Inverse limits and a property of J. L. Kelley. II (Q1432031)

This paper is a continuation of the author's earlier paper [Bol. Soc. Mat. Mex., III. Ser. 8, 83-91 (2002; Zbl 1035.54006)]. A permutation on \(n\) elements is a function from \(\{1,2,\ldots,n\}\) onto \(\{1,2,\ldots,n\}\). \(S_n\) denotes the set of all permutations on \(n\) elements. To each permutation \(\sigma \in S_{n}\), a map \(f_{\sigma}\) from \([0,1]\) onto \([0,1]\), determined by \(\sigma\), is assigned. \(f_{\sigma}\) is defined in the following way: for \(i \in \{1,2,\ldots,n\}\) let \(a_{i}=\frac{i-1}{n-1}\); \(f_{\sigma}\) is the linear extension of the map which takes \(a_{i}\) to \(a_{\sigma(i)}\). The set of all maps determined by permutations \(\sigma \in S_{n}\) is denoted by \({\mathcal S}_{n}\). In [Topology Appl. 126, 393--408 (2002; Zbl 1018.54027)] the author described all the continua which occur as an inverse limit of the inverse system \(\{[0,1],f_{\sigma}\}\) using a single bonding map \(f_{\sigma}\) corresponding to a member \(\sigma\) of \(S_{3}\) or \(S_{4}\) or \(S_{5}\), i.e. \(f_{\sigma} \in {\mathcal S}_{3}\cup{\mathcal S}_{4}\cup{\mathcal S}_{5}\). In this paper it is proved that these continua have the property of Kelley. It is also proved that the continuum arising as an inverse limit on \([0,1]\) using the single logistic map \(f_{\lambda}(x)=4\lambda x(1-x)\), where \(\lambda\) is the Feigenbaum limit, has the property of Kelley.