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 = K H , we give a description of minimal subflows and the Ellis group of its universal definable flowS G ( R )in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic toN G ( H ) ∩ K ( R ) . This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to G / G 00 . 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.