×

Smooth quasiregular maps with branching in \(\mathbf R^n\). (English) Zbl 1078.30015

A continuous mapping \(f: D \to {\mathbb R}^n,\;D \subset {\mathbb R}^n\) a domain, is \(K\)-quasiregular if \(f \in W^{1,n}_{\text{loc}}(D)\) and \(| f'(x)| \leq KJ(x,f)\) a.e. It was shown in [O. Martio, S. Rickman and J. Väisälä, Ann. Acad. Sci. Fenn., Ser. A I 488, 1–31 (1971; Zbl 0223.30018)] that if \(n \geq 3\) and \(f \in C^{n/(n-2)}\), then a quasiregular mapping is a local homeomorphism or constant. For \(n = 3\) the \(C^{n/(n-2)}\)-condition was shown to be sharp in [M. Bonk and J. Heinonen, Publ. Math. Inst. Hautes Étud. Sci. 100, 153–170 (2004; Zbl 1063.30021)]. For \(n \geq 5\) the authors show that there exists \(\varepsilon = \varepsilon (n) > 0\) and a \(C^{1+\varepsilon}\)-smooth quasiregular mapping \(f: {\mathbb R}^n \to {\mathbb R}^n\) whose branch set is homeomorphic to \({\mathbb R}^{n-2}\). The construction is complicated and uses the affine approximation properties of quasisymmetric mappings on compact sets [see J. Väisälä, Ann. Acad. Sci. Fenn. Ser. A I, Math. 11, 239–244 (1986; Zbl 0607.30019)].

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math., 96 (1956), 125-142. · Zbl 0072.29602
[2] C. J. Bishop, A quasisymmetric surface with no rectifiable curves, Proc. Amer. Math. Soc., 127 (1999), 2035-2040. · Zbl 0920.30016
[3] M. Bonk and J. Heinonen, Smooth quasiregular mappings with branching, Publ. Math., Inst. Hautes Études Sci., 100 (2004), 153-170. · Zbl 1063.30021
[4] A. V. Černavskii, Finite-to-one open mappings of manifolds, Mat. Sb. (N.S.), 65 (1964), 357-369. · Zbl 0129.15003
[5] A. V. Černavskii, Addendum to the paper “Finite-to-one open mappings of manifolds”, Mat. Sb. (N.S.), 66 (1965), 471-472.
[6] P. T. Church, Differentiable open maps on manifolds, Trans. Amer. Math. Soc., 109 (1963), 87-100. · Zbl 0202.54801
[7] G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann., 315 (1999), 641-710. · Zbl 0944.53004
[8] S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math., 163 (1989), 181-252. · Zbl 0704.57008
[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 0199.39701
[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 1005.30019
[11] J. Heinonen and S. Rickman, Quasiregular maps S3→S3 with wild branch sets, Topology, 37 (1998), 1-24. · Zbl 0895.30016
[12] J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, Duke Math. J., 113 (2002), 465-529. · Zbl 1017.30023
[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).
[14] M. Kiikka, Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms, Ann. Acad. Sci. Fenn., Ser. A I, Math., 8 (1983), 251-256. · Zbl 0565.30015
[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 0265.30027
[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 0223.30018
[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 0819.28004
[18] Y. G. Reshetnyak, Space mappings with bounded distortion, Sibirsk. Mat. Z., 8 (1967), 629-659.
[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 0667.30018
[20] S. Rickman, Quasiregular Mappings, Springer, Berlin (1993).
[21] S. Rickman, Construction of quasiregular mappings, in Quasiconformal mappings and analysis (Ann Arbor, MI 1995), Springer, New York (1998), pp. 337-345. · Zbl 0888.30018
[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 0326.30020
[23] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. (1970). · Zbl 0207.13501
[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 0478.57007
[25] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math., 5 (1980), 97-114. · Zbl 0403.54005
[26] P. Tukia and J. Väisälä, Quasiconformal extension from dimension n to n+1, Ann. of Math. (2),115 (1982), 331-348. · Zbl 0484.30017
[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 0533.30020
[28] J. T. Tyson and J.-M. Wu, Quasiconformal dimensions of self-similar fractals, Rev. Mat. Iberoamer., accepted for publication.
[29] J. Väisälä, Lectures onn-dimensional quasiconformal mappings, no. 229 in Lecture Notes in Mathematics, Springer, Berlin (1971). · Zbl 0221.30031
[30] J. Väisälä, A survey of quasiregular maps in Rn, in Proceedings of the International Congress of Mathematicians (Helsinki 1978), Acad. Sci. Fennica, Helsinki (1980), pp. 685-691.
[31] J. Väisälä, Bi-Lipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fenn., Ser. A I, Math., 11 (1986), 239-274. · Zbl 0607.30019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.