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A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map


Author: Judy Kennedy
Journal: Trans. Amer. Math. Soc. 326 (1991), 773-793
MSC: Primary 54F15; Secondary 54C10, 54F50, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1991-1010412-2
MathSciNet review: 1010412
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Abstract: A powerful theorem and construction of Wayne Lewis are used to build two homeomorphisms on the pseudoarc, each of which is semiconjugate to the tent map on the unit interval. The first homeomorphism is transitive, thus answering a question of Marcy Barge as to whether such homeomorphisms exist. The second homeomorphism admits wandering points. Also, it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points.


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  • [B1] M. Barge, Homoclinic intersections and indecomposability, Proc. Amer. Math. Soc. 101 (1987), 541-544. MR 908665 (88k:58119)
  • [B2] -, Horseshoe maps and inverse limits, Pacific J. Math. 121 (1986), 29-39. MR 815029 (87g:58077)
  • [B3] -, A method for constructing attractors, Ergodic Theory Dynamical Systems 8 (1988), 331-349. MR 961734 (90a:58100)
  • [B4] -, The topological entropy of homeomorphisms of Knaster continua, Houston J. Math. 13 (1987), 465-485. MR 929286 (89c:54077)
  • [BG] M. Barge and R. Gillette, Indecomposability and dynamics of invariant plane separating continua, Preprint. MR 1112800 (92k:54049)
  • [BM1] M. Barge and J. Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289 (1985), 355-365. MR 779069 (86h:58079)
  • [BM2] -, Dense orbits on the interval, Michigan Math. J. 34 (1987), 3-11. MR 873014 (88c:58031)
  • [BM3] -, Dense periodicity on the interval, Proc. Amer. Math. Soc. 94 (1985), 731-735. MR 792293 (87b:58068)
  • [BM4] -, The construction of global attractors, Preprint.
  • [Bi1] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. MR 0043451 (13:265b)
  • [Bi2] -, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742. MR 0027144 (10:261a)
  • [Bi3] -, Each homogeneous nondegenerate chainable continuum is a pseudorarc, Proc. Amer. Math. Soc. 10 (1959), 345-346. MR 0105072 (21:3818)
  • [Bi4] -, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 0043450 (13:265a)
  • [Bk] G. D. Birkhoff, Sur quelques courbes fermees remarquables, Bull. Soc. Math. France 60 (1932), 1-26. MR 1504983
  • [C] C. Carathéodory, Uber die Begrenzung einfach zusammenhangender Gebiete, Math. Am. 73 (1913), 323-370. MR 1511737
  • [CL1] M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the secondorder. I, The equation $ \ddot y - k(1 - {y^2})\dot y + y = b\lambda \,\cos (\lambda t + \alpha),k$ large, J. London Math. Soc. 20 (1945), 180-189. MR 0016789 (8:68g)
  • [CL2] -, Some fixed point theorems, Ann. of Math. 54 (1951), 1-37. MR 0042690 (13:148f)
  • [Ch] M. Charpentier, Sur quelques propriétiés des courbes de M. Birkhoff, Bull. Soc. Math. France 62 (1934), 193-224. MR 1505024
  • [D] R. L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings, Menlo Park, Calif., 1986. MR 811850 (87e:58142)
  • [Ha] O. H. Hamilton, A fixed point theorem for pseudoarcs and certain other metric continua, Proc. Amer. Math. Soc. 2 (1951), 173-174. MR 0039993 (12:627f)
  • [H] M. Handel, A pathological area preserving $ {C^\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), 163-168. MR 663889 (84f:58040)
  • [K] J. Kennedy, Stable extensions of homeomorphisms on the pseudoarc, Trans. Amer. Math. Soc. 310 (1988), 167-178. MR 939804 (89d:54023)
  • [Kr] J. Krasinkiewicz, Mapping properties of hereditarily indecomposable continua, Preprint. MR 688251 (84e:54037)
  • [KM] J. Krasinkiewicz and P. Minc, Mappings onto indecomposable continua, Bul. Acad. Pol. Sci. 25 (1977), 675-680. MR 0464184 (57:4119)
  • [L1] W. Lewis, Most maps of the pseudoarc are homeomorphisms, Proc. Amer. Math. Soc. 91 (1984), 147-154. MR 735582 (85g:54025)
  • [L2] -, Stable homeomorphisms of the pseudo-arc, Canad. J. Math. 31 (1977), 363-374. MR 528817 (80m:54053)
  • [OT] L. G. Oversteen and E. D. Tymchatyn, On hereditarily indecomposable continua, Geometric and Algebraic Topology, Banach Centre Publ., vol. 18, PWN, Warsaw, 1986, pp. 403-413.
  • [S] E. E. Slaminka, A Brouwer translation theorem for free homeomorphisms, Doctoral Dissertation, University of Michigan, 1984.
  • [W] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1010412-2
Keywords: Pseudoarc, wandering point, transitive homeomorphism, indecomposable continuum, tent map, semiconjugate, chainable continuum
Article copyright: © Copyright 1991 American Mathematical Society

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