The plan of today’s discussion is to use the tools (“orbit by orbit estimates” and a variant of Rokhlin disintegration theorem) from the previous post to study the following question (stated after the picture).
Let be a certain (fixed) level of the systole function on a connected component of a stratum of the moduli space of translation surfaces of genus . Denote by the set of such that all non-vertical saddle-connections have length and, for each , consider the set of translation surfaces with systole having the form for some , and . In other words, using the notation introduced in the previous post,
Geometrically, consists of the pieces of arcs of hyperbola below the threshold in the figure below:
In this notation, given a -invariant probability measure on , we want to compute the measure of in terms of , that is, we want to determine how “fat” is the set of translation surfaces that are “accessible” (via and movements) from .
In fact, a precise answer to this question will occupy this entire post and, in the next (and last) post of this series, we will use this answer to estimate the -measure of the set (of translation surfaces with two non-parallel saddle-connections of lengths ) as follows. Firstly, we will see that captures “almost all” translation surfaces in (for , i.e., , conveniently chosen). In particular, we will reduce the problem of measuring to the task of estimating . Then, we will see that the translation surfaces generating a translation surface have a pair of saddle-connections with a very small angle (with tending to zero as ) and it is not hard to see that this angle condition corresponds to a subset of with arbitrarily small “density” in .
We organize this post into two sections. In the first section we will explain how the variant of Rokhlin disintegration theorem from the previous post of this series can be used to describe the -measure of in terms of the measure of with respect to a certain obtained after disintegrating along certain pieces of -orbits. Then, in the second section, we will explain why the measure obtained from this disintegration process is a sort of “density” measure.
1. Construction of the “density measure” on
Let be a -invariant probability measure on a connected component of a stratum of the moduli space of translation surfaces of genus . Given such that , consider the set
of translation surfaces with systole accessible from from and movements, where is the set of translation surfaces with systole such that all non-vertical saddle-connections have length .
Note that we can write
where the symbol denotes a disjoint union: indeed, if with , then ; since and have systole and , it follows that ; in particular, and, from the definition of , we also deduce that .
Next, we observe that the infinitesimal generator of is tangent to . Indeed, since fixes the vertical direction while changing the horizontal direction only a little bit if is close to zero, we have that for all and sufficiently close to zero.
Remark 1 Here, we implicitly used that is a manifold when talking about tangent vectors to . While this is true if the translation surfaces in have no non-trivial automorphisms (symmetries), in general is an orbifold but not a manifold. However, this little technical problem is easy to overcome: one can either take finite covers of that are manifolds (e.g., one can mark horizontal separatrices of the translation surfaces to “kill” automorphisms), or one can write as a finite union of -invariant manifolds (as in our preprint with Artur and Jean-Christophe). In any event, for the sake of this post, we will pretend that is a manifold.
In particular, since is a codimension submanifold of (we are fixing the value of the systole, a codimension 1 condition, and the direction of the lenght-minimizing saddle-connections, another codimension 1 condition), we can select near any point a codimension submanifold locally transverse to the infinitesimal generator of . Using these ‘s, we can represent as
with , close to , and . At this point, we are ready to use the technology of the previous post to understand the measure on in terms of a density measure on . Indeed, we begin by noticing that has positive -measure. In fact, by Fubini’s theorem and the -invariance of , we have that
where is any probability measure on . By taking a “normalized” version of Haar measure of (say multiplied by an adequate everywhere positive density in order to get a probability measure) and by noticing that, for each with , the set of elements has non-empty interior in and thus positive -measure, we deduce that is positive as soon as the -measure of the set is positive.
Once we know that , we can apply the variant of Rokhlin’s theorem in the previous post to the to deduce that locally
where and is a finite measure on .
Next, we recall from the Subsection “The decomposition and Haar measure on ” of the previous post that . Therefore, we can write
From this formula, we can answer the question at the beginning of this post, i.e., we can compute the -measure of the set
In fact, by (1)
where and is the length of the interval . Now, we recall that the quantity was computed in Subsection “On the action of the diagonal subgroup ” of the previous post:
where . By plugging this into the integral expression for above, we get that:
Finally, the integral on the right hand-side was computed in the Subsection “The decomposition and Haar measure on ” of the previous post and its value is . In summary,
In other terms, the -measure of the subset of captured by is an explicit (and simple) function of the quantity where was obtained from after disintegration along pieces of -orbits.
