Semiconjugacies to angle-doubling

Boyland, Philip

doi:10.1090/S0002-9939-05-08381-4

Semiconjugacies to angle-doubling
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by Philip Boyland PDF

Proc. Amer. Math. Soc. 134 (2006), 1299-1307 Request permission

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 $D > 2$.

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