Smooth quasiregular mappings with branching

Mario Bonk; Juha Heinonen

Publications Mathématiques de l'IHÉS (2004)

  • Volume: 100, page 153-170
  • ISSN: 0073-8301

Abstract

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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 inn-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.

How to cite

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Bonk, Mario, and Heinonen, Juha. "Smooth quasiregular mappings with branching." Publications Mathématiques de l'IHÉS 100 (2004): 153-170. <http://eudml.org/doc/104198>.

@article{Bonk2004,
abstract = {We give an example of a $\mathcal \{C\}^\{3-\epsilon \}$-smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping inn-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.},
author = {Bonk, Mario, Heinonen, Juha},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {differentiable quasiregular mappings; branch set},
language = {eng},
pages = {153-170},
publisher = {Springer},
title = {Smooth quasiregular mappings with branching},
url = {http://eudml.org/doc/104198},
volume = {100},
year = {2004},
}

TY - JOUR
AU - Bonk, Mario
AU - Heinonen, Juha
TI - Smooth quasiregular mappings with branching
JO - Publications Mathématiques de l'IHÉS
PY - 2004
PB - Springer
VL - 100
SP - 153
EP - 170
AB - We give an example of a $\mathcal {C}^{3-\epsilon }$-smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping inn-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.
LA - eng
KW - differentiable quasiregular mappings; branch set
UR - http://eudml.org/doc/104198
ER -

References

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  1. 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. Zbl0548.30016MR731786
  2. 2. G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann., 315 (1999), 641–710. Zbl0944.53004MR1731465
  3. 3. S. K. Donaldson and D. Sullivan, Quasiconformal 4-manifolds, Acta Math., 163 (1989), 181–252. Zbl0704.57008MR1032074
  4. 4. H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York (1969). Zbl0176.00801MR257325
  5. 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). Zbl1045.30011MR1859913
  6. 6. R. Kaufman, J. T. Tyson, and J.-M. Wu, Smooth quasiregular maps with branching in R 4, Preprint (2004). 
  7. 7. M. Kiikka, Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms, Ann. Acad. Sci. Fenn., Ser. A I, Math., 8 (1983), 251–256. Zbl0565.30015MR731785
  8. 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. Zbl0265.30027MR352453
  9. 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. Zbl0923.30015MR1678028
  10. 10. P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, Cambridge (1995). Zbl0819.28004MR1333890
  11. 11. E. E. Moise, Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics, Vol. 47, Springer, New York (1977). Zbl0349.57001MR488059
  12. 12. J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. Math. (2), 72 (1960), 521–554. Zbl0108.18101MR121804
  13. 13. Yu. G. Reshetnyak, Space mappings with bounded distortion, Sibirsk. Mat. Z., 8 (1967), 629–659. Zbl0167.06601MR994644
  14. 14. Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, Vol. 73, American Mathematical Society, Providence, RI (1989). Zbl0667.30018MR994644
  15. 15. S. Rickman, Quasiregular Mappings, Springer, Berlin (1993). Zbl0816.30017MR1238941
  16. 16. S. Rickman and U. Srebro, Remarks on the local index of quasiregular mappings, J. Anal. Math., 46 (1986), 246–250. Zbl0603.30025MR861703
  17. 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. Zbl0326.30020MR396945
  18. 18. D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), 543–555, Academic Press, New York, 1979. Zbl0478.57007MR537749
  19. 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. Zbl0427.30019MR562672
  20. 20. J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer, Berlin (1971). Zbl0221.30031MR454009

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