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1.
In [10], the notion of homogeneous perfect sets as a generalization of Cantor type sets is introduced and their Hausdorff
and lower box-counting dimensions are studied. In this paper, we determine their exact packing and upper box-counting dimensions
based on the length of their fundamental intervals and the gaps between them. Some known results concerning the dimensions
of Cantor type sets are generalized.
This work was supported by NSFC (10571138). 相似文献
2.
Li-Min Shi 《Journal of Mathematical Analysis and Applications》2006,318(1):190-198
In this paper we obtain a lower bound for the Hausdorff dimension of recurrent sets and, in a general setting, we show that a conjecture of Dekking [F.M. Dekking, Recurrent sets: A fractal formalism, Report 82-32, Technische Hogeschool, Delft, 1982] holds. 相似文献
3.
I. S. Baek 《Acta Mathematica Hungarica》2003,99(4):279-283
A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor
dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will
show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set
with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
4.
We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system
of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping
infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran
to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff
dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated
function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition
recently introduced by Ngai and Wang.
相似文献
5.
Toby C. O'Neil 《Transactions of the American Mathematical Society》2007,359(11):5141-5170
For a compact set and a point , we define the visible part of from to be the set (Here denotes the closed line segment joining to .)
In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,
We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .
6.
§ 1 IntroductionThe book[1 ] and the references therein show thatthe structure of arithmetic sums ofCantor sets is relevantto natural questions in smooth dynamics.Palis and Takens[1 ] askedabout the structure of the sums of two Cantor sets and conjectured that“typically” theyhave either zero Lebesgue measure or contained intervals. In 1 997,Solomyka[2 ] showedthatfor eachγ∈ 0 ,12 ,the set Kγ+Kλ(where Kλ,Kγis the middle-α Cantorset forα=1 -2λ or 1 -2γ) of two centered Cantor s… 相似文献
7.
We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.
8.
In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite. 相似文献
9.
《Indagationes Mathematicae》2019,30(5):862-873
An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets. 相似文献
10.
Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two. 相似文献
11.
吴敏 《应用数学学报(英文版)》2000,16(2):140-148
Let{a.}.>1beasequenceofrealnumberssatisfyingEla.l~coandtiman=0.n=1n-cocoThenRademacherseriesZa.(l--Ze.(x))takeseverypreassignedrealvalueN(cardinaln=1numberofthecontinuum)timesforx6(0,1],whereE.(x)isthen-thdigitofthe(unique)non--terminating2--adicexpansionofx6(0,l].W.A.Beyer[1]showsthatif{a.}.>1Efi,{a.}411,thenforanyaCR,dimH{xE(0,1];Za.(1--26.(x))~a}~1;ifn=1co{a.}.21411,nlLmcoan~0,,Zla.--a.--if相似文献
12.
In this paper, we give the exact upper densities of Hausdorff measures of a class of symmetric perfect sets. 相似文献
13.
Uniform perfectness of self-affine sets 总被引:2,自引:0,他引:2
Feng Xie Yongcheng Yin Yeshun Sun 《Proceedings of the American Mathematical Society》2003,131(10):3053-3057
Let be affine maps of Euclidean space with each nonsingular and each contractive. We prove that the self-affine set of is uniformly perfect if it is not a singleton.
14.
Cheng-qin QU 《应用数学学报(英文版)》2013,29(1):117-122
We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets. 相似文献
15.
16.
In this paper,we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition(OSC).As applications,we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure. 相似文献
17.
Pedro Mendes 《Proceedings of the American Mathematical Society》1999,127(11):3305-3308
In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).
18.
SHEN Liang LMAM School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2006,49(9):1284-1296
Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2. 相似文献
19.
20.
We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献