I am gathering things that would be nice to check around Boshernitzan complete periodicity conjecture for the billiard flow in a direction perpendicular to one of the side of a triangle (a, b, c). Let me call vertical the direction that we are looking at in the unfolded surface. In the unfolded surface, there are either 1, 2 or 3 edges orthogonal to the vertical (at most one for each side of the initial triangle). When a + b + c is odd there are 3 of them and when it is even it could be 1 or 2 depending on the direction. Let me call these the horizontal edges.
when complete periodicity holds (especially in the a + b + c odd case)
test whether each vertical saddle connection intersects an horizontal edge
compute how many saddle connections there are on each side of each cylinder (the generic situation is just one on each side but we are far away from a generic situation)
test whether the core curve of each vertical cylinder intersects an horizontal edge (weaker than the above but less likely to be an "explanation" if the conjecture holds true)
for each minimal component, compute the SAF invariant (especially whether it is zero or not). See flatsurf/#297
I am gathering things that would be nice to check around Boshernitzan complete periodicity conjecture for the billiard flow in a direction perpendicular to one of the side of a triangle
(a, b, c). Let me call vertical the direction that we are looking at in the unfolded surface. In the unfolded surface, there are either 1, 2 or 3 edges orthogonal to the vertical (at most one for each side of the initial triangle). Whena + b + cis odd there are 3 of them and when it is even it could be 1 or 2 depending on the direction. Let me call these the horizontal edges.a + b + codd case)