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Transactions of the American Mathematical Society

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Pseudocircles in dynamical systems


Authors: Judy A. Kennedy and James A. Yorke
Journal: Trans. Amer. Math. Soc. 343 (1994), 349-366
MSC: Primary 58F15; Secondary 54F15, 54F50, 54H20, 58F20
DOI: https://doi.org/10.1090/S0002-9947-1994-1187029-5
MathSciNet review: 1187029
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct an example of a $ {C^\infty }$ map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the $ {C^1}$-metric) to the constructed map has a similar set.


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

  • [B] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951). MR 0043451 (13:265b)
  • [F1] L. Fearnley, The pseudo-circle is not homogeneous, Bull. Amer. Math. Soc. 75 (1969), 554-558. MR 0242126 (39:3460)
  • [F2] -, The pseudo-circle is unique, Bull. Amer. Math. Soc. 75 (1969), 398-401. MR 0246265 (39:7569)
  • [Ha] Michael Handel, A pathological area preserving $ {C^\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), 163-168. MR 663889 (84f:58040)
  • [He] Michael Herman, Construction of some curious diffeomorphisms of the Riemann sphere, J. London Math. Soc. 34 (1986), 375-384. MR 856520 (87m:58128)
  • [KR] J. Kennedy and J. T. Rogers, Jr., Orbits of the pseudocircle, Trans. Amer. Math. Soc. 296 (1986), 327-340. MR 837815 (87g:54076)
  • [Kr] J. Krasinkiewicz, Mapping properties of hereditarily indecomposable continua, preprint. MR 688251 (84e:54037)
  • [M] Richard Moeckel, Rotations of the closures of some simply connected domains, Complex Variables Theory Appl. 4 (1985), 285-294. MR 801645 (86k:30027)
  • [OT] L. G. Oversteegen and E. D. Tymchatyn, On hereditarily indecomposable continua, Geometric and Algebraic Topology, Banach Centre Publ., vol. 18, PWN, Moscow, 1986, pp. 403-413. MR 925879 (88m:54044)
  • [PR] Ch. Pommerenke and B. Rodin, Intrinsic rotations of simply connected regions. II, Complex Variables Theory Appl. 4 (1985), 223-232. MR 801639 (87k:30058)
  • [R] J. T. Rogers, The pseudo-circle is not homogeneous, Trans. Amer. Math. Soc. 148 (1970), 417-428. MR 0256362 (41:1018)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1187029-5
Keywords: Indecomposable invariant set, pseudocircle, $ {C^\infty }$ map, perturbable dynamical system
Article copyright: © Copyright 1994 American Mathematical Society

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