Timeline for Connectedness of space of ergodic measures
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 21, 2011 at 18:54 | comment | added | Vaughn Climenhaga | (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.) | |
| Dec 21, 2011 at 18:53 | comment | added | Vaughn Climenhaga | 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 $0$). | |
| Dec 21, 2011 at 18:13 | comment | added | Andrey Gogolev | 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. | |
| Dec 21, 2011 at 3:31 | comment | added | Vaughn Climenhaga | 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. | |
| Dec 21, 2011 at 2:05 | comment | added | Vaughn Climenhaga | 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. | |
| Dec 21, 2011 at 2:02 | vote | accept | Vaughn Climenhaga | ||
| Dec 21, 2011 at 1:52 | history | edited | Andrey Gogolev | CC BY-SA 3.0 |
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| Dec 21, 2011 at 1:47 | history | answered | Andrey Gogolev | CC BY-SA 3.0 |