It is a basic fact in the weak-* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-* topology. Furthermore, the set of ergodic measures is equal to the set of extremal points of the the set of invariant measure.

In the symbolic space, the set of ergodic measures are dense in the set of invariant measures. However, for the general dynamic model, there still lacks a clear picture of a characterization of ergodic measure.

According to my knowledge, there are few results to give more topological characterizations of ergodic measure. There is a very natural question to ask: for a given convex set $A$, can we realize it into a set of invariant measures for some dynamical system? To be precise, I am asking whether there exists a dynamical system whose invariant measure set is $A$.

It is obvious for a convex set in a finite dimension. an we say something about infinite dimensional convex sets? Any reference and comments will be appreciated in advance.