Inducing fixed points in the Stone-Čech compactification (Q1916445)

For a continuous map \(f\) of a Tikhonov space \(X\) into itself \(\beta f\) denotes the unique extension of \(f\) over the Čech-Stone compactification of \(X\). It is known that if \(X\) is paracompact and \(f\) is a fixed-point-free autohomeomorphism then \(\beta f\) is also fixed-point-free; see [\textit{E. K. van Douwen}, ibid. 51, 191-195 (1993; Zbl 0792.54037)] and [the reviewer with \textit{D. Y. Kim}, Commentat. Math. Univ. Carol. 29, 657-663 (1988; Zbl 0687.54011)]. It is also known that if one removes one point from the Cantor cube of weight \(\omega_1\) then the resulting space carries an autohomeomorphism without fixed points but its Čech-Stone extension has exactly one fixed point. In the paper under review a normal space is constructed with the same property. The main result of the paper says that for every number \(n > 1\) there exists a first countable strongly zero-dimensional subparacompact space \(X\) together with an autohomeomorphism \(f\) such that every orbit of \(f\) is of size \(n\) but \(\beta f\) has a fixed point.