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.


1 Answer 1


Let $X$ be a compact metric space and $T\colon X\to X$ be a continuous map. The set of $T$-invariant Borel probability measures $\mathcal{M}_T(X)$ is well known to be non-empty, convex, compact, and metrizable. Moreover, its extreme points $\mathcal{M}^e_T(X)$ coincide with ergodic invariant measures and every invariant measure can be represented as a barycenter of a unique measure supported on $\mathcal{M}^e_T(X)$ by the ergodic decomposition theorem. This can be found in Walter's Introduction to Ergodic Theory [Theorem 6.10 and Remark (2) page 153] or in Downarowicz's Entropy in Dynamical Systems (Cambridge University Press 2011). Hence, $\mathcal{M}_T(X)$ is a Choquet simplex for every compact dynamical system. It turns out that any abstract Choquet simplex is affinely homeomorphic to the set of invariant measures of some minimal dynamical system. This is a result of Downarowicz [The Choquet simplex of invariant measures for minimal flows. Israel J. Math. 74 (1991), no. 2-3, 241--256], which completes the line of research begun in 50's (see references therein).


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