## Real Analysis Exchange

### Dynamics of Certain Distal Actions on Spheres

#### 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

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

#### 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