Invariant sets and inverse limits
A map from a topological space to another is called monotone if each point-inverse is a continuum. A map \(f\) of \([0,1]\) onto itself is called Markov if \(f\) has a finite invariant set \(A\) containing both 0 and 1 and such that if \(x\) and \(y\) are consecutive members of \(A\) then \(f\mid [x,y]\) is monotone. If \(\sigma\in S_n\) is a permutation let \(f_\sigma: [0,1]\to [0,1]\) be defined as follows: For each \(i\), \(0\leq i\leq 1\) let \(a_i=(i-1)/n-1\), \(f_\sigma(a_i)= a_{\sigma(i)}\) and then \(f_\sigma\) is obtained by linear extension. Among other interesting results obtained, the author shows that if \(\sigma\) is a \(p\)-cycle where \(p\) is a prime, then \(\varprojlim \{[0,1], f_\sigma\}\) is an indecomposable continuum.
scientific article; zbMATH DE number 1900959