Periodic quasiregular mappings of finite order (Q1884102)

Let \(f\) be a sense-preserving, quasiregular map on \(\mathbb R^m\), \(m\geq 2\) which is also entire. The order of \(f\) is defined by \[ \rho=\limsup_{r\to\infty}\frac{\log\log (\max\{| f(x)| :| x| \leq r\})}{\log r}. \] \textit{O. Martio} and \textit{U. Srebro} [J. Anal. Math. 28, 20--40 (1975; Zbl 0317.30025)] proved that there exist \((m-1)\)-periodic maps \(f\) of order \(1\) and \(\infty\) and that \(1\) is a lower bound for the orders of such functions. In the present article the authors prove that for any given \(\rho>1\), there exists an \((m-1)\)-periodic \(K(m)\)-quasiregular map \(g\) of order \(\rho\), thus answering a question posed by Martio and Srebro in 1975. The construction of \(g\) (which is long and technical) shows that an equation \(g(x)=a\) has infinite number of solutions in each fundamental region of the function \(Z\) (Zorich's analogue of the exponential function). This fact answers negatively another question of Martio and Srebro.