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2019 Dynamics of Certain Distal Actions on Spheres
Riddhi Shah, Alok K. Yadav
Real Anal. Exchange 44(1): 77-88 (2019). DOI: 10.14321/realanalexch.44.1.0077

Abstract

Consider the action of \(SL(n+1,\mathbb{R})\) on \(\mathbb{S}^n\) arising as the quotient of the linear action on \(\mathbb{R}^{n+1}\setminus\{0\}\). We show that for a semigroup \(\mathfrak{S}\) of \(SL(n+1,\mathbb{R})\), the following are equivalent: \((1)\) \(\mathfrak{S}\) acts distally on the unit sphere \(\mathbb{S}^n\). \((2)\) the closure of \(\mathfrak{S}\) is a compact group. We also show that if \(\mathfrak{S}\) is closed, the above conditions are equivalent to the condition that every cyclic subsemigroup of \(\mathfrak{S}\) acts distally on \(\mathbb{S}^n\). On the unit circle \(\mathbb{S}^1\), we consider the ‘affine’ actions corresponding to maps in \(GL(2,\mathbb{R})\) and discuss the conditions for the existence of fixed points and periodic points, which in turn imply that these maps are not distal.

Citation

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Riddhi Shah. Alok K. Yadav. "Dynamics of Certain Distal Actions on Spheres." Real Anal. Exchange 44 (1) 77 - 88, 2019. https://doi.org/10.14321/realanalexch.44.1.0077

Information

Published: 2019
First available in Project Euclid: 27 June 2019

zbMATH: 07088964
MathSciNet: MR3951335
Digital Object Identifier: 10.14321/realanalexch.44.1.0077

Subjects:
Primary: ‎37B05‎, 54H20‎
Secondary: 43A60

Rights: Copyright © 2019 Michigan State University Press

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Vol.44 • No. 1 • 2019
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