It is well known that the set of invariant measures of a topological dynamical system is a non-empty metrisable Choquet simplex. In 1991, Downarowicz proved that all such simplices arise as the sets of invariant measures of a class of minimal subshifts. Hence, one can ask the following question: which non-empty Choquet simplices can be realised as the sets of invariant measures for minimal homeomorphisms on manifolds? In the case of one-dimensional manifolds, we observe that the geometry of manifolds restricts the available dynamics. In my talk, I will discuss which measurable dynamical system can be realised as a minimal homeomorphism on a manifold. This will answer the question of realisation of Choquet simplex on manifolds of higher dimension.

I will also talk about necessary and sufficient conditions for an ergodic measure in Choquet simplex to have a discrete spectrum. The criterion is imposed on generic points of such a measure.

The talk is based on the results obtained in joint works with Melih Emin Can, Jernej Činč, Till Hauser, Dominik Kwietniak, and Piotr Oprocha.