Timeline for Connectedness of space of ergodic measures
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 22, 2011 at 17:20 | answer | added | Vaughn Climenhaga | timeline score: 6 | |
| Dec 22, 2011 at 5:59 | comment | added | Vaughn Climenhaga | @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.) | |
| Dec 21, 2011 at 3:05 | comment | added | Gerald Edgar | Almost every simplex is the Poulsen simplex. What about this one? | |
| Dec 21, 2011 at 2:02 | vote | accept | Vaughn Climenhaga | ||
| Dec 21, 2011 at 1:51 | history | edited | Andrey Gogolev |
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| Dec 21, 2011 at 1:47 | answer | added | Andrey Gogolev | timeline score: 15 | |
| Dec 21, 2011 at 1:37 | comment | added | Vaughn Climenhaga | @Mark: Specification is a uniform version of topological transitivity, which in particular rules out taking disjoint unions. | |
| Dec 21, 2011 at 1:21 | comment | added | Mark | 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. | |
| Dec 21, 2011 at 0:34 | history | asked | Vaughn Climenhaga | CC BY-SA 3.0 |