∩K),这里假设.设数ξ和q+β(q)分别由■和■,β'(q)=-1所确定.本文研究集合K中使得函数f(x)的导数不存在的点集,使得函数f(x)的导数为零的点集,及使得函数f(x)的导数为无穷的点集的维数,本文结果表明上述定义的两个数可以给出这些维数的一个很好的刻画. 相似文献
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Cantor集的自乘积集的Hausdorff测度的下界 总被引:1,自引:0,他引:1
证明了三分Cantor集C的自乘积集C×C的Hausdorff测度满足 H~(log_3)~4(C×C)≥1.48329。 相似文献
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本文利用类比的方法,将Cantor集上定义的Cantor函数进行了推广.首先给出了正测度Cantor集及正测度Cantor函数的定义;然后通过严格的证明得到了正测度Cantor函数的一些性质,并给出了正测度Cantor函数的一些应用;最后通过实例说明,由于正测度Cantor函数构造的特殊性,可以用来作为一些命题的反例. 相似文献
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广义自相似集的维数研究 总被引:8,自引:0,他引:8
广义自相似集的维数研究华苏(清华大学应用数学系,北京100084)ONTHEDIMENSIONOFGENERALIZEDSELR-SIMILARSETS¥HUASU(DepartmentofAppliedMathematics,TsinghuaUni... 相似文献
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对于任意的正整数p≥3,用{an}n≥0表示p-进展式数字只取偶数的非负整数所构成的数列.我们给出了an的增长阶为logsp,其中s=[p/2],[·]为取上整函数.证明了{an/nlogs p}n≥1在[2s-2/p-1,2]中稠密.并从测度的角度对该稠密性加以阐释. 相似文献
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Ignacio Garcia Ursula Molter Roberto Scotto 《Proceedings of the American Mathematical Society》2007,135(10):3151-3161
We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.
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Roger L. Kraft 《Transactions of the American Mathematical Society》2000,352(3):1315-1328
Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .
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On the geometry of random Cantor sets and fractal percolation 总被引:1,自引:0,他引:1
Random Cantor sets are constructions which generalize the classical Cantor set, middle third deletion being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1]
e
. This answers a question of Dekking and Meester,(9) (b) The special case of fractal percolation arises when the substitution is as follows: The cube [0,1]
d
is divided intoM
d subcubes of side-lengthM
–, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep
c(M, d) ofp, marking the existence of crossings of [0,1]
d
contained in the limit set, satisfiesp
c(M, d)p
c(d) asM, wherep
c(d) is the critical probability of site percolation on a latticeL
d
obtained by adding certain edges to the hypercubic lattice
d
. This result generalizes in an unexpected way a finding of Chayes and Chayes,(4) who studied the special case whend=2. 相似文献
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Projections of random Cantor sets 总被引:1,自引:0,他引:1
K. J. Falconer 《Journal of Theoretical Probability》1989,2(1):65-70
Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases. 相似文献
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We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.
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Jun Wang 《Journal of Mathematical Analysis and Applications》2011,379(2):637-812
Let K(a) be the symmetrical Cantor set generated by ?0(x)=ax and ?1(x)=ax+(1−a), where 0<a<1/2. Let s be the Hausdorff dimension of K(a) and μ the Cantor measure. In this paper, under the hypothesis that a is slightly greater than 1/3, we obtain the explicit formulas of the upper and lower s-densities Θ?s(μ,x), for every point x∈K(a). Moreover, we describe the range of a key quantity τ(x) in these formulas. 相似文献
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We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R[Hdim(C)]+1, where [Hdim(C)] denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural sequence of refining coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of Rd is bi-Lipschitz embeddable in Rd+1.We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry. 相似文献