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Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point $(z,z)$, then $x=y$.

Now let $G$ be a locally compact group and let $H$ be a group of automorphisms of $G$. Let $K$ and $L$ be $H$-invariant closed subgroups of $G$ such that $L \le K$. The coset spaces $G/K$, $K/L$ and $G/L$ then carry the quotient topology.

  1. To show $H$ acts distally on $G/K$, is it enough to show that no $H$-orbit accumulates at the trivial coset?

  2. If $H$ acts distally on $G/K$ and on $K/L$, does it follow that $H$ acts distally on $G/L$?

If $L$ is normal in $K$ and $K$ is normal in $G$, it looks like the answer to both is yes, but otherwise the actions of $K$ and $L$ by conjugation could complicate things.

Edit: a third question which is somewhat related (in that it might help with the first two):

  1. Given a group $G$ acting by homeomorphisms on a compact space $X$, one can take the closure of $E$ of $G$ in $X^X$; as shown by Ellis, $E$ is a semigroup, which is compact group if and only if $G$ is distal. Does something like this work without the assumption that $X$ is compact? Is there still a useful notion of 'Ellis semigroup'?
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