arXiv 1408.21762014-08math.CAmath.PR

纤维的Hausdorff维数和填充维数与普遍连续映射的图

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

Richárd Balka, Udayan B. Darji, Márton Elekes

备注：: 42 pages
发表信息：: Adv. Math. 293 (2016), 221--274

摘要：

害羞和流行的概念推广的性质是零和充分的哈尔措施任意（不一定是局部紧）波兰集团。本文的主要目标是回答以下问题：我们可以说什么的Hausdorff和包装维的纤维的流行连续地图？设$K$是不可数紧度量空间.我们证明了流行的$f\in C(K,\mathbb{R}^d)$有许多纤维几乎最大的Hausdorff维数。这推广了Doughmann的一个定理，并得到了普遍的$f\in C(K,\mathbb{R}^d)$具有最大Hausdorff维数的图，推广了Bayart和Heurteaux的一个结果.我们得到类似的结果的包装尺寸。我们发现，对于流行的$f\in C([0,1]^m,\mathbb{R}^d)$的$y\in f([0,1]^m)$的集合，其中$\dim_H f^{-1}(y)=m$包含一个稠密的开集，具有关于占领措施$λ^m \circ f^{-1}$，其中$\dim_H$和$λ^m$表示Hausdorff维数和$m$维勒贝格测度，分别满测度。当$[0,1]^m$被任意满足开集条件的自相似集所取代时，我们也证明了类似的结果。我们不能用Lebesgue测度来代替上述声明中的占用测度：我们证明了正多个水平集是单例的函数$f\in C[0,1]$在$C[0,1]$中形成一个非害羞集。为了做到这一点，我们推广了Antunović，Burdzy，Peres和Ruscher的一个定理。作为一个补充的结果，我们证明了$f\in C[0,1]$的功能，$\dim_H f^{-1}(y)=1$的所有$y\in (\min f,\max f)$形成一个非害羞集$C[0,1]$。我们还证明了关于广义Hausdorff测度用正测度代替大Hausdorff维数的更尖锐的结果，从而回答了Fraser和Hyde的一个问题.

英文摘要：

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let $K$ be an uncountable compact metric space. We prove that the prevalent $f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of $y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open set having full measure with respect to the occupation measure $λ^m \circ f^{-1}$, where $\dim_H$ and $λ^m$ denote the Hausdorff dimension and the $m$-dimensional Lebesgue measure, respectively. We also prove an analogous result when $[0,1]^m$ is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions $f\in C[0,1]$ for which positively many level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions $f\in C[0,1]$ for which $\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in $C[0,1]$. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.

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