平均wiggly连续统的度量性质
Metric properties of mean wiggly continua
摘要：
本文研究了正密度尺度下波动集的豪斯多夫维数的上界。主要的技术成分是，对于每个连续统K，构造一个Borel概率测度μ，该测度μ具有以下性质：在每个球B（x，r）上，x在K中，该测度以r \exp（-g（x，r））的普适常数倍为界，其中g（x，r）> 0是一个显函数。连续统K在K中的点x处是平均波动的，其中g（x，r）随着r趋于0而对数增长到无穷大。平均波动连续统理论通过维数的乘积公式导出康托集的豪斯多夫维数的新估计。我们还证明了渐近平坦集的豪斯多夫维数为1和渐近无孔连续的最大维数。该理论的另一个应用是有理动力学中拓扑Collet-Eckmann映射的几何Bowen二分法。特别是，平均摆动连续体是动态自然的，因为它们作为二次多项式的朱利亚集出现，参数来自Mandelbrot集边界上的脱殊集。
英文摘要：
We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure μwith the property that on every ball B(x,r), with x in K, the measure is bounded by a universal constant multiple of r\exp(-g(x,r)), where g(x,r) > 0 is an explicit function. The continuum K is mean wiggly at exactly those points x in K where g(x, r) has a logarithmic growth to infinity as r goes to 0. The theory of mean wiggly continua leads, via the product formula for dimensions, to new estimates of the Hausdorff dimension for Cantor sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymptotically non-porous continua are of the maximal dimension. Another application of the theory is geometric Bowen's dichotomy for Topological Collet-Eckmann maps in rational dynamics. In particular, mean wiggly continua are dynamically natural as they occur as Julia sets of quadratic polynomials for parameters from a generic set on the boundary of the Mandelbrot set.
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