普遍连续映射图的维数
Dimensions of graphs of prevalent continuous maps
摘要：
设$K$是不可数紧度量空间，$C(K,\mathbb{R}^d)$表示赋最大范数的连续映射集$f\colon K \to \mathbb{R}^d$.本文的目的是确定流行的$f\in C(K,\mathbb{R}^d)$的图形的各种分形维数。作为本文的主要结果，我们证明了如果$K$有1000个孤立点，那么流行的$f\in C(K,\mathbb{R}^d)$图的下盒维数和上盒维数分别为$\underline{\dim}_B K+d$和$\overline{\dim}_B K+d$.这推广了Gruslys，Jonušas，Mijovic，Ng，Olsen和Petrykiewicz的一个定理。我们证明了流行的$f\in C(K,\mathbb{R}^d)$的图具有填充维数$\dim_P K+d$，推广了Darta，Darji和Elekes的结果. Darji和Elekes证明了流行的$f\in C(K,\mathbb{R}^d)$的图的豪斯多夫维数等于$\dim_H K+d$。本文根据Fraser和Hyde的方法给出了这一命题的一个较简单的证明。
英文摘要：
Let $K$ be an uncountable compact metric space and let $C(K,\mathbb{R}^d)$ denote the set of continuous maps $f\colon K \to \mathbb{R}^d$ endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$. As the main result of the paper we show that if $K$ has finitely many isolated points then the lower and upper box dimension of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ are $\underline{\dim}_B K+d$ and $\overline{\dim}_B K+d$, respectively. This generalizes a theorem of Gruslys, Jonušas, Mijovic, Ng, Olsen, and Petrykiewicz. We prove that the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ has packing dimension $\dim_P K+d$, generalizing a result of Balka, Darji, and Elekes. Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ equals $\dim_H K+d$. We give a simpler proof for this statement based on a method of Fraser and Hyde.
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