Proceedings of the American Mathematical Society

Author(s): Philip Boyland
Journal: Proc. Amer. Math. Soc. 134 (2006), 1299-1307.
MSC (2000): Primary 37E10
Posted: October 5, 2005
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Abstract: A simple consequence of a theorem of Franks says that whenever a continuous map,

, 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

is strictly larger than that of angle-doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-

circle maps with

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