Houston Problem Book Problem 33

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Problem 33 (HPB): Suppose $P$ is a subset of the positive integers with the property that, if $k \in P$, the $n \cdot k$ is in $P$ for each positive integer $n$. Does there exist a locally connected continuum $X$ and a mapping $f$ of $X$ onto $X$ such that $f^n$ has a fixed point if and only if $n \in P$? (Asked by W. Kuperberg, 11 October 1972)

Answer: Yes (M. COok, 20 February 1989)