Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F03, 58F11

Retrieve articles in all journals with MSC: 58F03, 58F11


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1219722-6
PII: S 0002-9939(1995)1219722-6
Article copyright: © Copyright 1995 American Mathematical Society