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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Bifurcation to badly ordered orbits in one-parameter families of circle maps, or angels fallen from the devil's staircase


Authors: Kevin Hockett and Philip Holmes
Journal: Proc. Amer. Math. Soc. 102 (1988), 1031-1051
MSC: Primary 58F08; Secondary 58F14
DOI: https://doi.org/10.1090/S0002-9939-1988-0934888-7
MathSciNet review: 934888
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Abstract: We discuss the structure of the bifurcation set of a one-parameter family of endomorphisms of $ {S^1}$ having two critical points and negative Schwarzian derivative. We concentrate on the case in which one of the endpoints of the rotation set is rational, providing a partial characterization of components of the nonwandering set having specified rotation number and the bifurcations in which they are created. In particular we find, for each rational rotation number $ p'/q'$ less than the upper boundary of the rotation set $ p/q$, infinitely many saddle-node bifurcations to badly ordered periodic orbits of rotation number $ p'/q'$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0934888-7
Article copyright: © Copyright 1988 American Mathematical Society