  We study a skew product transformation associated to an irrational rotation
of the circle [0,1]/~. This skew product keeps track of the number of times an
orbit of the rotation lands in the two complementary intervals of {0,1/2} in
the circle. We show that under certain conditions on the continued fraction
expansion of the irrational number defining the rotation, the skew product
transformation has certain dense orbits. This is in spite of the presence of
numerous non-dense orbits. We use this to construct laminations on infinite
type surfaces with exotic properties. In particular, we show that for every
infinite type surface with an isolated planar end, there is an infinite clique
of 2-filling rays based at that end. These 2-filling rays are relevant to
Bavard--Walker's loop graphs.
