Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Note on rotation set
HTML articles powered by AMS MathViewer

by Ryuichi Ito PDF
Proc. Amer. Math. Soc. 89 (1983), 730-732 Request permission

Abstract:

Let $f$ be an endomorphism of the circle of degree 1 and $\bar f$ be a lifting of $f$. We characterize the rotation set $\rho (\bar f)$ by the set of probability measures on the circle, and prove that if ${\rho _ + }(\bar f)\;({\rho _ - }(\bar f))$, the upper (lower) endpoint of $\rho (\bar f)$, is irrational, then ${\rho _ + }({R_\theta }\bar f) > {\rho _ + }(\bar f)\;({\rho _ - }({R_\theta }\bar f) > {\rho _ - }(\bar f))$ for any $\theta > 0$, where ${R_\theta }(x) = x + \theta$. As a corollary, if $f$ is structurally stable, then both ${\rho _ + }(\bar f)$ and ${\rho _ - }(\bar f)$ are rational.
References
    M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979).
  • Ryuichi Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 107–111. MR 591976, DOI 10.1017/S0305004100057984
  • S. Newhouse, J. Palis and F. Takens, Stable families of dynamical system. I: diffeomorphisms, I.M.P.A., Rio de Janeiro, Brazil, 1979, preprint.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F99
  • Retrieve articles in all journals with MSC: 58F99
Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 89 (1983), 730-732
  • MSC: Primary 58F99
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0719006-8
  • MathSciNet review: 719006