Smooth quasiregular maps with branching in 𝐑 n
Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 209-241

According to a theorem of Martio, Rickman and Väisälä, all nonconstant C n/(n-2) -smooth quasiregular maps in 𝐑 n , n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in 𝐑 3 . We prove that the order of smoothness is sharp in 𝐑 4 . For each n≥5 we construct a C 1+ϵ(n) -smooth quasiregular map in 𝐑 n with nonempty branch set.

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     author = {Kaufman, Robert and Tyson, Jeremy T. and Wu, Jang-Mei},
     title = {Smooth quasiregular maps with branching in $\mathbf {R}^n$},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {209--241},
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Kaufman, Robert; Tyson, Jeremy T.; Wu, Jang-Mei. Smooth quasiregular maps with branching in $\mathbf {R}^n$. Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 209-241. doi: 10.1007/s10240-005-0031-4

1. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math., 96 (1956), 125-142. | Zbl | MR

2. C. J. Bishop, A quasisymmetric surface with no rectifiable curves, Proc. Amer. Math. Soc., 127 (1999), 2035-2040. | Zbl | MR

3. M. Bonk and J. Heinonen, Smooth quasiregular mappings with branching, Publ. Math., Inst. Hautes Études Sci., 100 (2004), 153-170. | Zbl | MR | Numdam

4. A. V. Černavskii, Finite-to-one open mappings of manifolds, Mat. Sb. (N.S.), 65 (1964), 357-369. | Zbl | MR

5. A. V. Černavskii, Addendum to the paper “Finite-to-one open mappings of manifolds”, Mat. Sb. (N.S.), 66 (1965), 471-472. | Zbl

6. P. T. Church, Differentiable open maps on manifolds, Trans. Amer. Math. Soc., 109 (1963), 87-100. | Zbl | MR

7. G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann., 315 (1999), 641-710. | Zbl | MR

8. S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math., 163 (1989), 181-252. | Zbl | MR

9. W. H. J. Fuchs, Théorie de l'approximation des fonctions d'une variable complexe, Séminaire de Mathématiques Supérieures, no. 26 (Été 1967). Les Presses de l'Université de Montréal, Montréal, Que. (1968). | Zbl

10. J. Heinonen, The branch set of a quasiregular mapping, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing (2002), pp. 691-700. | Zbl | MR

11. J. Heinonen and S. Rickman, Quasiregular maps S 3→S 3 with wild branch sets, Topology, 37 (1998), 1-24. | Zbl

12. J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, Duke Math. J., 113 (2002), 465-529. | Zbl | MR

13. T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2001). | Zbl | MR

14. M. Kiikka, Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms, Ann. Acad. Sci. Fenn., Ser. A I, Math., 8 (1983), 251-256. | Zbl | MR

15. O. Martio and S. Rickman, Measure properties of the branch set and its image of quasiregular mappings, Ann. Acad. Sci. Fenn., Ser. A I, 541 (1973), 16. | Zbl | MR

16. 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 | MR

17. P. Mattila, Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1995). | Zbl | MR

18. Y. G. Reshetnyak, Space mappings with bounded distortion, Sibirsk. Mat. Z., 8 (1967), 629-659. | Zbl | MR

19. Y. G. Reshetnyak, Space mappings with bounded distortion, vol. 73 of Translations of Mathematical Monographs, American Mathematical Society, Providence (1989). Translated from the Russian by H. H. McFadden. | Zbl | MR

20. S. Rickman, Quasiregular Mappings, Springer, Berlin (1993). | Zbl | MR

21. S. Rickman, Construction of quasiregular mappings, in Quasiconformal mappings and analysis (Ann Arbor, MI 1995), Springer, New York (1998), pp. 337-345. | Zbl | MR

22. J. Sarvas, The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn., Ser. A I, Math., 1 (1975), 297-307. | Zbl | MR

23. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. (1970). | Zbl | MR

24. D. Sullivan, Hyperbolic geometry and homeomorphisms, in Geometric topology (Proc. Georgia Topology Conf., Athens, Ga. 1977), Academic Press, New York (1979), pp. 543-555. | Zbl | MR

25. P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math., 5 (1980), 97-114. | Zbl | MR

26. P. Tukia and J. Väisälä, Quasiconformal extension from dimension n to n+1, Ann. of Math. (2), 115 (1982), 331-348. | Zbl | MR

27. P. Tukia and J. Väisälä, Extension of embeddings close to isometries or similarities, Ann. Acad. Sci. Fenn., Ser. A I, Math., 9 (1984), 153-175. | Zbl | MR

28. J. T. Tyson and J.-M. Wu, Quasiconformal dimensions of self-similar fractals, Rev. Mat. Iberoamer., accepted for publication. | Zbl | MR

29. J. Väisälä, Lectures on n -dimensional quasiconformal mappings, no. 229 in Lecture Notes in Mathematics, Springer, Berlin (1971). | MR

30. J. Väisälä, A survey of quasiregular maps in R n , in Proceedings of the International Congress of Mathematicians (Helsinki 1978), Acad. Sci. Fennica, Helsinki (1980), pp. 685-691. | Zbl | MR

31. J. Väisälä, Bi-Lipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fenn., Ser. A I, Math., 11 (1986), 239-274. | Zbl | MR

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