Communications in Mathematical Physics

, Volume 195, Issue 1, pp 175–193 | Cite as

Diophantine Conditions Imply Critical Points on the Boundaries of Siegel Disks of Polynomials

  • James T. Rogers Jr.


Let f be a polynomial map of the Riemann sphere of degree at least two. We prove that if f has a Siegel disk G on which the rotation number satisfies a diophantine condition, then either the boundary B of G contains a critical point or B is a Lakes of Wada indecomposable continuum with one of the lakes containing a critical point. Consequently, if the boundary B of G has only 2 complementary domains, then B contains a critical point. We also show, without any assumption on the rotation number, that each proper nondegenerate subcontinuum of the boundary B of G is tree-like, and any other bounded complementary domain of B is a preperiodic component of the grand orbit of G. Finally, we establish some conditions under which B contains no periodic point.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • James T. Rogers Jr.
    • 1
  1. 1.Department of Mathematics, Tulane University, New Orleans, LA 70118, USA.¶E-mail: jim@math.tulane.eduUS

Personalised recommendations