Topological dynamics for groups definable in real closed field

We study the definable topological dynamics of groups definable in an o-minimal expansion of an arbitrary real closed field M. For a definable group G   which admits a compact-torsion-free decomposition G=HKG=HK, we give a description of the minimal subflow and Ellis group of the universal definable G(M)G(M)-flow SG,ext(M)SG,ext(M). This Ellis group is isomorphic to NG(H)∩K(R)NG(H)∩K(R), which extends the result of G. Jagiella from [7]. We also consider SL(2,M)SL(2,M) as an example, explaining the difference between the universal definable SL(2,R)SL(2,R)-flow, SG(R)SG(R) and the universal definable G(M)G(M)-flow, SG,ext(M)SG,ext(M) for an arbitrary model M≻RM≻R.