HAUSDORFF MEASURES OF A CLASS OF SIERPINSKI CARPETS

In this paper, a lemma as a new method to calculate the Hausdorff measure of fractal is given. And then the exact values of Hausdorff measure of a class of Sierpinski sets which satisfy balance distribution and dimension ≤ 1 are obtained.

Hausdorff Measure of Linear Cantor Set

We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and the first gap. This condition is easy to use. Key words linear Cantor set - Hausdorff measure - strong seperated condition CLC number O 174. 12 Foundation item: Supported by the National Natural Science Foundation of China (10171028)Biography: Ma Chao (1975-), male, Ph. D. candidate, research direction: fractal geometry.

ON THE EXACT HAUSDORFF MEASURE OF A CLASS OF SELF-SIMILAR SETS SATISFYING OPEN SET CONDITION

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.

AN APPROXIMATION METHOD TO ESTIMATE THE HAUSDORFF MEASURE OF THE SIERPINSKI GASKET

In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/(l + l/2n-3)s≤Hs(S)≤ Pn(S). An algorithm is presented to get Pn(S) for n ≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.

Estimation of the Hausdorff Measure of a Kind of Sierpinski Carpet

Suppose F0 is an arbitrary triangle and F is a kind of Sierpinski carpet generated by F0.We construct a projection mapping to obtain the lower bound of the Hausdorff measure of F ;meanwhile the upper bound of the Hausdorff measure of F is calculated by the general covering.

The Convergence of Statistically Self Similar Sets and the Upper Bound and Lower Bound of Hausdorff Measure

We constructed a class of self-similar sets and proved the convergence in this paper.Besides these,the upper bound and lower bound of Hausdorff measures of them were given too.

The Proof of the Sufficient and Necessary Conditions which Make the Hausdorff Measure of the Generalized Non-uniform Cantor Set'exist

The paper succeeds in the obtaining a class of generalized non-uniform Cantor set based on the iteration （1）： Si（x） = αix ＋ bi, x ∈ [0, 1], i = 1,2,…, m, where 0 〈 αi 〈 1, i = 1,2,…,m; bi ＋ αi 〉 0, i = 1,2,…,m- 1, b1 = 0 and αm ＋ bm = 1. Providing the sufficient and necessary conditions of its existence Hausdorff measure.

BOUNDS OF THE HAUSDORFF MEASURE OF SIERPINSKI CARPET

By means of the idea of [2]（Jia Baoguo,J.Math.Anal.Appl.In press） and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.

A NEW LOWER BOUND OF THE HAUSDORFF MEASURE OF THE SIERPINSKI GASKET

For the Sierpinski gasket, by using a sort of cover consisting of special regular hexagons, we define a new measure that is equivalent to the Hausdorff measure and obtain a lower bound of this measure. Moreover, the following lower bound of the Hausdroff measure of the Sierpinski gasket has been achieved H^s（S）≥0.670432,where S denotes the Sierpinski gasket, s = dimn（S） = log23, and H^s（S） denotes the s-dimensional Hausdorff measure of S. The above result improves that developed in .

HAUSDORFF MEASURE OF UNIFORM SELF-SIMILAR FRACTALS

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system （UIFS） on R^d. Denote by D the Hausdorff dimension, by H^D（E） the Hausdorff measure and by diam（E） the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H^O（E） = diam（E）^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H^D（E） 〈 diam（E）^D, if c is small enough.

THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON

In this paper, we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E- the self-similar sets generating in a unit regular pentagon on the plane. Under some conditions, we show the natural covering is the best one, and the Hausdorff measures of those sets are euqal to ｜E｜^S, where s = dimHE.

THE INTEGRAL FORMULA FOR CALCULATINGTHE HAUSDORFF MEASURE OF SOME FRACTAL SETS

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.

On the Lower Bound of the Hausdorff Measure of the Koch Curve

This paper gives a lower bound of the Hausdorff measure of the Koch curve bymeans of the mass distribution principle.

Subsets with Finite Measure of Multifractal Hausdorff Measures

Let ( be a Borel Probability measure on R^d. q, t,∈ R. Let H_(^(q,t) denote the multifractal Hausdorff measure. We prove that, when satisfies the so-called Federer condition, for a closed subset E∈R^n, such that H_(^(q,t) (E) > 0, there exists a compact subset F of E with 0 < H_(^(q,t) (F) <∞ , i.e, the finite measure subsets of multifractal Hausdorff measure exist.

Hausdorff Measure of Homogeneous Cantor Set

This paper gives the Hausdorff measure of a class of homogeneous Cantor sets.

Potential Analysis on Carnot Groups,Part Ⅱ:Relationship between Hausdorff Measures and Capacities

In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general results,we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G.Given any set E(?)G,B_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)G of Hausdorff dimension Q-d,where 0<d<Q,B_(α,p)(E)=0 holds if and only if αp(?)d. A version of O.Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).

Hausdorff Measures for a Class of Homogeneous Cantor Sets

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.

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W (t), t ∈ R}, {B(t), t ∈ R +} be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (X 1(t),…, X d (t)) and X 1(t),…, X d (t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q ? (0, ∞), the exact Hausdorff measures of the image X(Q) = {X(t): t ∈ Q} and the graph GrX(Q) = {(t, X(t)): t ∈ Q} are established.

Strong Mixing Subshift of Finite Type and Hausdorff Measure of Its Chaotic Set

We present a sufficient and necessary condition for the subshift of finite type to be a measure-preserving transformation or to be a strong mixing measure-preserving transformation with respect to the Hausdorff measure. It is proved that a strong mixing subshift of finite type has a chaotic set with full Hausdorff measure.

THE HAUSDORFF MEASURE FOR THE SAMPLE PATH OF THE WESTWATER PROCESS

It is proved that φ(s) = s2 log |logs| is the correct function to give the Hausdorff measureof a sample path of the Westwater process.