  A simple consequence of a theorem of Franks says that whenever a continuous
map, $g$, is homotopic to angle doubling on the circle it is semiconjugate to
it. We show that when this semiconjugacy has one disconnected point inverse,
then the typical point in the circle has a point inverse with uncountably many
connected components. Further, in this case the topological entropy of $g$ is
strictly larger than that of angle doubling, and the semiconjugacy has
unbounded variation. An analogous theorem holds for degree-$D$ circle maps with
$D > 2$.
