Available in electronic format
Available in print format
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
     

A Weakly Chainable Tree-Like Continuum without the Fixed Point Property

Author(s): Piotr Minc
Journal: Trans. Amer. Math. Soc. 351 (1999), 1109-1121.
MSC (1991): Primary 54F15; Secondary 54H25
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: An example of a fixed points free map is constructed on a tree-like, weakly chainable continuum.


References:

1.
W. L. Ayres, Some generalizations of the Scherrer fixed-point theorem, Fund. Math. 16 (1930), 332-336.

2.
H. Bell, On fixed point properties of plane continua, Trans. Amer. Math. Soc. 128 (1967), 539-548. MR 35:4888

3.
D.P. Bellamy, A tree-like continuum without the fixed-point property, Houston J.Math. 6 (1980), 1-13. MR 81h:54039

4.
-, The fixed point property in dimension one, in Continua with the Houston problem book (H. Cook et al., eds.), Marcel Dekker, New York, 1995, pp. 27-35. MR 96a:54056

5.
R.H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 13:265a

6.
-, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132. MR 38:5201

7.
K. Borsuk, Einige Sätze über stetige Streckenbilder, Fund. Math. 18 (1932), 198-213.

8.
-, Sur un continu acyclique qui se laisse transformer topologiquement en lui même sans points invariants, Fund. Math. 24 (1935), 51-58.

9.
-, A theorem on fixed points, Bull. Acad. Sci. Polon. 2 (1954), 17-20. MR 16:275h

10.
L. Fearnley, Characterizations of continuous images of pseudoarc, Trans. Amer. Math. Soc. 111 (1964), 380-399. MR 29:596

11.
J. B. Fugate and L. A. Mohler, A note on fixed points in tree-like continua, Topology Proc. 2 (1977), 457-460. MR 80k:54062

12.
C. L. Hagopian, A fixed point theorem for plane continua, Bull. Amer. Math. Soc. 77 (1971), 351-354. MR 42:8469

13.
-, The fixed point property for deformations of tree-like continua, preprint.

14.
O. H. Hamilton, A fixed point theorem for pseudoarcs and certain other metric continua, Proc. Amer. Math. Soc 2 (1951), 173-174. MR 12:627f

15.
S. Iliadis, Positions of continua in a plane and fixed points, Vestn. Moskov. Univ. 25 (1970), 66-70. MR 44:4726

16.
S. Kinoshita, On some contractible continua without fixed point property, Fund. Math. 40 (1953), 96-98. MR 15:642b

17.
R. Knill, Cones, products and fixed points, Fund. Math. 60 (1967), 35-46. MR 35:2270

18.
A. Lelek, On weakly chainable continua, Fund Math. 51 (1962), 271-282. MR 26:742

19.
W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361-394. MR 86a:54038

20.
R. Ma\'{n}ka, Association and fixed points, Fund. Math. 91 (1976), 105-121. MR 54:1183

21.
R. D. Mauldin (ed.), The Scottish Book: Mathematics from the Scottish Café, Birkhauser, Boston, 1981. MR 84m:00015

22.
P. Minc, A fixed point theorem for weakly chainable plane continua, Trans. Amer. Math. Soc. 317 (1990), 303-312. MR 90d:54067

23.
-, A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories, Topology and its Appl. 46 (1992), 99-106. MR 94a:54108

24.
-, A periodic points free homeomorphism of a tree-like continuum, Trans Amer Math. Soc. 348 (1996), 1487-1519. MR 96h:54029

25.
J. Mioduszewski, A functional conception of snake-like continua, Fund. Math. 51 (1962), 179-189. MR 26:1859

26.
G. Nöbeling, Eine Fixpunkteigenschaft der BaumKurven, Ergebnisse eines mathematischen Kolloquium 2 (1932), 19.

27.
L.G. Oversteegen and J.T. Rogers, Jr., Tree-like continua as limits of cyclic graphs, Topology Proc. 4 (1979), 507-515. MR 82c:54032

28.
-, An inverse limit description of an atriodic tree-like continuum and an induced map without a fixed point, Houston J.Math. 6 (1980), 549-564. MR 82j:54075

29.
-, Fixed-point-free maps on tree-like continua, Topology and its Appl. 13 (1982), 85-95. MR 83b:54044

30.
W. Scherrer, Über ungeschlossene stetige Kurven, Math. Zeitschr. 24 (1926), 125-130.

31.
K. Sieklucki, On a class of plane acyclic continua with the fixed point property, Fund. Math. 63 (1968), 257-278. MR 39:2139


Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54F15, 54H25

Retrieve articles in all Journals with MSC (1991): 54F15, 54H25


Additional Information:

Piotr Minc
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: mincpio@mail.auburn.edu

DOI: 10.1090/S0002-9947-99-02071-1
PII: S 0002-9947(99)02071-1
Keywords: Tree-like continuum, weakly chainable, fixed point property
Received by editor(s): May 20, 1996
Received by editor(s) in revised form: March 6, 1997
Additional Notes: This research was supported in part by NSF grant # DMS-9505108.
The author is grateful to Charles L. Hagopian for suggesting several valuable improvements
Copyright of article: Copyright 1999, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google