Question

Consider an ultrametic space $X=K^n$ (for the norm $\| x\| =\max_{i=1,n} (\mid {x_1}\mid,\dots,\mid{x_n}\mid)$ where $K$ is an ultrametric field. Let $B(1):=\lbrace x \in X \mid \|x\| \leq 1\rbrace$ be the unit ball. Equip $X$ with Haar measure. Is it possible to partition $B(1)$ into $k$ smaller balls $B_{a_1}(r_1),\dots, B_{a_k}(r_k)$ (where $B_a(r)$ is the closed ball of radius $r$ and center $a$), $k>1$?

Or put in a different way: is there a measure preserving bianalytic map taking $B(1)$ to the union of $B_{a_i}(r_i)$?

Thank you

Answer 1

If you can do the case of $K$ itself, decompose into $k$ smaller balls, then for $K^n$ you can decompose into $k^n$ smaller balls. For some ultrametric fields this works. Indeed, you say "Haar measure" so you must be assuming local compactness, and it works for any such field, since they are finite extensions of $p$-adic fields.