Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T16:26:47.369Z Has data issue: false hasContentIssue false
  • English
  • Français

On PZ type Siegel disks of the sine family

Published online by Cambridge University Press:  10 November 2014

GAOFEI ZHANG
GAOFEI ZHANG*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China email zhanggf@hotmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 973 - 1006
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

David, G.. Solutions de l’equation de Beltrami avec ∥𝜇∥∞ = 1. Ann. Acad. Sci. Fenn. Math. 13 (1988), 25–70.Google Scholar
de Faria, E. and De Melo, W.. Rigidity of critical circle mappings I. J. Eur. Math. Soc. (JEMS) 1 (1999), 339–392.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.Google Scholar
Khinchin, A. Ya.. Continued Fractions. Dover, New York, 1992.Google Scholar
McDonald, J. and Weiss, N.. A Course in Real Analysis. Academic Press, New York, 1999.Google Scholar
Petersen, C.. Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177 (1996), 163–224.Google Scholar
Petersen, C. and Zakeri, S.. On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. of Math. (2) 159(1) (2004), 1–52.Google Scholar
Yampolsky, M.. Complex bounds for renormalization of critical circle maps. Ergod. Th. & Dynam. Sys. 19(1) (1999), 227–257.Google Scholar
Yoccoz, J.-C.. Il n’y a pas de contre-exemple de Denjoy analytique. C. R. Acad. Sci. Paris 298 (1984), 141–144.Google Scholar
Zhang, G.. On the dynamics of e 2𝜋i𝜃 sin(z). Illinois J. Math. 49(4) (2005), 1171–1179.Google Scholar
Zhang, G.. An application of the topological rigidity of the sine family. Ann. Acad. Sci. Fenn. Math. 33 (2008), 81–85.Google Scholar
Zhang, G.. Polynomial Siegel disks are typically Jordan domains. Preprint, 2012, arXiv:1208.1881.Google Scholar