Herman’s Condition and Siegel Disks of Bi-Critical Polynomials

Abstract

Abstract. We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.

Singular values and bounded Siegel disks

“…If c has a higher multiplicity, the symmetry arguments are more involved: we refer to [CR,Proposition 26 and Paragraph 4.11.6] for the proof of the fact that ∆ = ∆. Now, as in the proof of Proposition A, the complement of ∆ can be uniformized to C \ D via a holomorphic map Φ, in order to get, by Schwarz reflection, an analytic map g defined in a neighborhood of S 1 , such that g| S 1 is an analytic circle map with no critical points (see [CR,Lemma 30]). As g is conjugate to f in a small neighborhood of the unit disk, using the fact that f : U → V has degree d, it can be shown that also g has degree exactly d on S 1 [CR,Lemma 30].…”

“…They will only be used in the proof of Proposition C, in Section 4. Most of them can be found or deduced from results in [CR,section 2], but we include the proof here for completeness and self-containment. PROPOSITION 2•6.…”

“…If c (the preimage of v in U ) is a simple critical point, this concludes the proof that = , because there are exactly two preimages and of and both are contained in . If c has a higher multiplicity, the symmetry arguments are more involved: we refer to [CR, for the proof of the fact that = .…”

“…Now, as in the proof of Proposition A, the complement of can be uniformised to C \ D via a holomorphic map , in order to get, by Schwarz reflection, an analytic map g defined in a neighbourhood of S 1 , such that g| S 1 is an analytic circle map with no critical points (see [CR,lemma 44 and section 2•3]). As g is conjugate to f in a small neighbourhood of the unit disk, using the fact that f : U → V has degree d, it can be shown that also g has degree exactly d on S 1 [CR,lemma 44]. Now suppose that g has a non-repelling periodic point x of period q in S 1 .…”

On the topology of the inverse limit of a branched covering over a Riemann surface

“…Now we investigate the case, in which W is a Siegel disk and its boundary ∂W is a Jordan curve. Several cases in which this occurs and ∂W contains a critical point, are listed and studied in [18], [24], and [7]. One such case is presented in the Main Theorem of [24].…”