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.