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Proceedings of the American Mathematical Society

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Note on rotation set


Author: Ryuichi Ito
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
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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 [Enhancements On Off] (What's this?)

  • [1] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979).
  • [2] Ryuichi Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 107–111. MR 591976, https://doi.org/10.1017/S0305004100057984
  • [3] S. Newhouse, J. Palis and F. Takens, Stable families of dynamical system. I: diffeomorphisms, I.M.P.A., Rio de Janeiro, Brazil, 1979, preprint.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0719006-8
Keywords: Rotation set
Article copyright: © Copyright 1983 American Mathematical Society