Real Analysis Exchange

Dynamics of Certain Distal Actions on Spheres

Riddhi Shah and Alok K. Yadav

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 27 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Topological dynamics distal action affine map fixed point


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.

Export citation