Iterative roots with circuits for piecewise continuous and globally periodic maps (Q428807)

Let \(f\) be a self-map of a subset \(D\) of the reals \(\mathbb{R}\). Such a map is called globally periodic with the prime period \(m\) if \(f^m =\text{id}\) and \(m\) is the smallest positive integer with this property. We say that \(f\) has a \(k\)-circuit for an integer \(k\geq 2\) if there is a sequence \((I_i)_{i=1}^{k}\) of pairwise disjoint intervals such that \(\bigcup _{i=1}^{k}I_i\subset D\), the restriction of \(f\) to each \(I_i\) is continuous, \(f(I_i)=I_{i+1}\) for \(i\in \{1,\ldots , k-1\}\) and \(f(I_k)=I_1\). Moreover, if \(D\) is an interval, then we say that \(f\) has a 1-circuit.NEWLINENEWLINEIn this paper, the general construction of piecewise continuous iterative roots (i.e., solutions \(g\) of the equation \(g^n=f\)) having circuits of piecewise continuous and globally periodic maps is presented. Moreover, two examples of such iterative roots for \(f:{\mathbb{R}} \setminus \{-1,2\} \to {\mathbb{R}} \setminus \{-1,2\}\) of the form \(f(x)=\frac{2x-7}{x+1}\) as well as some sufficient and necessary conditions for global periodicity of Möbius transformation (both in real and complex cases) are given.