Local Quasimöbius Mappings on a Circle

V. V. Aseev ; D. G. Kuzin

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 14 (2014) no. 1, pp. 3-18

Résumé

For a family of continuous light mappings of a circle $S$ into itself it is introduced the notion ${\mathcal D}$-normality which signifies that for every graphically convergent sequence its graphical limit looks like $(Z\times S)\cup \Gamma f$, where $Z$ — zero-dimensional compact set (possibly, empty), and $\Gamma f$ is a graph of either constant mapping or continuous light mapping. It is proved that every ${\mathcal D}$-normal and Möbius invariant family of the mappings of circle $S$ into itself consist of local $\omega$-quasimöbius mappings with unified distortion function $\omega$.

Détail

Keywords: quasiconformal mapping, quasisymmetric mappings, light mapping, graphical limit, graphical convergence, normal family of mappings
Mots-clés : quasimöbius mapping, local quasimöbius mapping, Möbius invariant families of mappings.

Bibliographie

[1] L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, $2^\text{nd}$ ed., American Mathematical Society, 2006 | MR | Zbl

[2] Kelingos J. A., “Boundary Correspondence under Quasiconformal Mappings”, Michigan Math. J., 13 (1966), 235–249 | DOI | MR | Zbl

[3] Tukia P., Väisälä J., “Quasisymmertic Embeddings of Metric Spaces”, Ann. Acad. Sci. Fenn., Ser. A1 Math., 5 (1980), 97–114 | DOI | MR | Zbl

[4] Heinonen J., Lectures on Analysis on Metric Spaces, Springer, N.Y., 2001 | MR | Zbl

[5] V. V. Aseev, Quasisymmertic embeddings and maps, restrictedly distorting modulus, Dep. in VINITI, 06.11.84 No 3209-84, Novosibirsk, 1984

[6] Väisälä J., “Quasimöbius Maps”, J. Analyse Math., 44 (1984/1985), 218–234 | DOI | MR

[7] P. P. Belinskiy, General Properties of Quasiconformal Mappings, Nauka, Novosibirsk, 1974 (in Russian)

[8] Gehring F. W., “The Caratheodory Convergence Theorem for Quasiconformal Mappings in Space”, Ann. Acad. Sci. Fenn. Ser. A1, 336:11 (1963), 1–21 | MR

[9] Väisälä J., “Invariants for Quasisymmetric, Quasimöbius and Bilipschitz Maps”, J. Anal. Math., 50 (1988), 201–233 | DOI | MR

[10] V. V. Aseev, “About convergence and stability of mappings, partially distorting modules”, Sib. Mat. Zh., 25:1 (1984), 19–29 (in Russian) | MR

[11] V. V. Aseev, “Normal families of topological embeddings”, Dinamika sploshnoy sredy, 76, 1986, 32–42 (in Russian) | Zbl

[12] Del Prete I., Di Iorio M., Hola L., “Graph Convergence of Set Valued Maps and its Relationship to Other Convergences”, J. Appl. Anal., 6:2 (2000), 213–226 | MR | Zbl

[13] Aseev V. V., “Quasi-Symmetric Embeddings”, J. Math. Sci. (New York), 108:3 (2002), 375–410 | DOI | MR | Zbl

[14] S. Stoilov, Teoriya Funktsiy Kompleksnogo Peremennogo, v. 2, Inostrannaya Literatura, M., 1962 (in Russian)

[15] R. Engelking, General Topology, Sigma Series in Pure Mathematics, Heldermann Verlag, 1989 | MR | Zbl

[16] K. Kuratovsky, Topology, v. 2, Academic Press, 1966 | MR

[17] N. Bourbaki, Topologie Générale, v. 3, General Topology. Topological groups. Numbers and Related Groups and Spaces, Hermann, 1971 | MR

[18] Lehto O., Virtanen K., Quasikonforme Abbildungen, Springer-Verlag, 1965 | MR | Zbl