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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.

MSC:

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

Citations:

Zbl 0317.30025
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References:

[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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