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로저스는 이 두 가지 방법 중 하나만 사용하여 Siegel圆의 경계를 분석할 수 있으며, 그 결과 Siegel圆의 경계가 Jordan 곡선을 이루는 경우에 한정된다고 결론지으며, 나머지는 indecomposable continuum인 것으로 나타난다.
그것은 또한 로저스가 제시한 두 가지 방법 중 하나만 사용하여 Siegel圆의 경계를 분석할 수 있으며, 그 결과 Siegel圆의 경계가 Jordan 곡선을 이루는 경우에 한정된다고 결론지으며, 나머지는 indecomposable continuum인 것으로 나타난다.
로저스는 또한 이 문제에 대한 더 깊은 이해를 위해 다음의 정리를 제안한다.
1.2 정리: polynomial f의 Siegel圆의 경계가 Jordan 곡선을 이루는 경우, 그경계에 있는 arc A가 있을 때, 두 internal ray G에 의해 형성되는 ray이 A에 닿으면, ∂G가 Jordan curve를 이룬다.
로저스는 또한 이 정리를 사용하여 polynomial f의 Siegel圆의 경계가 indecomposable continuum인 경우를 찾고자 한다.
마지막으로 로저스는 인과 관계가 없는 arc가 있는 경우에는 Siegel圆의 경계가 indecomposable continuum이 될 수 없다는 것을 보여준다.
한글 요약 끝
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arXiv:math/9210225v1 [math.CV] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 284-287IS THE BOUNDARY OF A SIEGEL DISK A JORDAN CURVE?James T. Rogers, Jr.Abstract. Bounded irreducible local Siegel disks include classical Siegel disks ofpolynomials, bounded irreducible Siegel disks of rational and entire functions, andthe examples of Herman and Moeckel.
We show that there are only two possibilitiesfor the structure of the boundary of such a disk: either the boundary admits a nicedecomposition onto a circle, or it is an indecomposable continuum.1. IntroductionLet f : C →C be a rational map of the Riemann sphere of degree at least two.The dynamics of f divides C into two disjoint sets: the stable set or Fatou set, andthe unstable set or Julia set.
On the Fatou set the dynamics of f is well behaved,while the dynamics of f on the Julia set is chaotic.The work of Sullivan [Su] completed the understanding of the dynamics of fon the Fatou set. Every component of the Fatou set is eventually periodic, andessentially five kinds of dynamical behavior are possible on these domains.
One ofthese behaviors is a Siegel disk.A component G of the Fatou set of f is a Siegel disk if f(G) = G, G contains aneutral fixed point w0, and fG is analytically conjugate to a rotation. Siegel [S]showed in 1942 that such disks exist.To say that w0 is a neutral fixed point means f(w0) = w0 and |f ′(w0)| = 1.Hence f ′(w0) = e2πiθ for some real number θ in [0, 1).
It is known that θ must beirrational, and much has been written in the effort to decide which irrationals yielda Siegel disk (see [B] or [M]).The dynamics of the Julia set is more subtle, and much is still unknown. Douadyand Sullivan [D1] have raised a very natural question: Is the boundary of a Siegeldisk a Jordan curve?
Herman [D2] has obtained an affirmative answer in specialcircumstances, but, in general, no answer is known.More generally, let us define a bounded local Siegel disk to be a pair (G, Fθ), whereG is a bounded simply connected domain in C, and Fθ : G →G is a conformal mapcomplex analytically conjugate to a rotation through the irrational angle θ suchthat Fθ extends continuously to the boundary of G. The fixed point w0 is againcalled a Siegel point. A bounded Siegel disk (G, Fθ) is irreducible if the boundaryof G separates the Siegel point w0 from ∞, but no proper closed subset of the1991 Mathematics Subject Classification.
Primary 30C35, 54F20.Key words and phrases. Siegel disk, Julia set, Fatou set, indecomposable continuum, primeend.Received by the editors November 21, 1991 and, in revised form, March 24, 1992This research was partially supported by a COR grant from Tulane Universityc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1
2J. T. ROGERS, JR.boundary has this property.
