论文标题
大型集合图像的维度
Dimension of Images of Large Level Sets
论文作者
论文摘要
令$ k $为自然数字。我们考虑$ k $ -times连续差异化的实值函数$ f：e \ to \ mathbb {r} $，其中$ e $是具有正长度的行的一定间隔。对于$ 0 <α<1 $ let $i_α（f）$表示\ y Mathbb {r} $的值$ y \ y y \ y \ preImage $ f^{ - 1}（y）$具有Hausdorff dimension $ \ dim f^{ - fim f^{ - 1}（-1}（y）\geα$。我们考虑$i_α（f）$的hausdorff尺寸，作为$ f $范围的所有$ c^k（e，\ mathbb {r}）$的所有$ k $ - times times-times complunly-timesly-differentilly-differentible-differentible-differentible-e $ e $ to $ e $ in $ e $ to $ \ to Mathbb {r} $。我们表明，$ \ dimi_α（f）$的尖锐上限为$ \ displayStyle \ frac {1-α} k $。
Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<α<1$ let $I_α(f)$ denote the set of values $y\in\mathbb{R}$ whose preimage $f^{-1}(y)$ has Hausdorff dimension $\dim f^{-1}(y) \ge α$. We consider how large can be the Hausdorff dimension of $I_α(f)$, as $f$ ranges over the set $C^k(E,\mathbb{R})$ of all $k$-times continuously-differentiable functions from $E$ into $\mathbb{R}$. We show that the sharp upper bound on $\dim I_α(f)$ is $\displaystyle\frac{1-α}k$.
