论文标题
Cantor套装具有高维预测
Cantor sets with high-dimensional projections
论文作者
论文摘要
1994年，J.COBB在$ \ Mathbb r^3 $中构建了一个Tame Cantor设置，其投影为$ 2 $ - 平面是一维的。我们表明，安托万的项链可以作为一个cantor套装的一个例子，所有预测都是一维且连接的。我们证明，每个Cantor设置为$ \ Mathbb r^n $，$ n \ geqslant 3 $，可以通过小的环境同位素移动，以便将结果的cantor放置在每个$（N-1）$ - 平面为$（n-2）$（n-2）$ - dimensional中。我们表明，如果$ x \ subset \ mathbb r^n $，$ n \ geqslant 2 $，是一种零维的压缩，其投影到某个平面$π\ subset \ subset \ subbb r^n $带有$ \ dimiss in \ in \ in \ in \ in \ {1，2，2，2，2，n-2，n-2，n-2，n-2，n-2，n-2，n-2，n-2，n-2，n-2，然后这扩展了D.R.Mcmillan，Jr。（1964）和D.G. Wright，J.J.Walsh（1982）的结果。我们使用定义路易·安托万（Louis Antoine）的序列的技术。
In 1994, J.Cobb constructed a tame Cantor set in $\mathbb R^3$ each of whose projections into $2$-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of whose projections are one-dimensional and connected. We prove that each Cantor set in $\mathbb R^n$, $n\geqslant 3$, can be moved by a small ambient isotopy so that the projection of the resulting Cantor set into each $(n-1)$-plane is $(n-2)$-dimensional. We show that if $X\subset \mathbb R^n$, $n\geqslant 2$, is a zero-dimensional compactum whose projection into some plane $Π\subset \mathbb R^n$ with $\dim Π\in \{1, 2, n-2, n-1\}$ is zero-dimensional, then $X$ is tame; this extends some particular cases of the results of D.R.McMillan, Jr. (1964) and D.G.Wright, J.J.Walsh (1982). We use the technique of defining sequences which comes back to Louis Antoine.
