On the existence of fundamental representatives of cyclic permutations in maps of an interval (Q1332607)

Let \(\sigma\) be a permutation of the set \(\{1,2, \dots, n\}\). We say that \(\sigma\) has a representative if there is a continuous map \(f\) of a compact interval \(I\) into itself, together with points \(p_ 1 < \cdots < p_ n\) all in \(I\) so that \(f(p_ i) = p_{\sigma(i)}\). This leads to the notion of forcing for permutations. \(\sigma\) forces \(\pi\) if for every representation of \(\sigma\) as shown above, there is a representative of \(\pi\), with the same \(f\). From literature the problem of forcing can be effectively decided if it is known that \(\sigma\) has a representative with special properties, the so-called fundamental representative. The contribution of the paper consists in giving a condition under which from the existence of some representative one can infer that a fundamental representative also exists.