Bounded irreducible local Siegel disks include classicalSiegel disks of polynomials as well as bounded irreducible Siegel disks of rationaland entire functions and even the exotic examples of Herman [H1] and Moeckel[Mo].We describe the structure of the boundaries of such domains by proving thefollowing theorem.Theorem 1.1. The boundary ∂G of a bounded irreducible local Siegel disk satisfiesexactly one of the following properties:(1) The inverse ϕ−1 of the Riemann map ϕ: D →G of the conjugation extendscontinuously to a map ψ: ∂G →∂D = S1, or(2) ∂G is an indecomposable continuum.An indecomposable continuum is a compact connected space which cannot bewritten as a union A ∪B with A and B connected closed proper subsets of X.Indecomposable continua are complicated spaces; nevertheless, Herman [H1] hasconstructed a bounded irreducible local Siegel disk whose boundary is a certainindecomposable continuum known as the pseudocircle.In case (1), the point inverses of ψ: ∂G →S1 are the impressions of prime endsof ϕ.
In particular, the point inverses of ψ are connected. A space with such adecomposition onto a circle can be written as a union A ∪B as described above, sothe two possibilities are mutually exclusive.
Moeckel [Mo] has constructed such anexample in which the point inverses of ψ are either points or straight line intervals.The Moeckel example shows that we cannot require ϕ−1 to extend to a homeo-morphism in (1), while the Herman example shows that (2) can occur. Thus theresult is the best possible for such local Siegel disks.The boundary of a Siegel disk is a Jordan curve if and only if the Riemann mapϕ: D →G of the conjugation extends to a homeomorphism of D onto G. This isequivalent, of course, to ϕ−1 : G →D extending to a homeomorphism of G ontoD.
Thus we may interpret the theorem to imply any counterexample must be asnice as possible or as complicated as possible.The theorem above implies that a weak additional hypothesis is enough to answerthe Douady-Sullivan question affirmatively.Theorem 1.2. Let A be an arc in the boundary of a Siegel disk of a polynomial ofdegree d ≥2.
If two internal rays from G land on A, then ∂G is a Jordan curve.Thus, any arc in a counterexample must be “hidden.” In particular, we have thefollowing corollary.Corollary 1.3. If the boundary of a Siegel disk of a polynomial of degree d ≥2 isarcwise connected, then ∂G is a Jordan curve.The Julia set of a polynomial f is the closure of the set of repelling periodicpoints of f, and the boundary of a Siegel disk is a subset of the Julia set.
Hencethe next theorem is in one sense a little surprising.Theorem 1.4. If the boundary ∂G is a Siegel disk of a polynomial of degree d ≥2contains a periodic point, then ∂G is an indecomposable continuum.This paper is an abstract of the results in [R3].
The paper [R3] contains a briefhistory of indecomposable continua occurring in the study of dynamical systemsand suggests that it is not so unexpected that we must deal with indecomposablecontinua in this situation.
IS THE BOUNDARY OF A SIEGEL DISK A JORDAN CURVE?32. The structure of the boundary of local Siegel disksLet (G, Fθ) denote a bounded irreducible local Siegel disk.
We need a numberof tools to complete the proof of the structure theorem.The first is a result of Pommerenke and Rodin [PR] about prime ends η andtheir impressions I(η).Theorem 2.1. Each local Siegel disk has a Pommerenke-Rodin number; i.e., thereexists a number d (not to be confused with the degree of a polynomial) with 0 ≤d ≤2such that, for prime ends η1 and η2 in ∂D,I(η1) ∩I(η2) ̸= ∅⇔|η1 −η2| ≤d .The distance |η1 −η2| on ∂D is given by the Euclidean metric on C; hence, forexample, |η1 −η2| = 2 if and only if η1 and η2 are diametrically opposite.
It followsthat d = 0 if and only if all impressions are pairwise disjoint, while d = 2 if andonly if each pair of impressions has a point in common.The second and most important tool is the theory of prime ends as related toindecomposable continua. The work of Rutt [Ru] is used, for instance, in prov-ing the following result of the author [R1], a result that enables us to recognizeindecomposable continua by analytic methods.Theorem 2.2.
If (G, Fθ) is a local Siegel disk, then ∂G is an indecomposablecontinuum if and only if there exists a prime end η of G such that the impressionI(η) = ∂G.The proof of the structure theorem is completed by a somewhat delicate analysisof the relationship between prime ends and indecomposable continua. The detailsappear in [R3].References[B]P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull.
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Soc11 (1984), 85–141.[D1]A. Douady, Syst`emes dynamiques holomorphes, Seminaire Bourbaki, expos´e 599, Ast´erisque,vol.
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Conference/Workshopon Continuum Theory and Dynamical Systems (to appear). [R3], Singularities in the boundaries of local Siegel disks, Ergodic Theory DynamicalSystems (to appear).[Ru]N.
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(2) 122 (1985), 401–418.Department of Mathematics, Tulane University, New Orleans, Louisianna 70118
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