  Polygonal billiards exhibit a rich and complex dynamical behavior. In recent
years polygonal billiards have attracted great attention due to their
application in the understanding of anomalous transport, but also at the
fundamental level, due to its connections with diverse fields in mathematics.
We explore this complexity and its consequences on the properties of particle
transport in infinitely long channels made of the repetitions of an elementary
open polygonal cell. Borrowing ideas from the Zemlyakov-Katok construction, we
construct an interval exchange transformation classified by the singular
directions of the discontinuities of the billiard flow over the translation
surface associated to the elementary cell. From this, we derive an exact
expression of a scattering map of the cell connecting the outgoing flow of
trajectories with the unconstrained incoming flow. The scattering map is
defined over a partition of the coordinate space, characterized by different
families of trajectories. Furthermore, we obtain an analytical expression for
the average speed of propagation of ballistic modes, describing with high
accuracy the speed of propagation of ballistic fronts appearing in the tails of
the distribution of the particle displacement. The symbolic hierarchy of the
trajectories forming these ballistic fronts is also discussed.
