A simple consequence of a theorem of Franks says that whenever a continuous map,
g
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
g
is strictly larger than that of angle-doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-
D
D
circle maps with
D
>
2
D > 2
.