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.

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在 -ADIC 领域的某些行动的远距离

考虑行动$\operatorname{GL}(n,\mathbb{Q}_{p})$在$p$-adic 单位球体${\mathcal{S}}_{n}$产生于线性作用$\mathbb{Q}_{p}^{n}\setminus \{0\}$. 我们证明对于半群的作用$\mathfrak{S}$的$\operatorname{GL}(n,\mathbb{Q}_{p})$在${\mathcal{S}}_{n}$, 以下是等价的: (1)$\mathfrak{S}$作用于远端${\mathcal{S}}_{n}$; (2) 关闭图像$\mathfrak{S}$在$\operatorname{PGL}(n,\mathbb{Q}_{p})$是一个紧群。在${\mathcal{S}}_{n}$，我们考虑“仿射”图$\overline{T}_{a}$对应于$T$在$\operatorname{GL}(n,\mathbb{Q}_{p})$和一个非零$a$在$\mathbb{Q}_{p}^{n}$令人满意的$\Vert T^{-1}(a)\Vert_{p}<1$. 我们证明存在一个紧开子群$V$，这取决于$T$, 这样$\overline{T}_{a}$对每个非零都是远端的$a\in V$当且仅当$T$作用于远端${\mathcal{S}}_{n}$. “仿射”地图的动态$p$-adic 单位球体与真实单位球体上的完全不同。