Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the set of all ergodic measures, exactly the extremal points. Moreover, the set $\mathrm{C}(\mathcal{M}^e)$ of convex combinations of elements of $\mathcal{M}^e$ is dense in $\mathcal{M}$ (in the weak topology).

What is known about which measures in $\mathcal{M}$ are (and are not) in $\mathrm{C}(\mathcal{M}^e)$? In particular, what is an example of a measure in $\mathcal{M} \setminus \mathrm{C}(\mathcal{M}^e)$? What about just a measure in $\mathcal{M} \setminus \mathcal{M}^e$?

I've tagged with 'reference-request' because, besides just answers to the above questions, I'd like some references to where this stuff has been studied.