Distality of Certain Actions on $p$-adic Spheres

Abstract

Consider the action of $GL(n,{\mathbb{Q}}_p)$ on the $p$-adic unit sphere ${\mathcal{S}}_n$ arising from the linear action on ${\mathbb{Q}}_p^n\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $GL(n,{\mathbb{Q}}_p)$ on ${\mathcal{S}}_n$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_n$. (2) the closure of the image of $\mathfrak{S}$ in $PGL(n,{\mathbb{Q}}_p)$ is a compact group. On ${\mathcal{S}}_n$, we consider the `affine' maps ${\overline{T}}_a$ corresponding to $T$ in $GL(n,{\mathbb{Q}}_p)$ and a nonzero $a$ in ${\mathbb{Q}}_p^n$ satisfying $\Vert T^{-1}(a){\Vert }_p<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that ${\overline{T}}_a$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_n$. The dynamics of `affine' maps on $p$-adic unit spheres is quite different from that on the real unit spheres.