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

Abstract

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.