Semiconjugacies to angle-doubling

Abstract

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 .