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Proceedings of the American Mathematical Society

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



Cycles in $ C\sp r$-twists

Author: C. R. Carroll
Journal: Proc. Amer. Math. Soc. 123 (1995), 927-934
MSC: Primary 58F03; Secondary 58F11
MathSciNet review: 1219722
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Abstract: Let $ {f_t},0 \leq t \leq 1$, be a continuous one-parameter family of $ {C^r}$ diffeomorphisms of the circle obtained by monotonically twisting away from $ {f_0}$. It is well known that for a dense set of parameter values $ {t, f_t}$ has a periodic orbit. To what extent does the distribution in the circle of these periodic orbits reflect the degree of differentiability r? We show that if $ r \geq 6,{\sup _t}{\left\Vert {{f_t} \circ f_0^{ - 1}} \right\Vert _{{C^r}}} < \infty $, and the rotation number of $ {f_0}$ is an irrational $ \alpha $ with bounded continued fraction expansion, then periodic orbits corresponding to small values of the parameter t echo the metric structure of $ {f_0}$ in the following sense: If the rotation number of the orbit is a convergent of $ \alpha $, then the orbit divides the circle into intervals of nearly equal $ \mu $-measure, where $ \mu $ is the invariant Borel probability measure of $ {f_0}$. The corresponding result for low differentiability is false.

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Article copyright: © Copyright 1995 American Mathematical Society

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