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.

References in corpus (3)

• The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth
• Distal actions and shifted convolution property
• Orbits of Distal Actions on Locally Compact Groups

Cited by in corpus (3)

• Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups
• On the distality and expansivity of certain maps on spheres
• Distal Actions of Automorphisms of Lie Groups $G$ on $\mathrm{Su}{\mathrm{b}}_{\mathrm{G}}$