m分Cantor尘的Hausdorff测度

Hausdorff measure of m-Cantor Dust

为得到一类相似分形的Hausdorff测度准确值.给出了m分Cantor尘的几何结构,利用几何度量关系对m分Cantor尘的Hausdorff测度准确值进行研究.证明了m分Cantor尘的Hausdorff测度准确为Hs(E)=1/((m-1)s)[(m-2k+1)2+(m-1)2]s/2,其中s=logm 4,m≥4,1≤k≤m.结果表明它是Cantor尘和Sierpinski地毯的Hausdorff测度的准确值的推广,4分Cantor尘和4分Sierpinski地毯的Hausdorff测度的准确值是其特例.

To obtain the exact Hausdorff measure value of a kind of m-Cantor dust. We present the geometry construction and discuss the exact Hausdorff measure of this kind of m-Cantor dust. Proving the theorem that the exact value of m-Cantor dust is H^s（E）=1/（m-1）^s[（m-2k＋1）^2＋（m-1）^2]^s/2, where s = logm 4, m ≥ 4,1 ≤ k ≤ m, through geometric metric relation. According the properties of similar fractal geometry, we obtain the exact Hausdorff measure value of a kind of m-Cantor dust, which is extension about the exact Hausdorff measure value of 4-Cantor dust and 4-Sierpinski carpet.