Geodesic
Existence of Fixed Points for Mappings of Finite Sets
Matematičeskie zametki, Tome 82 (2007) no. 2, pp. 201-206.

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We show that the existence theorem for zeros of a vector field (fixed points of a mapping) holds in the case of a “convex” finite set $X$ and a “continuous” vector field (a self-mapping) directed inwards into the convex hull $\operatorname{co}X$ of $X$. The main goal is to give correct definitions of the notions of “continuity” and “convexity”. We formalize both these notions using a reflexive and symmetric binary relation on $X$, i.e., using a proximity relation. Continuity (we shall say smoothness) is formulated with respect to any proximity relation, and an additional requirement on the proximity (we shall call it the acyclicity condition) transforms $X$ into a “convex” set. If these two requirements are satisfied, then the vector field has a zero (i.e., a fixed point).
Keywords: Brouwer fixed-point theorem, self-mapping, vector field on a finite set, convexity binary relation, proximity relation, acyclic set. 
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     title = {Existence of {Fixed} {Points} for {Mappings} of {Finite} {Sets}},
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V. I. Danilov; G. A. Koshevoy. Existence of Fixed Points for Mappings of Finite Sets. Matematičeskie zametki, Tome 82 (2007) no. 2, pp. 201-206. http://geodesic.mathdoc.fr/item/MZM_2007_82_2_a4/

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