Intuitively, we want to think of as a sort of “density measure” on and we wish to say that occupies most of the level of the systole function, but these properties are not automatic from Rokhlin’s disintegration theorem. On the other hand, as we are going to explain in the next section, we dispose of all elements to give a precise meaning for the intuition that is a density measure.
2. Why is a density measure?
The main result of this section is:
Of course, this proposition says that is a density measure in the sense that can be recovered from the -measures of slices near the level set of the systole function.
We will divide the proof of this proposition into two parts. Firstly, we will construct a regular part of the slice consisting of translation surfaces obtained by applying a “twisted Teichmüller flow” to and we will show that this regular part has -measure ; in particular, this computation of the -measure of the regular part of shows that . Secondly, we will show that the -measure of the singular part of the slice , i.e., the complement of the regular part, is and this will complete the proof of the desired proposition.
In order to organize the discussion of Proposition 2, we will treat the regular and singular parts of separately in the following two subsections below.
2.1. Regular part of slices
Let and consider the following “twisted Teichmüller flow”:
when . Note that is injective and has systole .
In this language, the regular part of the slice is:
For the computation of the -measure of , it is convenient to study the following measure on :
for any Borel set .
Proposition 3 doesn’t depend on : in fact, on .
Proof: As we saw in the previous section, if we write as with .
On the other hand, by definiton, if with , then
So, we will be able to compute if we can express in terms of the decomposition, i.e.,
Note that, for small enough, the vector satisfies and . By the results in Subsection “The decomposition and the Haar measure of ” of the previous post, it follows that has a decomposition . Furthermore, by a direct computation similar to the one in Subsection “The decomposition and the Haar measure of ” of the previous post for the computation of the density of Haar measure in -coordinates, one can check that, for close to zero, , and , so that
for any Borel set . It follows that as desired.
2.2. Singular part of slices
The main proposition of this subsection is:
Observe that, by definition, the singular part of is contained in the set of translation surfaces with a saddle-connection of length non-parallel to a length-minimizing one. For each , let denote the minimal angle between two non-parallel saddle-connections of length .
The proof of Proposition 5 follows from the following two lemmas:
Let us start with the case of big angles. By Fubini’s theorem and the -invariance of , we have that
for any Borel set , where is a Haar measure on . By applying this formula with , our task is reduced to estimate the quantity
for each . By definition, if and , then has systole (for small enough) and has a pair of non-parallel saddle-connections and of lengths and angle such that and have lengths between and . In other words, using the notations from Subsection “Euclidean norms under –action on ” from the previous post,
where and are non-parallel holonomy vectors of saddle-connections of with lengths between and and making angle . From the main result of the Subsection “Euclidean norms under –action on ” from the previous post, the Haar measures of each of the sets are where the implied constant depend only on , i.e., . Thus, the proof of Lemma 7 will be complete once we dispose of upper bounds on the number of pairs as above and, as a matter of fact, these bounds exist in the literature: indeed, among several other things, H. Masur (see also Theorem 5.4 of this paper of A. Eskin and H. Masur) showed that the number of saddle-connections of length of a translation surface with systole is uniformly bounded by a constant depending only on .
Now, let us deal with the case of small angles. Given , we can select small enough so that the -measure of the set of translation surfaces with two non-parallel saddle-connections with angle and lengths between and is . Next, we select small enough so that
for all . In particular, if we consider the subset of consisting of such that a length-minimizing saddle-connection makes angle with the vertical, then, from the fact that , one can check that the sets are mutually disjoint for . By combining this fact with the -invariance of , we deduce that
Since , we conclude that
Finally, since differs from only by the extra condition that a length-minimizing saddle-connection makes angle with the vertical, we can use the -invariance of to obtain that
Of course, this completes the proof of Lemma 6.
At this point, we can summarize the discussion of this post as follows. We wrote as and we used this to show that the -measure of the set of translation surfaces accessible from by and movements is
where is a “density” measure in the sense that
with denoting the slice .
Next time, we will use this information and the Siegel-Veech formula to complete our solution of Eskin-Kontsevich-Zorich regularity conjecture.