Concerning periodic points in mappings of continua

Author:
W. T. Ingram

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present some conditions which are sufficient for a mapping to have periodic points.

Theorem. *If* *is a mapping of the space* *into* *and there exist subcontinua* and *of* *such that* (1) *every subcontinuum of* has the fixed point property, (2) *and every subcontinuum of* *are in class* , (3) contains , (4) *contains* , *and* (5) *if* *is a positive integer such that* *intersects* , *then* , *then* *contains periodic points of* *of every period greater than* 1.

Also included is a fixed point lemma:

Lemma. *Suppose* *is a mapping of the space* *into* *and* *is a subcontinuum of* *such that* *contains* . *If* (1) *every subcontinuum of* *has the fixed point property, and* (2) *every subcontinuum of* *is in class* , then there is a point *of* *such that* .

Further we show that: If is a mapping of into and has a periodic point which is not a power of 2, then contains an indecomposable continuum. Moreover, for each positive integer , there is a mapping of into with a periodic point of period and having a hereditarily decomposable inverse limit.

**[1]**James F. Davis,*Atriodic acyclic continua and class*, Proc. Amer. Math. Soc.**90**(1984), 477-482. MR**728372 (85f:54067)****[2]**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. (to appear). MR**970910 (90g:54027)****[3]**Robert L. Devaney,*An introduction to chaotic dynamical systems*, Benjamin/Cummings, Menlo Park, Calif., 1986. MR**811850 (87e:58142)****[4]**O. H. Hamilton,*A fixed point theorem for pseudo arcs and certain other metric continua*, Proc. Amer. Math. Soc.**2**(1951), 173-174. MR**0039993 (12:627f)****[5]**W. T. Ingram,*An atriodic tree-like continuum with positive span*, Fund. Math.**77**(1972), 99-107. MR**0365516 (51:1768)****[6]**Daniel P. Kuykendall,*Irreducibility and indecomposability in inverse limits*, Fund. Math.**84**(1973), 265-270. MR**0326684 (48:5027)****[7]**Tien-Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), 985-992. MR**0385028 (52:5898)****[8]**Wayne Lewis,*Periodic homeomorphisms of chainable continua*, Fund. Math.**117**(1983), 81-84. MR**712216 (85c:54065)****[9]**Jack McBryde,*Inverse limits on arcs using certain logistic maps as bonding maps*, Master's Thesis, University of Houston, 1987.**[10]**Piotr Minc,*A fixed point theorem for weakly chainable plane continua*, preprint. MR**968887 (90d:54067)****[11]**Sam B. Nadler,*Examples of fixed point free maps from cells onto larger cells and spheres*, Rocky Mountain J. Math.**11**(1981), 319-325. MR**619679 (82f:54077)****[12]**David M. Read,*Confluent and related mappings*, Colloq. Math.**29**(1974), 233-239. MR**0367903 (51:4145)****[13]**Helga Schirmer,*A topologist's view of Sharkovsky's Theorem*, Houston J. Math.**11**(1985), 385-395. MR**808654 (87e:58178)****[14]**Michel Smith and Sam Young,*Periodic homeomorphisms on**-like continua*, Fund. Math.**104**(1979), 221-224. MR**559176 (81a:54038)****[15]**R. H. Sorgenfrey,*Concerning triodic continua*, Amer. J. Math.**66**(1944), 439-460. MR**0010968 (6:96d)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54F20,
54H20,
58F20

Retrieve articles in all journals with MSC: 54F20, 54H20, 58F20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962842-8

Keywords:
Periodic point,
fixed point property,
class ,
indecomposable continuum,
inverse limit

Article copyright:
© Copyright 1988
American Mathematical Society