Periodic points of a map of a system of intervals (Q1111216)

The authors prove the following main result. There exists a function \(g(n)\), \(n\) natural, having the following property. If \(I_ 1,...,I_ n\) are closed intervals in \({\mathbb R}\) and if \(f\) is a continuous map \({\mathbb R}\to {\mathbb R}\) such that \(f(I_ 1\cup...\cup I_ n)\supset I_ 1\cup...\cup I_ n,\) then there exists a periodic point of \(f\) in \(I_ 1\cup...\cup I_ n\) having period \(\leq g(n)\). If \(n\leq 4\) then \(g(n)=n\). The authors also discuss the multidimensional case.