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. http://www.numdam.org/articles/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. | MR | Zbl

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

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

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

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. | MR | Zbl

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

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

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. | MR | Zbl

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. | MR | Zbl

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). | MR | Zbl

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

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. | MR | Zbl

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. | MR | Zbl

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

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

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. | MR | Zbl

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

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

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. | MR | Zbl

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

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. | MR | Zbl

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

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

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. | MR | Zbl

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

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. | MR | Zbl

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

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