Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdorff dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdorff dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdorff dimension.

中文翻译：

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分形立方体的尺寸下降现象

令E为度量空间。我们介绍的连通指数的概念ê，这是不平凡的连接组件的工会的维数Ë。我们表明，分形立方体的连通性指数ê严格小于的维数Ë只要Ë拥有一个简单的连接组件。因此，连通性指数是一个新的Lipschitz不变量。此外，我们研究了连通性指数和拓扑Hausdorff维数之间的关系。