Grunwaldzki 2/4\
    50-384 Wrocław, Poland 
author:
- Krzysztof Krupiński
title: Definable topological dynamics
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Introduction
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Topological dynamics was introduced to model theory by Newelski in [@Ne1; @Ne2] and then further developed by various authors, e.g. in [@GiPePi], [@Ja], [@YaLo], [@ChSi], [@KrPi] and [@KrPiRz]. There are several natural categories to develop topological dynamics in model theory. The most natural are the categories of definable and externally definable “objects”. So far, however, mostly the externally definable category has been studied (the definable one was investigated under the extra assumption of definability of types, which makes both categories the same). In this paper, we develop basic topological dynamics in the category of definable flows, without the definability of types assumption.

Recall that a [*$G$-flow*]{} is a pair $(G,X)$, where $G$ is a group acting on a compact, Hausdorff space by homeomorphisms. We always consider discrete flows, i.e. with no topology on $G$ (or, if one prefers, with the discrete topology on $G$). A [*$G$-ambit*]{} is a $G$-flow $(G,X,x_0)$ with a distinguished point $x_0$ whose $G$-orbit is dense. With the obvious notion of a homomorphism of $G$-ambits, a universal $G$-ambit always exists and is unique; this universal ambit is exactly $(G,\beta G, e)$ 