Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Indecomposable continua and the Julia sets of polynomials


Authors: John C. Mayer and James T. Rogers
Journal: Proc. Amer. Math. Soc. 117 (1993), 795-802
MSC: Primary 58F23; Secondary 30C10, 30D05, 54F15, 54H20
DOI: https://doi.org/10.1090/S0002-9939-1993-1145423-7
MathSciNet review: 1145423
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find several necessary and sufficient conditions for the Julia set $ J$ of a polynomial of degree $ d \geqslant 2$ to be an indecomposable continuum. One condition that may be easier to check than others is the following: Suppose $ J$ is connected; then $ J$ is an indecomposable continuum iff the impression of some prime end of the unbounded complementary domain of $ J$ has interior in $ J$.


References [Enhancements On Off] (What's this?)

  • [B] B. Bielefeld, ed., Conformal dynamics problem list, preprint #1990/1, Institute for Mathematical Sciences, SUNY-Stony Brook.
  • [Br] B. L. Brechner, On stable homeomorphisms and imbeddings of the pseudo arc, Illinois J. Math. 22 (1978), 630-661. MR 0500860 (58:18371)
  • [BGM] B. L. Brechner, M. D. Guay, and J. C. Mayer, Rotational dynamics on cofrontiers, Continuum Theory and Dynamical Systems (M. Brown, ed.), Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 39-48. MR 1112801
  • [CL] E. F. Collingwood and A. J. Lohwater, Theory of Cluster sets, Cambridge Tracts in Math. and Math. Physics, vol. 56, Cambridge Univ. Press, Cambridge, 1966. MR 0231999 (38:325)
  • [DH1] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes (première partie), Publ. Math. Orsay 2 (1984), 1-75. MR 762431 (87f:58072a)
  • [DH2] -, Étude dynamique des polynômes complexes (deuxième partie), Publ. Math. Orsay 4 (1985), 1-154.
  • [H] C. L. Hagopian, A fixed point theorem for plane continua, Bull. Amer. Math. Soc. 77 (1971), 351-354. MR 0273591 (42:8469)
  • [HY] J. Hocking and G. Young, Topology, Addison-Wesley, Reading, MA, 1961. MR 0125557 (23:A2857)
  • [K1] J. Krasinkiewicz, On the composants of indecomposable plane continua, Bull. Polon. Acad. Sci. 20 (1972), 935-940. MR 0314011 (47:2563)
  • [K2] -, On internal composants of indecomposable plane continua, Fund. Math. 84 (1974), 255-263. MR 0339101 (49:3864)
  • [MO] J. C. Mayer and L. G. Oversteegen, Denjoy meets rotation on an indecomposable cofrontier , preprint. MR 1235352 (94d:54068)
  • [M] J. Milnor, Dynamics in one complex variable: introductory lectures, preprint #1990/5, Institute for Mathematical Sciences, SUNY-Stony Brook. MR 1721240 (2002i:37057)
  • [P] G. Piranian, The boundary of a simply connected domain, Bull. Amer. Math. Soc. 64 (1958), 45-55. MR 0100090 (20:6526)
  • [R1] J. T. Rogers, Jr., Intrinsic rotations of simply connected regions and their boundaries, Complex Variable Theory Appl. (to appear). MR 1269622 (95g:30009)
  • [R2] -, Indecomposable continua, prime ends, and Julia sets, preprint.
  • [R3] -, Singularities in the boundaries of local Siegel disks, preprint.
  • [Ru] N.E. Rutt, Prime ends and indecomposability, Bull. Amer. Math. Soc. 41 (1935), 265-273. MR 1563071

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F23, 30C10, 30D05, 54F15, 54H20

Retrieve articles in all journals with MSC: 58F23, 30C10, 30D05, 54F15, 54H20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1145423-7
Keywords: Julia set, indecomposable continuum, internal composant, prime end, simple dense canal, Lake of Wada, complex analytic dynamics, conformal dynamics
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society