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

Kennedy, Judy

doi:10.1090/S0002-9947-1991-1010412-2

A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map
HTML articles powered by AMS MathViewer

by Judy Kennedy PDF

Trans. Amer. Math. Soc. 326 (1991), 773-793 Request permission

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.

References

Marcy Barge, Homoclinic intersections and indecomposability, Proc. Amer. Math. Soc. 101 (1987), no. 3, 541–544. MR 908665, DOI 10.1090/S0002-9939-1987-0908665-6
Marcy Barge, Horseshoe maps and inverse limits, Pacific J. Math. 121 (1986), no. 1, 29–39. MR 815029
Marcy Barge, A method for constructing attractors, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 331–349. MR 961734, DOI 10.1017/S0143385700004491
Marcy Barge, The topological entropy of homeomorphisms of Knaster continua, Houston J. Math. 13 (1987), no. 4, 465–485. MR 929286
Marcy Barge and Richard M. Gillette, Indecomposability and dynamics of invariant plane separating continua, Continuum theory and dynamical systems (Arcata, CA, 1989) Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 13–38. MR 1112800, DOI 10.1090/conm/117/1112800
Marcy Barge and Joe Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289 (1985), no. 1, 355–365. MR 779069, DOI 10.1090/S0002-9947-1985-0779069-7
Marcy Barge and Joe Martin, Dense orbits on the interval, Michigan Math. J. 34 (1987), no. 1, 3–11. MR 873014, DOI 10.1307/mmj/1029003477
Marcy Barge and Joe Martin, Dense periodicity on the interval, Proc. Amer. Math. Soc. 94 (1985), no. 4, 731–735. MR 792293, DOI 10.1090/S0002-9939-1985-0792293-8

The construction of global attractors

R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451
R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. MR 27144
R. H. Bing, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345–346. MR 105072, DOI 10.1090/S0002-9939-1959-0105072-6
R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653–663. MR 43450
George D. Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1–26 (French). MR 1504983
C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913), no. 3, 323–370 (German). MR 1511737, DOI 10.1007/BF01456699
M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order. I. The equation $\ddot y-k(1-y^2)y+y=b\lambda k\;\textrm {cos} (\lambda t+a), k$ large, J. London Math. Soc. 20 (1945), 180–189. MR 16789, DOI 10.1112/jlms/s1-20.3.180
M. L. Cartwright and J. E. Littlewood, Some fixed point theorems, Ann. of Math. (2) 54 (1951), 1–37. With appendix by H. D. Ursell. MR 42690, DOI 10.2307/1969308
M. Charpentier, Sur quelques propriétés des courbes de M. Birkhoff, Bull. Soc. Math. France 62 (1934), 193–224 (French). MR 1505024
Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
O. H. Hamilton, A fixed point theorem for pseudo-arcs and certain other metric continua, Proc. Amer. Math. Soc. 2 (1951), 173–174. MR 39993, DOI 10.1090/S0002-9939-1951-0039993-2
Michael Handel, A pathological area preserving $C^{\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), no. 1, 163–168. MR 663889, DOI 10.1090/S0002-9939-1982-0663889-6
Judy Kennedy, Stable extensions of homeomorphisms on the pseudo-arc, Trans. Amer. Math. Soc. 310 (1988), no. 1, 167–178. MR 939804, DOI 10.1090/S0002-9947-1988-0939804-4
J. Krasinkiewicz, Mapping properties of hereditarily indecomposable continua, Houston J. Math. 8 (1982), no. 4, 507–516. MR 688251
Józef Krasinkiewicz and Piotr Minc, Mappings onto indecomposable continua, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 7, 675–680 (English, with Russian summary). MR 464184
Wayne Lewis, Most maps of the pseudo-arc are homeomorphisms, Proc. Amer. Math. Soc. 91 (1984), no. 1, 147–154. MR 735582, DOI 10.1090/S0002-9939-1984-0735582-4
Wayne Lewis, Stable homeomorphisms of the pseudo-arc, Canadian J. Math. 31 (1979), no. 2, 363–374. MR 528817, DOI 10.4153/CJM-1979-041-1

On hereditarily indecomposable continua

A Brouwer translation theorem for free homeomorphisms

Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108