关于自相似集的Hausdorff测度的一个判据及其应用

On a Criterion of the Hausdorff Measure of the Self-similar Sets and Applications

讨论了满足开集条件的自相似集．对于此类分形，用自然覆盖类估计它的Ｈａｕｓｄｏｒｆｆ测度只能得到一个上限，因而如何判断某一个上限就是它的 Ｈａｕｓｄｏｒｆｆ测度的准确值是一个重要的问题．本文给出了一个判据．作为应用，统一处理了一类自相似集，得到了平面上的一个Ｃａｎｔｏｒ集—Ｃａｎｔｏｒ尘的Ｈａｕｓｄｏｒｆｆ测度的准确值，并重新计算了直线上的Ｃａｎｔｏｒ集以及一个 Ｓｉｅｒｐｉｎｓｋｉ地毯的 Ｈａｕｓｄｏｒｆｆ测度．

The self-similar sets satisfying the open condition have been studied. An esti- mation of the fractal, by a class of natural covers, can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important. A criterion has been given in this paper. As applications, we have together discussed some self-similar sets, and obtained the exact value of the Hausdorff measure of a Cantor dust.