The unit ball is not auto similar.

Question

I want read a prove that the unit ball $B$ is not auto similar. I mean that there is not similarities $f_1,...,f_n$ with contracting constants <1, such that $$B=\bigcup_{i=1}^n f_i[B] $$

Anyone knows where to find this proof?

Thanks!

Answer 1

The statement is false in $\mathbb R$ and many other metric spaces as well. So let's assume we're working in $n$-dimensional Euclidean space for $n>1$.

As is well known, a circle in the plane is uniquely determined by three points and a sphere in space by four points. More generally, an $n-1$-sphere in ${\mathbb R}^n$ is uniquely determined by finitely many points ($n+1$, I believe). As a result, $f_i(B)$ can only contain $n$ points of $B$; otherwise it would contain the entire boundary of $B$ contradicting that $f_i$ is a contraction.

Thus, $$\bigcup_{i=1}^m f_i(B)$$ can contain at most finitely many points of the boundary of $B$.