  In this work we completely describe the dynamics of triangle tiling
billiards. In the first part of this work, we propose a geometric approach of
dynamics by introducing natural foliations associated to it. In the second
part, we exploit the relationship between triangle tiling billiards and a
family of fully flipped $3$-interval exchange transformations on the circle. We
give a combinatorial approach of dynamics via renormalization. By uniting the
two approaches, we prove several conjectures on the dynamics of triangle tiling
billiards. First, we prove the Tree Conjecture and the 4n+2 Conjecture, both
concerning the symbolic dynamics of periodic trajectories, and both stated by
Baird-Smith, Davis, Fromm and Iyer. Second, we study a family of exceptional
trajectories which are closely related to the orbits of minimal Arnoux-Rauzy
maps. We prove that all of these exceptional trajectories pass by all tiles,
which confirms our own conjecture with P. Hubert on their non-linear escape.
Moreover, we use tiling billiards to prove the convergence, up to rescaling, of
arithmetic orbits of the Arnoux-Yoccoz map to the Rauzy fractal, conjectured by
Hooper and Weiss. All of these conjectures have been stated in print in the
last three years.
