Pacific Journal of Mathematics

$u$-mappings on trees.

M. M. Marsh

Article information

Pacific J. Math., Volume 127, Number 2 (1987), 373-387.

First available in Project Euclid: 8 December 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54F20
Secondary: 54B25 54F50: Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]


Marsh, M. M. $u$-mappings on trees. Pacific J. Math. 127 (1987), no. 2, 373--387.

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  • [rop. 7] , it follows that / is universal. Suppose that X hasexactly m interior edges which arefolded by/. We assume that whenever /': Z -* Y is a w-mapping of a tree Z ontoY such that Z hasfewer than m interior edges which arefolded by/',then /' is universal. By way of contradiction, we assume that / is notuniversal. Let g: X -> Y bea mapping such that f(x) g(x) for each x e X.
  • [3,r2] ,r2] ,r2] ,r2] -* U,F] In a manner similar to that outlined in Example 2, it is easy to check that / has the desired properties. We also notice that a restriction of the mapping / would yield an example of a non-universal mapping which satisfies properties (1),(2),(3), and (5),but not (4). Let Xf = X - (39 b2]. Then the mapping f\x,: X' -> Y has the desired properties. Examples of non-universal mappings which do not satisfy property (1) or do not satisfy property (5) can also be given.
  • [1] D. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math., 6 (1979), 1-14.
  • [2] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly, 76 (1969), 119-132.
  • [3] C. A. Eberhart and J. B. Fugate, Weakly confluent maps on trees, General Topology and Modern Analysis, Academic Press, NewYork (1981), 209-215.
  • [4] O. H. Hamilton, A fixed point theorem for pseudo arcs and certain other metric continua, Proc. Amer. Math. Soc,2 (1951), 173-174.
  • [5] W. Holsztynski, Universal mappings andfixed point theorems, Bull. Acad. Polon. Sci., XV (1967), 433-438.
  • [6] M. M. Marsh, Fixed Point Theorems for Certain Tree-Like Continua, Dissertation, Unversity of Houston (1981).
  • [7] M. M. Marsh, A fixed point theorem for inverse limits of fans, Proc. Amer. Math. Soc, 91 (1984), 139-142.
  • [8] S. B. Nadler, Jr., Universal mappings and weakly confluent mappings, CX (1980), # 3 , 221-235.