Abstract
The main theme of this paper is that almost fixed point properties of discrete structures and fixed point properties of (topological) spaces are interdeducible via a suitable category which contains both graphs and spaces as objects. To carry out the program, we have to consider (almost) fixed points of multifunctions, and for this we need a preliminary discussion of power structures for graphs and simplicial complexes. Specific applications developed are: a “digital convexity” (discrete) version of Kakutani's fixed point theorem for convex-valued multifunctions; and fixed point properties of dendrites in terms of those of finite discrete trees.
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Smyth, M.B., Tsaur, R. AFPP vs FPP. Applied Categorical Structures 11, 95–116 (2003). https://doi.org/10.1023/A:1023017025800
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DOI: https://doi.org/10.1023/A:1023017025800