1 results
DISTALITY OF CERTAIN ACTIONS ON
$p$-ADIC SPHERES
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 109 / Issue 2 / October 2020
- Published online by Cambridge University Press:
- 29 July 2019, pp. 250-261
- Print publication:
- October 2020
-
- Article
- Export citation
-
Consider the action of
$\operatorname{GL}(n,\mathbb{Q}_{p})$ 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 
$\operatorname{GL}(n,\mathbb{Q}_{p})$ 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 
$\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On 
${\mathcal{S}}_{n}$, we consider the ‘affine’ maps 
$\overline{T}_{a}$ corresponding to 
$T$ in 
$\operatorname{GL}(n,\mathbb{Q}_{p})$ 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.
