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.

SOME PROBLEMS ON FRACTAL GEOMETRY AND TOPOLOGICAL DYNAMICAL SYSTEMS

This paper is a continuation of [14], some new problems on fractal geometry and topological dynamical systems have been posed.

Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

A Local Property of Hausdorff Centered Measure of Self-Similar Sets

We analyze the local behavior of the Hausdorff centered measure for selfsimilar sets. If E is a self-similar set satisfying the open set condition, then Cs(E∩B(x,r)) ≤(2r)s for all x ∈ E and r > 0, where Csdenotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.

A THEORETICAL FRAMEWORK FOR THE CALCULATION OF HAUSDORFF MEASURE-SELF—SIMILAR SET SATISFYING OSC

A theoretical framework for the calculation of Hausdorff measure of self-similar sets satisfying OSC has been established.

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

为 Sierpinski 垫板，由使用由特殊常规等边六角形组成的一种盖子，我们定义等价于 Hausdorff 措施的一项新措施并且获得这项措施更在下固定。而且， theSierpinski 垫板的 Hausdroff 措施的下列更低的界限被完成了 H^s ≥ 0.670432 在 S 表示 Sierpinski 垫板的地方， s =dim_H = log_23，和 H^s 表示 S 的 s 维的 Hausdorff 措施。上述结果改进发展了在那[2 ] 。

THE EXACT MEASURES OF A CLASS OF SELF-SIMILAR SETS ON THE PLANE

Let S  R2 be the attractor of the iterated function system {f1, f2, f3} iterating on the unit equilateral triangle S0, where fi(x) = λix+bi, i = 1,2,3, x = (x1,x2), b1 = (0,0), b2 = (1λ2,0), b3 = (1-2λ 3, √ 3 2 (1-λ3)). This paper determines the exact Hausdorff measure, centred covering measure and packing measure of S under some conditions relating to the contraction parameter.

A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems

In this work,by virtue of the properties of weakly almost periodic points of a dynamical system(X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,that is,M(T) = X,then the following statements are equivalent:(1)(X,T) is ergodic mixing;(2)(X,T) is topologically double ergodic;(3)(X,T) is weak mixing;(4)(X,T) is extremely scattering;(5)(X,T) is strong scattering;(6)(X×X,T×T) is strong scattering;(7)(X×X,T×T) is extremely scattering;(8) For any subset S of N with upper density 1,there is a c-dense Fσ-chaotic set with respect to S.As an application,the authors show that,for the sub-shift σA of finite type determined by a k×k-(0,1) matrix A,σA is strong mixing if and only if σA is totally transitive.