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Singularities in the boundaries of local Siegel disks

Published online by Cambridge University Press:  19 September 2008

James T. Rogers Jr
James T. Rogers Jr
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
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Abstract

Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle or it is an indecomposable continuum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 803 - 821
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[B]Barge, M.. Homoclinic intersections and indecomposability. Proc. Amer. Math. Soc. 101 (1987), 541–544.CrossRefGoogle Scholar
[BG]Barge, M. & Gillette, R.. Indecomposability and dynamics of invariant plane separating continua. Contemp. Math. 117 (1991), 13–38.CrossRefGoogle Scholar
[Biel]Bielefeld, B., ed. Conformal dynamics problem list. Preprint # 1990/1, Institute for Mathematical Sciences, SUNY-Stony Brook.Google Scholar
[Bi]Birkhoff, G. D.. Sur quelques courbes fermées remarquables. Bull. Soc. Math. France 60 (1932), 1–26.Google Scholar
[Bl]Blanchard, P.. Complex analytic dynamics on the Riemann sphere. Bull Amer. Math. Soc. 11 (1984), 85–141.Google Scholar
[Ch]Charpentier, M.. Sur quelques propriét´es des courbes de M. Birkhoff. Bull. Soc. Math. France 62 (1934), 193–224.Google Scholar
[CL]Collingwood, E. F. & Lohwater, A. J.. Theory of Cluster sets, Cambridge Tracts in Math. and Math. Physics 56. Cambridge University Press, Cambridge, 1966.CrossRefGoogle Scholar
[CL1]Cartwright, M. L. & Littlewood, J. E.. On non-linear differential equations of the second order. I, The equation , k large. J. London Math. Soc. 20 (1945), L80–189.Google Scholar
[CL2]Cartwright, M. L. & Littlewood, J. E.. Some fixed point theorems. Ann. Math. 54 (1951), 1–37.CrossRefGoogle Scholar
[D1]Douady, A.. Systemes dynamiques holomorphes. Séminaire Bourbaki, exposé 599. Astérisque 105–106 (1983), 39–63.Google Scholar
[D2]Douady, A.. Disques de Siegel et anneaux de Herman. Séminaire Bourbaki, exposé 677. Astérisque (1986–1987).Google Scholar
[DH1]Douady, A. & Hubbard, J. H.. Étude dynamique des complexes (deuxième partie). Publ. Math. D'Orsay 4 (1985), 1–154.Google Scholar
[Ha]Handel, M.. A pathological area preserving C∞ diffeomorphism of the plane. Proc. Amer. Math. Soc. 86 (1982), 163–168.Google Scholar
[H1]Herman, M. R.. Construction of some curious diffeomorphisms of the Riemann Sphere. J. London Math. Soc. 34 (1986), 375–384.Google Scholar
[H2]Herman, M. R.. Are there critical points on the boundary of singular domains? Comm. Math. Phys. 99 (1985), 593–612.Google Scholar
[H3]Herman, M. R.. Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of Cn over a fixed point. Proc. Eighth Int. Cong. Math. Phys., World Sci., 1986, pp. 138–198.Google Scholar
[HJ]Hocking, J. & Young, G.. Topology. Addison-Wesley, Reading, MA, 1961.Google Scholar
[H-O-V]Hubbard, J. & Oberste-Vorth, R.. Hénon mappings in the complex domain.Google Scholar
[KY]]Kennedy, J. & Yorke, J.. The forced damped pendulum and the Wada Property.Google Scholar
[K]Kuratowski, K.. Sur les coupres irréducibles du plan. Fund. Math. 6 (1924), 130–145.Google Scholar
[LeC]Le Calvez, P.. Propriétés des attracteurs de Birkhoff. Ergod. Th. Dynam. Si. 8 (1988), 241–310.CrossRefGoogle Scholar
[MR]Mayer, J. C. & Rogers, J. T. JrIndecomposable continua and the Julia sets of polynomials. Proc. Amer. Math. Soc. (to appear).Google Scholar
[Maz]Mazurkiewicz, S.. Sur les continus absolument indécomposable. Fund. Math. 16 (1930), 151–159.CrossRefGoogle Scholar
[M]Milnor, J.. Dynamics in one complex variable: introductory lectures. Preprint # 1990/5, Institute for Mathematical Sciences, SUNY-Stony Brook.Google Scholar
[Mo]Moeckel, R.. Rotations of the closures of some simply connected domains. Complex Variables Theory Appl. 4 (1985), 223–232.Google Scholar
[P]Plykin, R.. Sources and sinks for A-diffeomorphisms. USSR Math. Sb. 23 (1974), 233–253.Google Scholar
[Po]Pommerenke, Ch.. On the boundary continuity of conformal maps. Pac. J. Math. 120 (1985), 423–430.CrossRefGoogle Scholar
[PR]Pommerenke, Ch. & Rodin, B.. Intrinsic rotations of simply connected regions, II. Complex Variables Theory Appl. 4 (1985), 223–232.Google Scholar
[Rod]Rodin, B.. Intrinsic rotations of simply connected regions. Complex Variables Theory Appl. 2 (1984), 319–326.Google Scholar
[R1]Rogers, J. T. Jr. Intrinsic rotations of simply connected regions and their boundaries. Complex Variables Theory Appl. (to appear).Google Scholar
[R2]Rogers, J. T.. Indecomposable continua, prime ends and Julia sets. Proc. Conference/Worshop on Continua Theory and Dynamical Systems (to appear).Google Scholar
[R3]Rogers, J. T.. Rotations of simply connected regions and circle-like continua. In: Continuum Theory and Dynamical Systems. Brown, M., ed. Contemporary Mathematics 117, American Mathematical Society, Providence, RI, 1991, pp. 139–148.Google Scholar
[Ru]Rutt, N. E.. Prime ends and indecomposability. Bull. Amer. Math. Soc. 41 (1935), 265–273.Google Scholar
[S]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.Google Scholar
[Su]Sullivan, D.. Quasiconformal homeomorphisms and dynamics I, solution of the Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985), 401–418.Google Scholar
[W]Walker, R.. Basin boundaries with irrational rotations on prime ends. Trans. Amer. Math. Soc. 324 (1991), 303–317.CrossRefGoogle Scholar
[Wi]Williams, R. F.. One-dimensional nonwandering sets. Topology 6 (1967), 473–487.CrossRefGoogle Scholar