景流的动力学和测量的几何学
Dynamics of the scenery flow and geometry of measures
摘要：
我们采用遍历理论机械的风景流来解决经典的几何测度理论问题的欧几里得空间。我们的主要结果包括一个尖锐的版本的圆锥密度定理，我们表明，是密切相关的整流。此外，我们表明，维数论的测量理论的孔隙度可以减少到其集理论的版本，豪斯多夫和包装尺寸产生相同的最大尺寸多孔，甚至平均多孔措施，极值措施存在，可以选择满足广义的自相似性的概念。这些都是早期发现的在一些特殊情况下成立的现象的尖锐的一般公式。
英文摘要：
We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.
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