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In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition without proof:

Proposition 1.4. Let $G$ be a closed subgroup of $S_\infty$ and suppose $G$ acts continuously on the Polish space $X$. Then the following are equivalent for any $x \in X$:

  1. the orbit $G \cdot x$ is dense $G_\delta$;
  2. $G \cdot x$ is dense and turbulent.

They say that this can easily be proved directly but I am quite stuck. So any hints on how to prove this are appreciated.

Note that we call a point $x$ in $X$ turbulent if for every open neighbourhood $U$ of $x$ and every symmetric open neighbourhood $V$ of the identity of $G$ the local orbit $\mathcal{O}(x,U,V)$ is somewhere dense, i.e. $\mathrm{Int}(\mathrm{Cl}(\mathcal{O}(x,U,V))) \neq \emptyset$. Here, $\mathcal{O}(x,U,V) = \{ y \in X : \exists g_0,\dots,g_k \in V \forall i < k (g_i g_{i-1}) \dots g_0 \cdot x \in U \text{ and } g_k g_{k-1} \dots g_0 \cdot x = y \}$. This notion is $G$-invariant, i.e., we can speak about tubulent orbits.

In the paper linked above you can find several characterisations of turbulence in Proposition 3.2. One useful characterisation might be that $G \cdot x$ is turbulent if and only if it is $G_\delta$. However, I still don't succeed in proceeding from here.

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    $\begingroup$ You might like adding the definition of turbulent orbit. $\endgroup$ – Pedro Sánchez Terraf May 7 '16 at 20:22
  • $\begingroup$ And, perhaps, a link to the paper. I found one with a very similar title, by Kechris and another author, not Becker. $\endgroup$ – Pedro Sánchez Terraf May 8 '16 at 23:47
  • $\begingroup$ @PedroSánchezTerraf sorry, I confused the authors. However, I've linked the arXiv version of the paper now and stated the definition of turbulence. $\endgroup$ – namsap May 9 '16 at 6:51

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