Inverse limits which are the pseudoarc (Q2725453)

Let \(C_s(I,I)\) denote the space of surjective continuous maps of the compact interval \(I\). For a function \(f\) in this class, let \((\widehat{I}, \widehat{f})\) denote the inverse limit with respect to the countable sequence of maps \(f_n=f\) \((n\geq 1)\). It is shown that the set of \((\widehat{I}, \widehat{f})\), which are homeomorphic to the pseudoarc [\textit{B. Knaster}, Fundam. Math. 3, 247-286 (1922; JFM 48.0212.01); \textit{J. A. Kennedy}, Lect. Notes Pure Appl. Math. 170, 103-126 (1995; Zbl 0828.54026)], is nowhere dense in \(C_s(I,I)\). Furthermore, it is shown that \((\widehat{I}, \widehat{f})\) is not homeomorphic as above, if \(f\) has a periodic point, not of odd period.