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.

A Negative Answer to a Conjecture on Upper Convex Density

Let E be a self-similar set satisfying the open set condition. Professor Xu conjectures in his doctoral degree thesis that if H^8（E） 〈｜E｜^8, then for any x ∈ E, the inequality ^-D^3C（E,x）〉H^8（E）/｜E｜^8holds, where 3 = dimH（E）. The above conjecture is negatively answered in this＇paper.

An Equivalent Condition for the Self Similar Sets on the Real Line to Have Best Coverings

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.： Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17（2）, 493-502 （2004）].

A Note on Upper Convex Density

For a self-similar set E satisfying the open set condition,upper convex density is an important concept for the computation of its Hausdorff measure,and it is well known that the set of relative interior points with upper convex density 1 has a full Hausdorff measure.But whether the upper convex densities of E at all the relative interior points are equal to 1？ In other words,whether there exists a relative interior point of E such that the upper convex density of E at this point is less than 1？ In this paper,the authors construct a self-similar set satisfying the open set condition,which has a relative interior point with upper convex density less than 1.Thereby,the above problem is sufficiently answered.

Separation Properties for MW-Fractals

Corresponding to the irreducible 0 - 1 matrix (aij)n×n, take similitude contraction mappings ij for each aij=1, in Rd with ratio 0【rij【1. There are unique nonempty compact sets F1,…, Fn satisfying for each 1in, Fi = ∪j=1 aij=1n ij=（Fj）. We prove that open set condition holds if and only if Fi is an s-set for some 1in, where s is such that the spectral radius of matrix (rijs)n×n is 1.