Timeline for Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Current License: CC BY-SA 4.0
6 events
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| Nov 22, 2023 at 6:35 | comment | added | Sam Nead | The abelian case (and the needed reference to the "Choquet Simplex") is discussed in Section 4.4 of the forthcoming book "Translation surfaces" by Athreya and Masur. You will have to ask the authors for a copy (or ask the AMS to hurry up a bit!). | |
| Nov 20, 2023 at 7:53 | comment | added | Sam Nead | Oh, and then you need the fact that the space of transverse measures is always a simplex. I will look for a reference for all of this. | |
| Nov 20, 2023 at 7:50 | comment | added | Sam Nead | You remarked that you only need a linear bound. So you could proceed as follows. Suppose that $S$ is a surface and $F$ is a singular foliation. If $F$ is orientable then you are done. If not, then $F$ induces a (branched) double cover $S' \to S$ where it pulls back to an orientable foliation $F'$. Lastly, the covering induces an embedding of the space of transverse measures on $F$ into the space of transverse measures on $F'$. | |
| Nov 19, 2023 at 21:34 | comment | added | Alejo García Sassi | Thank you! The collapsing argument is very much on point, and the fact the resulting surface is homeomorphic appears for example here We now must understand ergodic transverse measures for singular foliations. Katok gives a bound for $\mathcal{C}^1$ flows with saddle singularities (i.e. coming from orientable laminations) here. Any reference for the non-orientable case is welcome! | |
| Nov 17, 2023 at 18:29 | history | edited | Sam Nead | CC BY-SA 4.0 |
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| Nov 17, 2023 at 18:22 | history | answered | Sam Nead | CC BY-SA 4.0 |