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.