Distal actions and ergodic actions on compact groups (Q1043769)

An automorphism \(\alpha\) (or a group of automorphisms \(\Gamma\)) of a compact metrizable abelian group \(X\) is said to be distal if the closure of the orbit under \(\alpha\) (or the action) of any point other than the identity does not contain the identity. If the image of \(\Gamma\) in the automorphism group of \(X\) is itself compact, then the action is automatically distal. The question considered here is whether distality for each element of \(\Gamma\) implies distality for \(\Gamma\) (which is on the face of it a much more stringent condition unless \(\Gamma\) is cyclic). This `local-to-global' result is shown under the hypothesis that \(\Gamma\) has a finite descending series of closed normal subgroups ending with the trivial group and with the property that each successive quotient is compactly generated with relatively compact conjugacy classes, or if \(X\) is a solenoid (that is, a connected compact group with finite topological dimension). For an action of a nilpotent group on a solenoid, it is shown as a result that ergodicity of the action implies the existence of an ergodic element.