Definable topological dynamics and real Lie groups

We investigate definable topological dynamics of groups definable in an o-minimal expansion of the field of reals. Assuming that a definable group G admits a model-theoretic analogue of Iwasawa decomposition, namely the compact-torsion-free decomposition G KH, we give a description of minimal subflows and the Ellis group of its universal definable flow SGpRq in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to NGpHq X KpRq. This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to G{G00. We further extend the results to universal topological covers of definable groups, interpreted in a two-sorted structure containing the o-minimal sort R and a sort for an abelian group.