Skip to main content Accessibility help
×
Home

Random iterations of rational functions

  • John Erik Fornaess (a1) and Nessim Sibony (a2)

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.

Copyright

References

Hide All
[BC]Benedicks, M. & Carleson, L.. On iterations of 1−ax2 on (−1, 1). Ann. Math. 122 (1985), 125.
[BI]Billingsley, P.. Probability and Measure. Wiley, New York, 1986.
[BR]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Math. 6 (1966), 103144.
[GU1]Guckenheimer, J.. Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70 (1979), 133160.
[GU2]Guckenheimer, J.. Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer Verlag, Berlin, 1986.
[Hö]Hörmander, L.. The analysis of linear partial differential operators I. Springer Verlag, Berlin, New York, 1983.
[KI]Kifer, Y.. Ergodic theory of random transformations. Progress in Probability and Statistics, vol. 10. Birkhauser, 1986.
[MSS]Mañé, R., Sad, P. & Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Ec. Norm. Sup. 16 (1983), 193217.
[Mil]Milnor, J.. Dynamics in one complex variable: introductory lectures. SUNY Stonybrook 1990, Preprint.
[MI]Misuriewicz, M.. Structure of mappings of an interval with zero entropy. Publ. Math. IHES 53, (1981) 516.
[OH]Ohno, T.. Asymptotic behaviour of dynamical systems with random parameters. Publ. R.I.M.S. Kyoto Univ. 19 (1983), 8398.
[RE]Rees, M.. Positive measure sets of ergodic rational maps. Ann. Sci. Ec. Norm. Sup. 19 (1986), 383407.
[RU1]Ruelle, D.. Small random perturbations of dynamical systems and the definition of attractors. Commun. Math. Phys. 82 (1981), 137151.
[RU2]Ruelle, D.. Elements of Differentiate Dynamics and Bifurcation Theory. Academic Press, New York, 1989.
[SI]Singer, S.. Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35 (1978), 260267.
[SU1]Sullivan, D.. Itération des fonctions analytiques complexes. C. R. Acad. Sci. Paris 294 (1982), 301303.
[SU2]Sullivan, D.. Quasiconformal homeomorphisms and dynamics I. Ann. Math. 122 (1985), 401418.

Random iterations of rational functions

  • John Erik Fornaess (a1) and Nessim Sibony (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed