Timeline for How does the mixing time of a geodesic flow on a surface vary with the genus?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 16, 2010 at 0:22 | vote | accept | CommunityBot | ||
| Dec 16, 2010 at 0:22 | history | bounty ended | Steve Huntsman | ||
| Dec 9, 2010 at 22:11 | answer | added | Sam Nead | timeline score: 4 | |
| Dec 9, 2010 at 20:27 | answer | added | Steve Huntsman | timeline score: 0 | |
| Dec 9, 2010 at 18:13 | comment | added | Sam Nead | @Steve - Thank you for the figure. So you are using the regular, right angled, $8g−4$-gon. Let's call it $P$. If you draw $P$ in the most symmetric fashion in the disc model of $H^2$ then the center $O \in P$ will be distance $\log(g)$ (more or less) from the boundary of $P$. So for subsets of the unit tangent "close to" the point $O$ you'll have to flow for at least that long before any mixing at all happens. | |
| Dec 9, 2010 at 15:24 | comment | added | Steve Huntsman | Also, can you flesh out your comment on the injectivity radius in an answer? My differential geometry is very rusty and it's not obvious to me how to compute this or how it determines the rate of mixing. | |
| Dec 9, 2010 at 13:45 | comment | added | Steve Huntsman | @Sam--thanks. I mean $8g-4$: see imgur.com/XDADm.png for the edge identification. | |
| Dec 9, 2010 at 10:30 | comment | added | Sam Nead | @Steve -- do you mean the regular $4g$-gon? The rate of mixing has to depend on $g$, because the surface that the $4g$-gon gives has injectivity radius an increasing function of $g$. (It grows like $\log(g)$.) | |
| Dec 9, 2010 at 0:59 | comment | added | Steve Huntsman | Let's say for concreteness that the surface of constant negative curvature is given by a identifying appropriate edges of a regular $8g-4$-gon in the disk model with metric $ds^2 = dz d\bar{z}/(1-|z|^2)^2$. | |
| Dec 8, 2010 at 23:58 | comment | added | Bill Thurston | There are many different metrics of curvature -1 on a surface of genus g. Do you have particular metrics in mind? For surfaces of genus 2, the mixing time can go to infinity as you vary the metric: if there are two genus-one subsurfaces separated by a long thin tube, the portion of the unit tangent bundle on one side takes a long time to mix with the portion on the other side. | |
| Dec 8, 2010 at 23:47 | history | bounty started | Steve Huntsman | ||
| Dec 6, 2010 at 21:08 | history | asked | Steve Huntsman | CC BY-SA 2.5 |
