Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense.
Here is an attempt to define topologically primitive in the same spirit.
A topological system of imprimitivity is a set $P$ of pairwise disjoint open subsets of $X$, such that $P$ is preserved setwise by $G$ and the union of $P$ is dense. Say such a system is non-trivial if $|P| > 1$ and the elements of $P$ are not all singletons.
The action of $G$ is topologically primitive if no non-trivial system of imprimitivity exists. Equivalently, $G$ is topologically transitive, and for every non-empty open subset $U$ of $X$, there exists $g \in G$ such that $U$ and $gU$ are neither equal nor disjoint.
Has anything been proved about topologically primitive actions, or a similar notion? I am particularly interested in actions on totally disconnected spaces, where the space itself is flexible enough to allow many systems of imprimitivity.
Topological systems of imprimitivity come up naturally, for example, when $X$ is the space of ends of a tree on which $G$ acts. Another natural family of examples to consider is if $G$ is a totally disconnected topological group, $H$ is a closed subgroup of $G$ and $X$ is the coset space $G/H$: the action is abstractly primitive if and only if $H$ is a maximal subgroup, and topologically primitive if and only if there is no open subgroup of $G$ strictly between $H$ and $G$.
