# Furstenberg decomposition for non-compact spaces

Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal orbit of $G$ on $X \times X$ has an accumulation point on the diagonal, and minimal if every orbit is dense.

A theorem of Furstenberg (The Structure of Distal Flows, 1963) says that if $G$ is a locally compact group and $X$ is a compact metrizable $G$-space such that the action of $G$ is minimal and distal, then $X$ is built out of isometric extensions: that is, there is a (potentially transfinite) sequence $(X_{\beta})_{\beta \le \alpha}$ of compact $G$-spaces and surjective morphisms $X_{\beta+1} \rightarrow X_{\beta}$ so that $X_0$ is the one-point space, $X_{\alpha} = X$, $X_{\beta}$ is the inverse limit of $(X_{\gamma})_{\gamma < \beta}$ when $\beta$ is a limit ordinal, and for each successor ordinal $\beta+1$ there is a $G$-invariant real-valued function on $X_{\beta+1} \times X_{\beta+1}$ that restricts to a compatible metric on each fibre of $X_{\beta+1} \rightarrow X_{\beta}$. (Conversely, any $G$-space built out isometric extensions in this way is distal; this direction is much easier and doesn't need any special hypotheses.)

Subsequent authors have generalized this result by removing various hypotheses and obtaining a similar conclusion about the decomposition of distal minimal $G$-spaces; sometimes the $G$-invariant metrics need to be replaced with some version of equicontinuity, but this is just an inevitable consequence of allowing $G$-spaces that are not first-countable. One constant though seems to be that $X$ needs to be compact. Are there any similar results if $X$ is instead some 'nice' topological space, e.g. a Polish space or a locally compact Hausdorff space, but not necessarily compact? Alternatively, are there some instructive examples of distal minimal $G$-spaces that don't break into equicontinuous pieces?

I am mainly interested in the case when $X = G/K$ for some closed subgroup $K$ of a locally compact or Polish group $G$, if that makes life easier (so in particular, the action is transitive).

Here is a counter example: take $G=\text{SL}_2(\mathbb{R})$ and $X=\mathbb{R}^2-\{0,0\}$. It is easy to check that the action is distal, and that there is no non-trivial factor carrying an invariant metric (the only proper factor is $\mathbb{P}^1(\mathbb{R})$).

More generally (I think), taking any simple Lie group $G$ and $X=G/U$ for a unipotent subgroup $U$, you will get a distal action with no isometric factors. This follows from the fact that $U$ has closed orbits on $G/U$ and Theorem 6.2 in https://arxiv.org/pdf/1408.4217.pdf.

However, the following answers positively the case $X=G/K$ where $G$ is locally compact and $K<G$ a compact subgroup.

Base: Fix a non constant function in $L^2(G/K)$ and pull it back to a right-$K$-invariant function in $L^2(G)$. Let $K_1$ be the stabilizer of this function wrt the right $G$ action. Thus $K<K_1<G$. By the fact that the function is in $L^2(G)$, $K_1$ is compact. The Hilbert metric of $L^2(G)$ restrict to a metric on the orbit, thus there is a $G$-invariant metric on $G/K_1$.

Step: Assume having a compact group $K_\alpha\gneq K$. Fix a non constant function in $L^2(K_\alpha/K)$ and pull it back to $L^2(K_\alpha)$. Let $K_{\alpha+1}$ be the stabilizer of this function wrt the right $K_\alpha$ action. Thus $K<K_{\alpha+1}\lneq K_\alpha$. The Hilbert metric of $L^2(K_\alpha)$ restrict to a metric on the orbit, thus there is a $K_\alpha$-invariant metric on $K_\alpha/K_{\alpha+1}$. Equivalently, there is a $G$-invariant metric on the fibers of $G/K_{\alpha+1}\to G/K_\alpha$.

Limit: Having a limit ordinal $\beta$ such that $K_\alpha$ is constructed for every $\alpha\lneq \beta$, define $K_\beta$ to be the closure of $\cup_{\alpha\lneq \beta} K_\alpha$.

Termination: With a lot of patient (depending on the cardinality of $G$) you get $\alpha$ such that $K_\alpha=K$.

• Thank you, those are informative counterexamples. The situation with compact K is in line with what I'd expect, as the action of K on G/K is the critical thing and actions of compact groups are very special. For G a t.d.l.c. group, the possibilities for a distal action on G/K are limited if K is compactly generated (see arxiv.org/abs/1610.06696). Of course, this doesn't exclude things like SL_2(Q_p) modulo a unipotent subgroup. Mar 3, 2017 at 8:36