We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.
We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.
As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually `thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension.
Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.
On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets
设$K$是$\rd$中具有正豪斯多夫维数的紧集.利用分数布朗运动，证明了在$K$上的一组普遍的连续函数中，图的豪斯多夫维数等于$\dim_{\mathcal H}(K)+1$ .这是可能的最大值。这一结果推广了J.M.弗雷泽和J.T. Hyde在{ \it Fractal and Related Fields~2}会议上曝光。讨论了$α$ -Hölderian函数的情形。
New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新网络阿尔弗雷德·雷尼数学研究所）；Institute of Mathematics and Informatics, Eszterházy Károly Catholic University（埃斯泰尔戈伊卡罗利天主教大学数学与信息学研究所）；Institute of Mathematics, ELTE Eötvös Loránd University（罗兰大学数学研究所）
