Orbits of Distal Actions on Locally Compact Groups

Abstract

We discuss properties of orbits of (semi)group actions on locally compact groups G. In particular, we show that if a compactly generated locally compact abelian group acts distally on G then the closure of each of its orbits is a minimal closed invariant set (i.e. the action has [MOC]). We also show that for such an action distality is preserved if we go modulo any closed normal invariant subgroup and hence [MOC] is also preserved. We also show that any semigroup action on G has [MOC] if and only if the corresponding actions on a compact invariant metrizable subgroup K and on the quotient space G/K has [MOC].

15 pages

References in corpus (2)

• The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth
• Distal actions and shifted convolution property

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• Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub${}_G$ and Applications to Lattices in Lie Groups