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DISTALITY OF CERTAIN ACTIONS ON $p$-ADIC SPHERES

Part of: Semigroups Lie groups Topological dynamics Linear algebraic groups and related topics

Published online by Cambridge University Press:  29 July 2019

RIDDHI SHAH Open the ORCID record for RIDDHI SHAH [Opens in a new window]  and
ALOK KUMAR YADAV Open the ORCID record for ALOK KUMAR YADAV [Opens in a new window]
RIDDHI SHAH
Affiliation:
School of Physical Sciences,Jawaharlal Nehru University, New Delhi 110067, India email riddhi.kausti@gmail.com, rshah@jnu.ac.in
ALOK KUMAR YADAV*
Affiliation:
School of Physical Sciences,Jawaharlal Nehru University, New Delhi 110067, India email alokmath1729@gmail.com
*
✉alokmath1729@gmail.com
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Abstract

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.

Keywords

distal actionsaffine mapsp-adic unit spheres

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 22E35: Analysis on $p$-adic Lie groups
Secondary: 20M20: Semigroups of transformations, etc. 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 2 , October 2020 , pp. 250 - 261
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

R. Shah would like to acknowledge the support of DST (SERB) through the MATRICS research grant. A. K. Yadav would like to acknowledge the support for a research assistantship from DST-PURSE grant in Jawaharlal Nehru University.

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