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The Brouwer fixed-point theorem implies that any continuous automorphism mapping the closed disk to itself must have a fixed point.

Does anyone know of a path-connected (and otherwise “well-behaved”) topological space such that any continuous automorphism mapping it to itself must have at least two fixed points?

I add the “path-connected” constraint because without it, it is very easy to construct a space with this property. Simply take the union of a point and a finite line segment (that does not contain the point) in 2-space, and this object satisfies the desired property because the point must be mapped to itself (making one fixed point) and the closed interval must be mapped to itself and also contain one fixed point (by Brouwer).

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    $\begingroup$ Constant functions always have one fixed point - so there are no such spaces. $\endgroup$ Oct 18 '19 at 21:58
  • $\begingroup$ What is an automorphism of a topological space? $\endgroup$
    – bof
    Oct 18 '19 at 23:25
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If your topological space is a graph (as in edges connecting vertices), then any continuous automorphism must send vertices of degree $3$ or more to vertices of the same degree. Therefore you can construct spaces for which any automorphism has at least $N$ fixed points, by just ensuring that the graph has enough vertices of distinct degrees.

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