Comptes Rendus
no. 2
Complex analysis/Topology
The branch set of a quasiregular mapping between metric manifolds
[L'ensemble de branchement d'une application quasi régulière entre variétés métriques]

Présenté par : the Editorial Board

Chang-Yu Guo1 ; Marshall Williams2
1 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
2 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 155-159.

Dans cette note, nous annonçons de nouveaux résultats quant à la porosité dénombrable quantitative de l'ensemble des branchements d'une application quasi régulière dans un cadre très général d'espaces métriques. Comme applications de nos résultats, nous répondons à une conjecture récente de Fässler et al., à un problème ouvert de Heinonen–Rickman et à une question ouverte de Heinonen–Semmes.

In this note, we announce some new results on quantitative countable porosity of the branch set of a quasiregular mapping in very general metric spaces. As applications, we solve a recent conjecture of Fässler et al., an open problem of Heinonen–Rickman, and an open question of Heinonen–Semmes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.022
Chang-Yu Guo 1 ; Marshall Williams 2

1 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
2 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA 
@article{CRMATH_2016__354_2_155_0,
     author = {Chang-Yu Guo and Marshall Williams},
     title = {The branch set of a quasiregular mapping between metric manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {155--159},
     publisher = {Elsevier},
     volume = {354},
     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.022},
     language = {en},
}
TY  - JOUR
AU  - Chang-Yu Guo
AU  - Marshall Williams
TI  - The branch set of a quasiregular mapping between metric manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 155
EP  - 159
VL  - 354
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2015.10.022
LA  - en
ID  - CRMATH_2016__354_2_155_0
ER  - 
%0 Journal Article
%A Chang-Yu Guo
%A Marshall Williams
%T The branch set of a quasiregular mapping between metric manifolds
%J Comptes Rendus. Mathématique
%D 2016
%P 155-159
%V 354
%N 2
%I Elsevier
%R 10.1016/j.crma.2015.10.022
%G en
%F CRMATH_2016__354_2_155_0
Chang-Yu Guo; Marshall Williams. The branch set of a quasiregular mapping between metric manifolds. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 155-159. doi : 10.1016/j.crma.2015.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.022/

[1] M. Bonk; J. Heinonen Smooth quasiregular mappings with branching, Publ. Math. Inst. Hautes Études Sci., Volume 100 (2004), pp. 153-170

[2] K. Fässler; A. Lukyanenko; K. Peltonen Quasiregular mappings on subRiemannian manifolds, J. Geom. Anal. (2015) (forthcoming) | DOI

[3] K. Grove; P. Petersen Bounding homotopy types by geometry, Ann. Math. (2), Volume 128 (1988) no. 1, pp. 195-206

[4] K. Grove; P. Petersen; J.Y. Wu Geometric finiteness theorems via controlled topology, Invent. Math., Volume 99 (1990) no. 1, pp. 205-213

[5] C.Y. Guo Mappings of finite distortion between metric measure spaces, Conform. Geom. Dyn., Volume 19 (2015), pp. 95-121

[6] C.Y. Guo, M. Williams, Porosity of the branch set of discrete open mappings with controlled linear dilatation, preprint, 2015.

[7] C.Y. Guo, S. Nicolussi Golo, M. Williams, Quasiregular mappings between subRimannian manifolds, preprint, 2015.

[8] J. Heinonen The branch set of a quasiregular mapping, Beijing, 2002, Higher Education Press Limited Company, Beijing (2002), pp. 691-700

[9] J. Heinonen; P. Koskela Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Volume 181 (1998), pp. 1-61

[10] J. Heinonen; S. Rickman Quasiregular maps S3→S3 with wild branch sets, Topology, Volume 37 (1998) no. 1, pp. 1-24

[11] J. Heinonen; S. Rickman Geometric branched covers between generalized manifolds, Duke Math. J., Volume 113 (2002) no. 3, pp. 465-529

[12] J. Heinonen; S. Semmes Thirty-three yes or no questions about mappings, measures, and metrics, Conform. Geom. Dyn., Volume 1 (1997), pp. 1-12

[13] J. Heinonen; D. Sullivan On the locally branched Euclidean metric gauge, Duke Math. J., Volume 114 (2002) no. 1, pp. 15-41

[14] O. Martio; S. Rickman; J. Väisälä Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 448 (1969), pp. 1-40

[15] O. Martio; S. Rickman; J. Väisälä Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 488 (1971), pp. 1-31

[16] L.F. McAuley; E.E. Robinson On Newman's theorem for finite-to-one open mappings on manifolds, Proc. Amer. Math. Soc., Volume 87 (1983) no. 3, pp. 561-566

[17] J. Onninen; K. Rajala Quasiregular mappings to generalized manifolds, J. Anal. Math., Volume 109 (2009), pp. 33-79

[18] P. Petersen A finiteness theorem for metric spaces, J. Differ. Geom., Volume 31 (1990) no. 2, pp. 387-395

[19] S. Rickman Quasiregular Mappings, Ergeb. Math. Grenzgeb. (3), vol. 26, Springer, Berlin, 1993

[20] J. Sarvas The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 1 (1975) no. 2, pp. 297-307

[21] S. Semmes Finding curves on general spaces through quantitative topology, with applications for Sobolev and Poincaré inequalities, Sel. Math. New Ser., Volume 2 (1996), pp. 155-295

[22] J. Väisälä Modulus and capacity inequalities for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 509 (1972), pp. 1-14

[23] M. Williams, Definition of quasiregularity in metric measure spaces, preprint, 2015.

[24] M. Williams, Bi-Lipschitz embeddability of BLD branched spaces, preprint, 2015.

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Quasiconformal extension for harmonic mappings on finitely connected domains

Iason Efraimidis

C. R. Math (2021)

Conformal dimension and combinatorial modulus of compact metric spaces

Matias Carrasco Piaggio

C. R. Math (2012)

Some comments on Laakso graphs and sets of differences

Alexandros Margaris; James C. Robinson

C. R. Math (2020)