Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T06:05:51.849Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

Published online by Cambridge University Press:  19 September 2008

Michael Benedicks  and
Lai-Sang Young
Michael Benedicks
Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Lai-Sang Young
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA, and Department of Mathematics, UCLA, Los Angeles, CA 90024, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 1 , March 1992 , pp. 13 - 37
Copyright
Copyright © Cambridge University Press 1992

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

[BC1]Benedicks, M. & Carleson, L.. On iterations of 1 −ax 2 on (−1, 1). Ann. Math. 122 (1985), 1–25.CrossRefGoogle Scholar
[BC2]Benedicks, M. & Carleson, L.. The dynamics of the Hénon map. Preprint (1989), 1–82.Google Scholar
[BL]Blokh, A. M. & Lyubich, M.. Measurable dynamics of S-unimodal maps of the interval. Preprint (1990).Google Scholar
[C]Collet, P.. Preprint Inst. Mittag-Leffler (1984).Google Scholar
[CE]Collet, P. & Eckmann, J.-P.. On the abundance of aperiodic behavior for maps on the interval. Commun. Math. Phys. 73 (1980), 115–160.CrossRefGoogle Scholar
[J1]Jakobson, M.. Topological and metric properties of 1-dimensional endomorphisms. Sov. Math. Dokl. 19 (1978), 1452–1456.Google Scholar
[J2]Jakobson, M.. Absolutely continuous invariant measures for one-parameter families of onedimensional maps. Commun. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[K]Kifer, Yu.. Ergodic Theory of Random Transformations. Birkhäuser, 1986.CrossRefGoogle Scholar
[KK]Katok, A. & Kifer, Yu.. Random perturbations of transformations of an interval. J. D'analyse Math. 47 (1986), 194–237.CrossRefGoogle Scholar
[M]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES 53 (1981), 17–51.CrossRefGoogle Scholar
[N]Nowicki, T.. Symmetric S-unimodal mappings and positive Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 611–616.CrossRefGoogle Scholar
[Re]Rees, M.. Positive measure sets of ergodic rational maps. Ann. Scient. Ec. Norm. Sup 4 e 19 (1986), 383–407.CrossRefGoogle Scholar
[Ry]Rychlik, M.. Another proof of Jakobson's theorem and related results. Ergod. Th. & Dynam.Sys. 8 (1988), 93–109.CrossRefGoogle Scholar
[TTY]Thieullen, P., Tresser, C. & Young, L. S.. Positive exponent for generic 1-parameter families of 1-d maps. In preparation.Google Scholar