Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a zero of the $1$-form are called saddle connnections, and are important because (among other things) for any fixed direction $\theta$, periodic trajectories on $S(P)$ with argument $\theta$ occur in cylinders bounded by saddle connections. Unfortunately, I do not know any way to compute saddle connections, or even their lenghts, other than high school trigonometry, which is all but useless except in the most trivial cases.

Question: Is there a standard technique which, given a flat structure $S$, angle $\theta$, and zeros $p,q\in S$ of the $1$-form on $S$ allows one to determine the saddle connections from $p$ to $q$ with argument $\theta$, and estimate their lengths? Even better, is there a way to compute the decomposition of $S$ into cylinders in the direction $\theta$?

Context: Consider the isosceles triangles $T_{nk}$ with base angles $\frac{2^n}{2^{n+k}-1}\pi$, where $k\ge 4$. These triangles arise naturally in the study of periodic trajectories on nearly isosceles triangles. I would like to show that as $n\to \infty$, the minimal length of saddle connections (excluding certain relatively easy-to-describe exceptions) between the vertex point and itself in $S(T_{nk})$ with argument $\pi/2$ or $\pi/2-\frac{2^n}{2^{n+k}-1}\pi$ diverges, or alternatively that the lengths of the cylinders do. I have (indirect) theoretical and computational reasons to believe this is true. This should imply a conjecture of Schwartz and Hooper that no neighborhood of the Veech triangles $V_{2^k}$ can be covered by a finite union of orbit tiles.


I am pretty sure this is unknown. The interesting case is when everything lies in some number field. Then for any $L$ it is possible (with a computer) to compute all saddle connections in a given direction of length at most $L$, but in general, one does not know how to rule out that longer saddle connections exist.

This is actually quite an important problem for the general theory of flat surfaces. Progress here would help produce much better algorithms for computing things like the SL(2,R) orbit closure of a flat surface.

  • $\begingroup$ Thank you for the response. Computing saddle connections of bounded length might be sufficient for my problem, if it could be done somehow uniformly for every flat surface in the family I'm interested in. Is there a characterization of (at least some) flat structures for which this is possible? $\endgroup$ – Alex Becker May 4 '13 at 6:19
  • $\begingroup$ @Alex Becker: Sorry, I do not know of any general method, other than doing it by computer, one surface at a time. If the flat surface is Veech, then you could just use the Veech group, but this will not be true for all surfaces in a family. Your problem is equivalent to computing periodic trajectories of an Interval Exchange Transformation. If the IET somehow collapses to a direct sum of interval exchanges of at most 2 intervals, then it is easy. If you have an irreducible IET of at least 4 invervals, then doing it explicitly and uniformly over a family is probably hopeless. $\endgroup$ – Alex Eskin May 4 '13 at 6:34
  • $\begingroup$ Darn. Here I thought I was so close, having reduced this from the case of an arbitrary direction to just 2 possible directions. $\endgroup$ – Alex Becker May 4 '13 at 6:38

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