Iterative roots of multifunctions (Q6614522)

Given a map \(F:X\to X\) on a nonempty set \(X\) and an integer \(n\ge 1\), a map \(G:X\to X\) such that \(G^n=F\) on \(X\), where \(G^n\) is the \(n\)-th iterate of \(G\) defined by \(G^n=G \circ G^{n-1}\) and \(G^0=\mathrm{id}\), is said to be an \(n\)-th order iterative root of \(F\). The problem of the existence of iterative roots of a function has been investigated by many researchers in various settings. In this long paper, the authors study the problem when \(F\) is a multifunction, that is \(F\) is a function from \(X\) to the power set \(2^X\). The results are too complex to be presented here in detail, however several theorems are proved showing that under some rather complex conditions on the function \(F\), it has not iterative roots. Various examples illustrate the results of the paper.