Dynamics of Certain Distal Actions on Spheres

TL;DR

This paper characterizes when semigroups of linear transformations act distally on spheres, showing equivalences with compactness of their closures, and explores affine actions on the circle regarding fixed and periodic points.

Contribution

It establishes a precise criterion linking distality of semigroup actions on spheres to the compactness of their closures, and analyzes affine circle actions for fixed and periodic points.

Findings

Semigroup acts distally iff its closure is a compact group.

Closed semigroups with distal actions have all cyclic subsemigroups acting distally.

Affine actions on the circle can have fixed or periodic points, preventing distality.

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.