Mapping Zr into Zs with Maximal Contraction
Mapping Zr into Zs with Maximal Contraction
复制标题
将 Zr 映射到具有最大收缩的 Zs
DOI：
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发表时间：
1998
影响因子：
0.8
通讯作者：
R. Stong
中科院分区：
文献类型：
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作者：
R. Stong
抽象的。给定任何双射f：Zr→f：Zs，其中s≥ r，简单的体积比较表明，必须有一个普适常数K>0（仅取决于r和s）和无穷多对点x，y∈Zr，使得||f（x）-f（y）||> K|| X-Y|| r/s。这就限制了任何这样的双射可以实现多大的收缩。相反，我们证明了对任意s≥ r，存在一个双射f：Zr→Zs和一个常数C>0，使得对所有x，y∈Zr，我们有||f（x）-f（y）||<C|| X-Y|| r/s。换句话说，有一个双射f：Zr→Zs，它尽可能地缩小任何两点之间的距离，直到一个常数因子。这推广了分形图像处理中的一种构造，并肯定地回答了Michael Freedman的一个问题。
Abstract. Given any bijection f:Zr→f:Zs with s≥ r , easy volume comparisons show that there must be a universal constant K>0 (depending only on r and s ) and infinitely many pairs of points x,y∈Zr such that || f(x)-f(y)|| > K|| x-y||r/s . This puts a bound on how much contraction can be achieved for any such bijection. We show that, conversely, for any s≥ r there is a bijection f:Zr→Zs and a constant C>0 such that for all x,y∈Zr we have || f(x)-f(y)|| <C|| x-y||r/s . Phrased differently there is a bijection f:Zr→Zs which shrinks the distance between the images of any two points as much as possible, up to a constant factor. This generalizes a construction in fractal image processing and answers in the affirmative a question of Michael Freedman.