gen. Bohr comp.\], \[theorem: factorization by H(uM)\] and \[theorem: strong amenability\]. We also use [@ChSi Theorem 5.6] to get a variant of this theorem in our definable context (see Corollary \[corollary: from ChSi\]).

In Section \[section: Polish structures\], we explain how to treat various invariants (e.g. the Ellis group or the $\Delta$-definable generalized Bohr compactification of $G$) as Polish structures, which suggests that in some situations one could expect to get structural and topological theorems about these invariants via application of theorems on small Polish structures. We make a few observations in this direction and formulate some questions.

Preliminaries {#section: preliminaries}
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Detailed preliminaries concerning topological dynamics in model theory were given in several papers, so here we only recall a few things. For more details, see e.g. [@KrPi Section 1]. A good reference for classical topological dynamics is for example [@Au] or [@Gl].

The [*Ellis semigroup*]{} of the flow $(G,X)$, denoted by $E(X)$, is the closure of the collection of functions $\{\pi_g \colon g \in G\}$ (where $\pi_g\colon X \to X$ is given by $\pi_g(x)=gx$) in the space $X^X$ equipped with the product topology, with composition as semigroup operation.

This semigroup operation is continuous in the left coordinate, $E(X)$ 