Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T09:17:51.847Z Has data issue: false hasContentIssue false
  • English
  • Français

Holomorphic families of quasi-Fuchsian groups

Published online by Cambridge University Press:  19 September 2008

K. Astala  and
M. Zinsmeister
K. Astala
Affiliation:
University of Helsinki, Hallituskatu 15, Helsinki, Finland
M. Zinsmeister
Affiliation:
The University of Bordeaux, Bordeaux, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We produce a holomorphic family of infinitely generated quasi-Fuchsian groups such that the Hausdorff dimension of the limit set L (Гλ) is identical to 1 for small λ, but strictly greater than 1 for λ ˜ 1. In particular, this shows that Hausdorff dimension does not depend real analytically on the parameter λ, contrary to the case of finitely generated groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 207 - 212
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AZ1]Astala, K. & Zinsmeister, M.. Fuchsian groups and the Mostow rigidity property. CR Acad. Sci. Paris 311 (1990), 301–306.Google Scholar
[AZ2]Astala, K. & Zinsmeister, M.. Teichmüller spaces and BMOA. Math. Annalen 289 (1991), 613–625.CrossRefGoogle Scholar
[Be]Bers, L.. Finite dimensional Teichmüller spaces and generalizations. Bull. Amer. Math. Soc. 5 (1981), 131–172.CrossRefGoogle Scholar
[Bo]Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. IHES 50 (1979), 11–25.CrossRefGoogle Scholar
[F]Fernandez, J.. Domains with strong barrier. Revista Math. Iber. 5 (1989), 47–65.CrossRefGoogle Scholar
[G]Garnett, J.. Bounded Analytic Functions. Academic: New York 1981.Google Scholar
[J]Jørgensen, T.. On discrete groups of Möbius transformations. Amer. J. Math. 92 (1976), 739–749.CrossRefGoogle Scholar
[RR]Rubel, L. A. & Ryff, J. V.. The bounded weak-star topology and the bounded analytic functions. J. Fund. Anal. 5 (1970), 167–183.CrossRefGoogle Scholar
[R]Ruelle, D.. Repellors for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
[Se]Semmes, S.. Quasiconformal mappings and chord-arc curves. Trans. Amer. Math. Soc. 306 (1988), 233–263.CrossRefGoogle Scholar
[Su1]Sullivan, D.. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6 (1982), 57–73.CrossRefGoogle Scholar
[Su2]Sullivan, D.. Quasiconformal homeomorphisms and dynamics II. Acta Math. 155 (1985), 243–260.CrossRefGoogle Scholar
[Z]Zinsmeister, M.. Domaines de Carleson. Mich. Math. J. 36 (1989), 213–220.CrossRefGoogle Scholar