Real Analysis Exchange

Dynamics of Certain Distal Actions on Spheres

Riddhi Shah and Alok K. Yadav

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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.

Article information

Source
Real Anal. Exchange, Volume 44, Number 1 (2019), 77-88.

Dates
First available in Project Euclid: 27 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.rae/1561622433

Digital Object Identifier
doi:10.14321/realanalexch.44.1.0077

Mathematical Reviews number (MathSciNet)
MR3951335

Zentralblatt MATH identifier
07088964

Subjects
Primary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions

Keywords
Topological dynamics distal action affine map fixed point

Citation

Shah, Riddhi; Yadav, Alok K. Dynamics of Certain Distal Actions on Spheres. Real Anal. Exchange 44 (2019), no. 1, 77--88. doi:10.14321/realanalexch.44.1.0077. https://projecteuclid.org/euclid.rae/1561622433


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