Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the weak* topology.
Now $\mathcal{M}$ is a Choquet simplex, and hence connected. The geometry of its extreme points is a little more subtle. These extreme points are precisely the ergodic measures. Let $\mathcal{M}^e$ denote the collection of ergodic measures in $\mathcal{M}$. Note that $\mathcal{M}^e$ has some nice properties; for instance, there is a natural embedding from the space of Hölder continuous functions into $\mathcal{M}^e$ that takes $\phi$ to its unique equilibrium state $\mu_\phi$. The image of the embedding is the collection of Gibbs measures (for Hölder potentials).
Of course, there are many ergodic measures that do not arise as equilibrium states of Hölder continuous functions, and so I wonder which nice properties of the collection of Gibbs measures extend to $\mathcal{M}^e$. In particular: Is $\mathcal{M}^e$ connected? Path connected? I expect that it is, and that moreover this should happen whenever $X$ is a compact metric space and $f\colon X\to X$ is a continuous map satisfying the specification property, but I don't know a reference and don't yet see how to approach a proof.
Best Answer
Hi Vaughn,
It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology. The proof is very neat and takes only a page or so. Here is the paper:
Sigmund, Karl "On the connectedness of ergodic systems." Manuscripta Math. 22 (1977), no. 1, 27–32.
I don't know about generalizations. Sigmund's proof does not generalize directly.