Geodesic
Fixed Points can Characterise Curves
Canadian mathematical bulletin, Tome 21 (1978) no. 2, pp. 207-211

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The concept of a firm fixed point of a selfmap of a metric space is introduced. Loosely speaking a fixed point is firm if it cannot be moved to a point nearby with the help of a map which is arbitrarily close to the given map. It is shown that a continuum always admits a selfmap with a firm fixed point if the continuum contains a triod and if the vertex of the triod has a neighbourhood which is a dendrite. This condition holds in particular for local dendrites. Hence a local dendrite is an arc or a simple closed curve if and only if it does not admit a selfmap which has a firm fixed point. 
Schirmer, Helga. Fixed Points can Characterise Curves. Canadian mathematical bulletin, Tome 21 (1978) no. 2, pp. 207-211. doi: 10.4153/CMB-1978-035-0
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     title = {Fixed {Points} can {Characterise} {Curves}},
     journal = {Canadian mathematical bulletin},
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     year = {1978},
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     number = {2},
     doi = {10.4153/CMB-1978-035-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1978-035-0/}
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