Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T02:41:33.993Z Has data issue: false hasContentIssue false
  • English
  • Français

Another proof of Jakobson's Theorem and related results

Published online by Cambridge University Press:  19 September 2008

Marek Ryszard Rychlik
Marek Ryszard Rychlik
Affiliation:
University of Washington, Department of Mathematics, GN-50, Seattle, Washington 98195USA and Institute of Mathematics, University of Warsaw, Poland
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.

The author shows that any family C2-close to fα(x) = 1 − αx2(2 − ε ≤ α ≤ 2) satisfies Jakobson's theorem: For a positive measure set of α the transformation fα has an absolutely continuous invariant measure. He also indicates some generalizations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 1 , March 1988 , pp. 93 - 109
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Benedicks, M. & Carleson, L.. On iterations of 1 − ax 2 on (−1, 1). Ann. of Math. (2) 122 (1985), no. 1, 1–25.Google Scholar
[2]Jakobson, M. V.. Absolutely continuous Invariant Measures for one-parameter families of one-dimensional maps. Comm. in Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[3]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. I.H.E.S. Publications Mathématiques, # 53 (1981), 17–52.Google Scholar
[4]Rees, M.. Positive measure sets of ergodic rational maps. Preprint.Google Scholar
[5]Rychlik, M.. Bounded variation and invariant measures. Studia Math. t. 76, (1983), 69–80.Google Scholar