# Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.

Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically transitive if for any open $U,V⊆X$ there is $t∈T$ with $tU\cap V\neq\emptyset$, and $X$ is called pointwise minimal if for any $x∈X$, the orbit closure $\overline{Tx}$ is a minimal flow.

It is known (I think due to Ellis) that if $T$ is countable, then any flow $X$ which is both topologically transitive and pointwise minimal is in fact minimal. My question is about if $T$ is uncountable. Are there examples of flows which are topologically transitive, point-wise minimal, and not minimal?

• Do you want $T$ to carry a topology, and do you want the action $T\times X\to X$ be continuous? And what is a minimal flow? – Sebastian Goette Jan 12 '16 at 17:48
• We can assume $T$ is discrete. A flow is minimal if every orbit is dense. If $X$ is metrizable and topologically transitive, then a Baire category argument shows that $X$ admits a transitive point, a point $x\in X$ whose orbit is dense. Hence in this setting pointwise minimal would immediately imply minimal. So if an example of what I'm looking for exists, then $X$ cannot be metrizable. – Andy Jan 12 '16 at 18:43
• If you admit a generic point (say the action is ergodic), then you have a point whose orbit closure is everything, hence the flow itself is minimal if it pointwise minimal and ergodic. – Asaf Jan 12 '16 at 21:30
• So in practice, you can decompose $X$ (up to null sets) to ergodic components (EDIT - probably I'm assuming here that $T$ is amenable), each of those is minimal (by the previous comment). Now the question is different ergodic components may mix by the topological transitivity. – Asaf Jan 12 '16 at 21:32
• @Asaf I'm not sure that the flow admitting an invariant measure would help; even if $X$ were ergodic wrt some invariant measure it may be so large (i.e. non-metrizable) as to not admit a generic point. – Andy Jan 13 '16 at 15:02