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Fixed point theorems for arc-preserving mappings of uniquely arcwise-connected continua. (English) Zbl 0848.54029

The authors prove that a map of a uniquely arcwise-connected continuum into itself with the property that the image of every arc is an arc or a point, has a fixed point. This result generalizes a theorem of the second author asserting that uniquely arcwise-connected continua have the fixed point property with respect to homeomorphisms [ibid. 52, 451-456 (1975; Zbl 0307.54035)]. The significance of these results is underscored by examples of uniquely arcwise-connected continua without the fixed point property by G. S. Young [ibid. 11, 880-884 (1961; Zbl 0102.37806)], the second author and L. G. Oversteegen [ibid. 95, 476-482 (1985; Zbl 0592.54030)] and by M. Sobolewski [Bull. Pol. Acad. Sci., Math. 34, 307-313 (1986; Zbl 0617.54029)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

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