Geodesic
The fixed-point property for deformations of tree-like continua
Fundamenta Mathematicae, Tome 155 (1998) no. 2, pp. 161-176

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Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.
DOI : 10.4064/fm-155-2-161-176
Keywords: fixed point, arc-component, deformation, tree-like continuum, Borsuk ray, dog-chases-rabbit argument

Charles L. Hagopian 1

1 
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Charles L. Hagopian. The fixed-point property for deformations of tree-like continua. Fundamenta Mathematicae, Tome 155 (1998) no. 2, pp. 161-176. doi: 10.4064/fm-155-2-161-176

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