Periodic quasiregular mappings of finite order. (English) Zbl 1052.30024

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 \[ \rho=\limsup_{r\to\infty}\frac{\log\log (\max\{| f(x)| :| x| \leq r\})}{\log r}. \] O. Martio and 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.


30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30D99 Entire and meromorphic functions of one complex variable, and related topics
30C85 Capacity and harmonic measure in the complex plane


Zbl 0317.30025
Full Text: DOI EuDML


[1] Hayman, W. K.: Subharmonic Functions, vol. 2. London Mathematical Society Monographs 20. Academic Press, London, 1989. · Zbl 0699.31001
[2] Martio, O., Rickman, R. and Väisälä, J.: Topological and metric properties of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I 488 (1971), 31 pp.[3] Martio, O. and Srebro, U.: Periodic Quasimeromorphic Mappings in Rn. J. Analyse Math. 28 (1975), 20-40. · Zbl 0317.30025
[3] Rickman, S.: Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 26. Springer-Verlag, Berlin, 1993. · Zbl 0816.30017
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