Dense and turbulent orbits

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.

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