Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials
Let \(f\) be a polynomial map of the Riemann sphere of degree at least two. Suppose that \(f\) has a Siegel disk \(G\) whose rotation number \(\alpha\) satisfies a diophantine condition, that is, there are numbers \(r> 0\) and \(k\geq 2\) such that \[ |\alpha- \tfrac pq|> \tfrac {r}{q^k} \] for every rational number \(\frac pq\). Then either the boundary \(B\) of \(G\) contains a critical point or \(B\) is an indecomposable continuum with three properties: (1) \(B\) has at least three complementary domains, and \(B\) is the boundary of each of them; (2) each bounded complementary domain of \(B\) is a component of the grand orbit of \(G\) and so a bounded component of the Fatou set, and (3) one of the bounded complementary domains of \(B\) contains a critical point. It is noted by the author, however, that the second possibility remains open: whether such a Siegel disk can exist is unknown.
scientific article; zbMATH DE number 1316945