Connectedness of space of ergodic measures

Question

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.

Answer 1

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.

Answer 2

I'll flesh out the consequences of Gerald's comment in a (CW-ed) answer. Lindenstrauss, Olsen, and Sternfeld showed in 1978 that if $S_1$ and $S_2$ are compact metrisable simplices such that the extremal points of $S_i$ are dense in $S_i$ for $i=1,2$, then there is an affine homeomorphism from $S_1$ to $S_2$; the unique (up to affine homeomorphism) compact metrisable simplex with the property that its extremal points are dense is called the Poulsen simplex.

In that same paper, it was shown that the Poulsen simplex has the property that its set of extremal points is arc-connected. Since $\mathcal{M}$ is a compact metrisable simplex whenever $X$ is a compact metric space and $f\colon X\to X$ is continuous, and the extremal points of $\mathcal{M}$ are precisely the ergodic measures $\mathcal{M}^e$, it follows that $\mathcal{M}^e$ is arc-connected whenever it is dense in $\mathcal{M}^e$. In particular, the strong specification property introduced by Bowen implies that periodic orbit measures are dense in $\mathcal{M}^e$ (Sigmund 1974), and since such measures are ergodic, this implies that $\mathcal{M}$ is the Poulsen simplex, and hence $\mathcal{M}^e$ is arc-connected, whenever $(X,f)$ has strong specification.

So that's not quite as constructive a proof as the approach following (Sigmund 1977) as suggested in Andrey's answer and the comment following, but it's certainly simpler to write down based on existing results.