Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T01:07:26.126Z Has data issue: false hasContentIssue false
  • English
  • Français

On the measurable dynamics of z → ez

Published online by Cambridge University Press:  19 September 2008

Etienne Ghys ,
Lisa R. Goldberg  and
Dennis P. Sullivan
Etienne Ghys
Affiliation:
Université des Sciences de LilleI 59655- Villeneuve d'Asq Cedex, France
Lisa R. Goldberg
Affiliation:
CUNY, 33 W 42nd St., New York, NY 10036, USA
Dennis P. Sullivan
Affiliation:
CUNY and IHES, 35, route de Chartres, 91 440-Bures sur Yvette, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the measure theoretic properties of the complex exponential map E(z) = ez.An particular, we show that the equivalence relation generated by E is recurrent and that E has no quasi-conformal deformations. This enables us to give some information concerning Devaney's semi-conjugacy between E and the shift map on sequences of integers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 329 - 335
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Ahlfors, L. V.. Complex Analysis. McGraw Hill, 1966.Google Scholar
[2]Ahlfors, L. V.. Lectures on Quasiconformal Mappings. D. Van Nostrand Company, Inc., 1966.Google Scholar
[3]Ahlfors, L. V.. Conformal Invariants, Topics in Geometric Function Theory. McGraw Hill, 1973.Google Scholar
[4]Ahlfors, L. & Bers, L.. Riemann's mapping theorem for variable metrics. Annals of Math. 72, (1960), 385–404.CrossRefGoogle Scholar
[5]Devaney, R.. Structural instability of Exp(z). To appear.Google Scholar
[6]Devaney, R. & Krych, . Dynamics of Exp (z). Ergod. Th. Dynam. Syst. 4, (1984), 35–52.CrossRefGoogle Scholar
[7]Fatou, P.. Mémoire sur les equations fonctionnelles. B.S.M.F. 47, (1919), 161–271; 47 (1920), 33–94 and 208–314.Google Scholar
[8]Fatou, P.. Sur l'iteration des fonctions transcendantes entières. Ada Math. 47, (1926), 337–370.Google Scholar
[9]Halmos, P.. Ergodic Theory.Google Scholar
[10]Julia, G.. Itération des applications fonctionelles. J. Math. Pures et Appl., (1918), 47–245.Google Scholar
[11]Lehto, O. & Virtanen, K. I.. Quasiconformal Mappings in the Plane. Springer-Verlag, 1973.CrossRefGoogle Scholar
[12]Misiurewicz, M.. On iterates of e z. Ergod. Th. & Dynam. Syst. 1, (1981), 103–106.CrossRefGoogle Scholar
[13]Spanier, E. H.. Algebraic Topology. McGraw-Hill, 1966.Google Scholar
[14]Sullivan, D.. Quasiconformal homeomorphisms and dynamics, I. To appear.Google Scholar