# Connectedness of space of ergodic measures

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.

• I'm not familiar with all relevant terms (particularly the specification property), but one example in which you probably aren't interested is as follows: take a finite disjoint union of compact metric spaces, each equipped with a continuous, uniquely ergodic self-map. Then the space of ergodic measures for the induced map on the union is a finite, discrete space. – Mark Dec 21 '11 at 1:21
• @Mark: Specification is a uniform version of topological transitivity, which in particular rules out taking disjoint unions. – Vaughn Climenhaga Dec 21 '11 at 1:37
• Almost every simplex is the Poulsen simplex. What about this one? – Gerald Edgar Dec 21 '11 at 3:05
• @Gerald: $\mathcal{M}^e$ is a dense subset of $\mathcal{M}$ whenever $X$ has specification, and since as I understand it the Poulsen simplex is characterised up to affine homeomorphism (among compact metrisable simplices) by the condition that extreme points are dense, I believe we are in fact dealing with the Poulsen simplex here. In particular, the 1978 paper of Lindenstrauss, Olsen, and Sternfeld shows that if extreme points are dense then they are arc-connected, which gives an even more complete answer. Thanks for the suggestion! (I hadn't heard of the Poulsen simplex before.) – Vaughn Climenhaga Dec 22 '11 at 5:59

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.

• Thanks! I knew about Sigmund's 1970 paper where he shows that the set of ergodic measures is residual, but I didn't know about the 1977 paper. – Vaughn Climenhaga Dec 21 '11 at 2:05
• Upon a little further reflection, I think it does generalise to the case with specification. Periodic measures are dense (as per his 1970 paper), and the periodic measure on $O(x)$ is the unique equilibrium state corresponding to the upper semi-continuous potential function that is $0$ on $O(x)$ and $-\infty$ everywhere else. To connect two periodic measures with an arc of ergodic measures, it suffices to connect the corresponding potential functions with an arc of Holder continuous potentials and consider the unique equilibrium states for these measures. – Vaughn Climenhaga Dec 21 '11 at 3:31
• So the endpoints of your arc are not Holder continuous. Are you sure that you have continuity of the map $\varphi\mapsto\mu_\varphi$ at the endpoints? That would be a different proof. – Andrey Gogolev Dec 21 '11 at 18:13
• Yes, endpoints are not Holder (although the equilibrium state is still unique). Continuity of $\phi\mapsto\mu_\phi$ doesn't follow from general principles, but you should get it pretty easily from the Gibbs property for $\mu_\phi$ when $\phi$ is Holder (observing that the weight on any Bowen ball that doesn't intersect the periodic orbit goes to $0$). – Vaughn Climenhaga Dec 21 '11 at 18:53
• (Continuity at the endpoints, I mean -- where you were talking about. Continuity on the rest of the path is a general property of equilibrium states for Holder potentials.) – Vaughn Climenhaga Dec 21 '11 at 18:54

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.