2
$\begingroup$

I need a proper reference to the following obvious fact:

An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $U$ is dense) if and only if it the orbit $Gx$ of some point $x\in K$ is dense in $K$.

For a cyclic group this characterization is proved here. I hope that some textbook in topological dynamics should contain such a basic fact.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

See Theorem 9.20 of "Topological Dynamics" by Gottschalk and Hedlund. It states that, for systems $(X,G)$ whose phase space is non-empty complete separable metric, point transitivity (a point having a dense orbit) and topological transitivity (every non-empty open set having dense orbit) are equivalent.

Improve this answer
$\endgroup$
3
  • $\begingroup$ The space should be non-empty for the equivalence to hold. $\endgroup$
    – YCor
    Commented Apr 4, 2020 at 12:26
  • $\begingroup$ @YCor I was working with the convention that excludes empty topological spaces but let me explicitly add that. $\endgroup$
    – Burak
    Commented Apr 4, 2020 at 12:29
  • $\begingroup$ Sure it's often implicit and quite pernicious as soon as you use operations such as considering $X^N$ ($N$-fixed points for some normal subgroup $N$ of $G$) as dynamical system too. $\endgroup$
    – YCor
    Commented Apr 4, 2020 at 13:32

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .