Connectedness of space of ergodic measures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:24:21Z http://mathoverflow.net/feeds/question/83981 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83981/connectedness-of-space-of-ergodic-measures Connectedness of space of ergodic measures Vaughn Climenhaga 2011-12-21T00:34:27Z 2011-12-24T22:05:58Z <p>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.</p> <p>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 H&ouml;lder 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 H&ouml;lder potentials).</p> <p>Of course, there are many ergodic measures that do not arise as equilibrium states of H&ouml;lder 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.</p> http://mathoverflow.net/questions/83981/connectedness-of-space-of-ergodic-measures/83985#83985 Answer by Andrey Gogolev for Connectedness of space of ergodic measures Andrey Gogolev 2011-12-21T01:47:35Z 2011-12-21T01:52:45Z <p>Hi Vaughn,</p> <p>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:</p> <p>Sigmund, Karl "On the connectedness of ergodic systems." Manuscripta Math. 22 (1977), no. 1, 27–32.</p> <p>I don't know about generalizations. Sigmund's proof does not generalize directly.</p> http://mathoverflow.net/questions/83981/connectedness-of-space-of-ergodic-measures/84103#84103 Answer by Vaughn Climenhaga for Connectedness of space of ergodic measures Vaughn Climenhaga 2011-12-22T17:20:04Z 2011-12-22T17:20:04Z <p>I'll flesh out the consequences of Gerald's comment in a (CW-ed) answer. Lindenstrauss, Olsen, and Sternfeld showed <a href="http://www.ams.org/mathscinet-getitem?mr=500918" rel="nofollow">in 1978</a> 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 <em>Poulsen simplex</em>.</p> <p>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 $\mathcal{M}^e$ (<a href="http://www.ams.org/mathscinet-getitem?mr=352411" rel="nofollow">Sigmund 1974</a>), 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.</p> <p>So that's not quite as constructive a proof as the approach following <a href="http://www.ams.org/mathscinet-getitem?mr=447528" rel="nofollow">(Sigmund 1977)</a> as suggested in Andrey's answer and the comment following, but it's certainly simpler to write down based on existing results.</p>