Semiconjugacies to angle-doubling

Abstract

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 $D > 2$.