Concerning periodic points in mappings of continua
HTML articles powered by AMS MathViewer
- by W. T. Ingram PDF
- Proc. Amer. Math. Soc. 104 (1988), 643-649 Request permission
Abstract:
In this paper we present some conditions which are sufficient for a mapping to have periodic points. Theorem. If $f$ is a mapping of the space $X$ into $X$ and there exist subcontinua $H$ and $K$ of $X$ such that (1) every subcontinuum of $K$ has the fixed point property, (2) $f[K]$ and every subcontinuum of $f[H]$ are in class $W$, (3) $f[K]$ contains $H$, (4) $f[H]$ contains $H \cup K$, and (5) if $n$ is a positive integer such that ${(f|H)^{ - n}}(K)$ intersects $K$, then $n = 2$, then $K$ contains periodic points of $f$ of every period greater than 1. Also included is a fixed point lemma: Lemma. Suppose $f$ is a mapping of the space $X$ into $X$ and $K$ is a subcontinuum of $X$ such that $f[K]$ contains $K$. If (1) every subcontinuum of $K$ has the fixed point property, and (2) every subcontinuum of $f[K]$ is in class $W$, then there is a point $x$ of $K$ such that $f(x) = x$. Further we show that: If $f$ is a mapping of $[0,1]$ into $[0,1]$ and $f$ has a periodic point which is not a power of 2, then $\lim \{ [0,1],f\}$ contains an indecomposable continuum. Moreover, for each positive integer $i$, there is a mapping of $[0,1]$ into $[0,1]$ with a periodic point of period ${2^i}$ and having a hereditarily decomposable inverse limit.References
- James F. Davis, Atriodic acyclic continua and class $W$, Proc. Amer. Math. Soc. 90 (1984), no. 3, 477–482. MR 728372, DOI 10.1090/S0002-9939-1984-0728372-X
- James F. Davis and W. T. Ingram, An atriodic tree-like continuum with positive span which admits a monotone mapping to a chainable continuum, Fund. Math. 131 (1988), no. 1, 13–24. MR 970910, DOI 10.4064/fm-131-1-13-24
- 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
- W. T. Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972), no. 2, 99–107. MR 365516, DOI 10.4064/fm-77-2-99-107
- D. P. Kuykendall, Irreducibility and indecomposability in inverse limits, Fund. Math. 80 (1973), no. 3, 265–270. MR 326684, DOI 10.4064/fm-80-3-265-270
- T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
- Wayne Lewis, Periodic homeomorphisms of chainable continua, Fund. Math. 117 (1983), no. 1, 81–84. MR 712216, DOI 10.4064/fm-117-1-81-84 Jack McBryde, Inverse limits on arcs using certain logistic maps as bonding maps, Master’s Thesis, University of Houston, 1987.
- Piotr Minc, A fixed point theorem for weakly chainable plane continua, Trans. Amer. Math. Soc. 317 (1990), no. 1, 303–312. MR 968887, DOI 10.1090/S0002-9947-1990-0968887-X
- Sam B. Nadler Jr., Examples of fixed point free maps from cells onto larger cells and spheres, Rocky Mountain J. Math. 11 (1981), no. 2, 319–325. MR 619679, DOI 10.1216/RMJ-1981-11-2-319
- David R. Read, Confluent and related mappings, Colloq. Math. 29 (1974), 233–239. MR 367903, DOI 10.4064/cm-29-2-233-239
- Helga Schirmer, A topologist’s view of Sharkovsky’s theorem, Houston J. Math. 11 (1985), no. 3, 385–395. MR 808654
- Michel Smith and Sam W. Young, Periodic homeomorphisms on $T$-like continua, Fund. Math. 104 (1979), no. 3, 221–224. MR 559176, DOI 10.4064/fm-104-3-221-224
- R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439–460. MR 10968, DOI 10.2307/2371908
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 643-649
- MSC: Primary 54F20; Secondary 54H20, 58F20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0962842-8
- MathSciNet review: 962842