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Uniformly quasiregular mappings of Lattès type. (English) Zbl 0897.30008
Summary: Using an analogy of the Lattès’ construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the \(n\)-sphere \(\overline{\mathbb{R}}^n \) whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is \({\mathbb{S}}^{n-1}\) and its complement in \({\mathbb{S}}^{n}\) consists of two superattracting basins. In the chaotic case we study the invariant conformal structures and show that Lattès type rational mappings are either rigid or form a 1-parameter family of quasiconformal deformations.

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
37F99 Dynamical systems over complex numbers
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[1] Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263 – 297. · Zbl 0806.30027
[2] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[3] A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 205 – 222. · Zbl 0856.30017
[4] Tadeusz Iwaniec and Gaven Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241 – 254. · Zbl 0860.30019
[5] O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 488 (1971), 31. · Zbl 0223.30018
[6] O. Martio and U. Srebro, Periodic quasimeromorphic mappings in \({\mathbb R}^n\), J. d’Analyse Math. 28 (1975), 20-40. · Zbl 0317.30025
[7] Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. · Zbl 0816.30017
[8] Jukka Sarvas, The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 297 – 307. · Zbl 0326.30020
[9] Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318 – 346. · Zbl 0603.30026
[10] P. Tukia and J. Väisälä, Quasiconformal extension from dimension \? to \?+1, Ann. of Math. (2) 115 (1982), no. 2, 331 – 348. · Zbl 0484.30017
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