  Dilation surfaces are geometric surfaces modelled after the complex plane
whose structure group is generated by the groups of translations and dilations.
For any dilation surface, for any direction $\theta$ in $S^1$, there exists a
foliation on the surface called the directional foliation in direction
$\theta$. In this Thesis, we prove a structure theorem for the directional
foliations on dilation surfaces using a decomposition theorem established by
C.J. Gardiner in the 1980s. We show that given a directional foliation on any
dilation surface, there exists a decomposition of the surface into finitely
many subsurfaces on which the foliation structure is in one of four possible
cases: completely periodic, Morse-Smale, minimal or Cantor-like. We further
prove that in the last two cases, the first return map on a segment transversal
to the foliation is semi-conjugated to a minimal interval exchange
transformation. As a corollary, we obtain an analogous result for affine
interval exchange transformations. Throughout the thesis, we accompany our
results with an explicit example of a dilation surface called the Disco
surface. We analyze the directional foliations on the Disco surface that
exhibit non-trivially recurrent behaviour and explain geometrically why these
foliations accumulate to a Cantor set.
