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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Semiconjugacies to angle-doubling


Author: Philip Boyland
Journal: Proc. Amer. Math. Soc. 134 (2006), 1299-1307
MSC (2000): Primary 37E10
Published electronically: October 5, 2005
MathSciNet review: 2199172
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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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Additional Information

Philip Boyland
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32605-8105
Email: boyland@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08381-4
PII: S 0002-9939(05)08381-4
Keywords: Circle dynamics
Received by editor(s): November 15, 2004
Published electronically: October 5, 2005
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society