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Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, which is an interval exchange transformation (IET). We need to find an upper bound for the amount of ergodic measures that exist for $R_{\Lambda}$, by some function of $g$.

We firmly believe the answer for this question to be affirmative and we think the desired function should be linear with $g$, but we also think that the answer could be found in the existent literature.

Can anybody throw some light for the (non?)-existence of such a bound?

Thanks in advance!


What we already searched.

-- Interval exchange transofrmations. We found here that the amount of ergodic measures an IET can have, is bounded by the minimal amount of intervals needed to express such map. There are works of Veech, Viana, Yoccoz, Masur, which give extensive explanations about IETs. In particular, Veech, Viana, Yoccoz, talk about the zippered rectangle construction, in which they realise interval exchange transformations as the return map of a singular foliation in what they call translation surfaces. For this particular example they find explicit results for the genus of the surface, but we do not know if there is a different construction that yields different surfaces for the same return map.

-- Trichotomy Geod. Laminations- Sing. Foliations - Train Tracks: Calegari talks about this trichotomy in his book Foliations and the geometry of 3-manifolds, and Erlandsson and Souto relate geodesic lamniations to weighted train tracks in Mirzakhani's Curve Counting and Geodesic Currents. But none of them relate them to interval exchange transformations.

-- Katok and other mathematicians discuss the amount of ergodic measures for certain singular foliations in surfaces, but again there are some pieces missing in order to obtain what we need.

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Let's restrict attention to the case where the components of $S - \Lambda$ are all discs (with three or more cusps). (In particular, I assume that $\Lambda$ has no leaves which are simple closed curves.) Following Thurston, we choose singular foliations of these discs, with the foliations being horocyclic in the cusps of the disks. These foliations-of-discs fit together to give a foliation $F$ of $S$. Note that $F$ is transverse to $\Lambda$. (If you arrange matters carefully, you can ensure that $F$ contains your loop $I$.) We now collapse all leaf segments of $F - \Lambda$. It is a (difficult) exercise to check that the resulting quotient $S /{\sim}$ is again a surface and is in fact homeomorphic to $S$. Also, the lamination $\Lambda$ descends to give a foliation $\Lambda/{\sim}$ of $S /{\sim}$. The spaces of transverse measures on $\Lambda$ and on $\Lambda/{\sim}$ are naturally isomorphic.

The collapsing argument can more-or-less be found in the (very readable) book Automorphisms of surfaces after Nielsen and Thurston. I would say that the application to counting ergodic transverse measures on laminations is "well-known to the experts".

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  • $\begingroup$ 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! $\endgroup$ Nov 19, 2023 at 21:34
  • $\begingroup$ 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 $F'$. $\endgroup$
    – Sam Nead
    Nov 20, 2023 at 7:50
  • $\begingroup$ 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. $\endgroup$
    – Sam Nead
    Nov 20, 2023 at 7:53
  • $\begingroup$ 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!). $\endgroup$
    – Sam Nead
    Nov 22, 2023 at 6:35

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