Smooth quasiregular mappings with branching
Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 153-170.

We give an example of a 𝒞 3-ϵ -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in n-space has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.

@article{PMIHES_2004__100__153_0,
     author = {Bonk, Mario and Heinonen, Juha},
     title = {Smooth quasiregular mappings with branching},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {153--170},
     publisher = {Springer},
     volume = {100},
     year = {2004},
     doi = {10.1007/s10240-004-0024-8},
     mrnumber = {2102699},
     zbl = {1063.30021},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-004-0024-8/}
}
TY  - JOUR
AU  - Bonk, Mario
AU  - Heinonen, Juha
TI  - Smooth quasiregular mappings with branching
JO  - Publications Mathématiques de l'IHÉS
PY  - 2004
SP  - 153
EP  - 170
VL  - 100
PB  - Springer
UR  - http://www.numdam.org/articles/10.1007/s10240-004-0024-8/
DO  - 10.1007/s10240-004-0024-8
LA  - en
ID  - PMIHES_2004__100__153_0
ER  - 
%0 Journal Article
%A Bonk, Mario
%A Heinonen, Juha
%T Smooth quasiregular mappings with branching
%J Publications Mathématiques de l'IHÉS
%D 2004
%P 153-170
%V 100
%I Springer
%U http://www.numdam.org/articles/10.1007/s10240-004-0024-8/
%R 10.1007/s10240-004-0024-8
%G en
%F PMIHES_2004__100__153_0
Bonk, Mario; Heinonen, Juha. Smooth quasiregular mappings with branching. Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 153-170. doi : 10.1007/s10240-004-0024-8. http://www.numdam.org/articles/10.1007/s10240-004-0024-8/

1. B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in R n , Ann. Acad. Sci. Fenn., Ser. A I, Math., 8 (1983), 257-324. | MR | Zbl

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

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

4. H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York (1969). | MR | Zbl

5. 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

6. R. Kaufman, J. T. Tyson, and J.-M. Wu, Smooth quasiregular maps with branching in R 4, Preprint (2004).

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

8. 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 pp. | MR | Zbl

9. O. Martio, U. Srebro, and J. Väisälä, Normal families, multiplicity and the branch set of quasiregular maps, Ann. Acad. Sci. Fenn., Ser. A I, Math., 24 (1999), 231-252. | MR | Zbl

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

11. E. E. Moise, Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics, Vol. 47, Springer, New York (1977). | MR | Zbl

12. J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. Math. (2), 72 (1960), 521-554. | MR | Zbl

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

14. Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, Vol. 73, American Mathematical Society, Providence, RI (1989). | MR | Zbl

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

16. S. Rickman and U. Srebro, Remarks on the local index of quasiregular mappings, J. Anal. Math., 46 (1986), 246-250. | MR | Zbl

17. 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

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

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

20. J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer, Berlin (1971). | MR

Cité par Sources :