Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T00:27:22.866Z Has data issue: false hasContentIssue false
  • English
  • Français

Random iterations of rational functions

Published online by Cambridge University Press:  19 September 2008

John Erik Fornaess  and
Nessim Sibony
John Erik Fornaess
Affiliation:
Princeton University, Princeton NJ 08540, USA
Nessim Sibony
Affiliation:
Université de Paris-Sud, Mathématique Bâtiment 425, 91405 Orsay Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the asymptotic behavior of iterates of rational functions with small perturbations. In presence of attractive cycles we show that almost surely, in the parameter space, the iterates converge to a given neighborhood of the attractive cycles. When there is no attractive cycle, we prove an ergodic theorem with respect to Lebesgue measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 4 , December 1991 , pp. 687 - 708
Copyright
Copyright © Cambridge University Press 1991

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

[BC]Benedicks, M. & Carleson, L.. On iterations of 1−ax2 on (−1, 1). Ann. Math. 122 (1985), 1–25.CrossRefGoogle Scholar
[BI]Billingsley, P.. Probability and Measure. Wiley, New York, 1986.Google Scholar
[BR]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Math. 6 (1966), 103–144.CrossRefGoogle Scholar
[GU1]Guckenheimer, J.. Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70 (1979), 133–160.CrossRefGoogle Scholar
[GU2]Guckenheimer, J.. Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer Verlag, Berlin, 1986.Google Scholar
[Hö]Hörmander, L.. The analysis of linear partial differential operators I. Springer Verlag, Berlin, New York, 1983.Google Scholar
[KI]Kifer, Y.. Ergodic theory of random transformations. Progress in Probability and Statistics, vol. 10. Birkhauser, 1986.Google Scholar
[MSS]Mañé, R., Sad, P. & Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Ec. Norm. Sup. 16 (1983), 193–217.CrossRefGoogle Scholar
[Mil]Milnor, J.. Dynamics in one complex variable: introductory lectures. SUNY Stonybrook 1990, Preprint.Google Scholar
[MI]Misuriewicz, M.. Structure of mappings of an interval with zero entropy. Publ. Math. IHES 53, (1981) 5–16.CrossRefGoogle Scholar
[OH]Ohno, T.. Asymptotic behaviour of dynamical systems with random parameters. Publ. R.I.M.S. Kyoto Univ. 19 (1983), 83–98.Google Scholar
[RE]Rees, M.. Positive measure sets of ergodic rational maps. Ann. Sci. Ec. Norm. Sup. 19 (1986), 383–407.CrossRefGoogle Scholar
[RU1]Ruelle, D.. Small random perturbations of dynamical systems and the definition of attractors. Commun. Math. Phys. 82 (1981), 137–151.CrossRefGoogle Scholar
[RU2]Ruelle, D.. Elements of Differentiate Dynamics and Bifurcation Theory. Academic Press, New York, 1989.Google Scholar
[SI]Singer, S.. Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35 (1978), 260–267.CrossRefGoogle Scholar
[SU1]Sullivan, D.. Itération des fonctions analytiques complexes. C. R. Acad. Sci. Paris 294 (1982), 301–303.Google Scholar
[SU2]Sullivan, D.. Quasiconformal homeomorphisms and dynamics I. Ann. Math. 122 (1985), 401–418.CrossRefGoogle Scholar