Why is the existence of periodic orbits (in triangular billiards) so hard to prove?

Question

I don't get why proving that every triangular billiard has a periodic orbit should be that hard. I mostly understand the partial results on the matter, mainly

1. Every triangle with rational angles works
2. Every triangle with angles all less than 100º works

Let $\Delta=\{\,(x,y,z)\mid x+y+z=\pi\,\}$ be the space of all triangles $T_{x,y,z}$. By the second result we know that $$X=\{\,(x,y,z)\mid T_{x,y,z}\text{ has a periodic orbit}\,\}$$ is dense in $\Delta$. Can we prove it is also open? It should make sense that, if $T_{a,b,c}$ has a periodic orbit, then every triangle $T_{\alpha,\beta,\gamma}$ suficiently close to it should also have a periodic orbit (of the same period). I haven't found anywhere a definition of "closeness" of triangles where used: I suspect that it is the euclidean distance in $\Delta$.

The paper (by Schwarz) proving the second results takes, for every orbit, an infinite word whose letters are the sides of the triangle in the order of "bounce": 12321... is the orbit that starts in edge 1, then bounces on the edge 2 and so on. He then uses the word to "unfold" the triangle so that the orbit becomes a straight line. They use this like a standard technique in the field. In the paper, a word (which produces an unfolding) is stable (i.e nice) if you can get a periodic orbit from it.

If an unfolding works for a triangle, it also works for nearby triangles (*). That's why I'm asking if $X$ can be proved to be open.

(*) There seems to be a problem for the triangles close to the 30-60-90 triangle: there is no unfolding that works for every triangle in any neigborhood of it, but "bad" points should be rare and easy to take care of (with the argument of Schwarz)

I suspect the problem comes down to the "bifurcation of orbits": even if the initial conditions (starting point and direction) are close, the behaviour of the orbits can be wildly different. But why is it so difficult to analyze? I get the feeling that there shouldn't be that many orbit behaviours and, if there are, should already be classified.

Thanks!