Hostname: page-component-5db58dd55d-ggg9q Total loading time: 0 Render date: 2026-05-25T13:26:33.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Fibonacci maps re(aℓ)visited

Published online by Cambridge University Press:  19 September 2008

Gerhard Keller  and
Tomasz Nowicki
Gerhard Keller
Affiliation:
University of Erlangen, FRG
Tomasz Nowicki
Affiliation:
University of Warsaw, Poland†
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ ∈ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 99 - 120
Copyright
Copyright © Cambridge University Press 1995

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[ADU]Aaronson, J., Denker, M. and Urbanski, M.. Ergodic theory for Markov fibered systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.CrossRefGoogle Scholar
[BL]Blokh, A.M. and Lyubich, M.Yu.. Measurable dynamics of S-unimodal maps of the interval. Ann. Sc. École Norm. Sup. 24 (1991), 545–573.CrossRefGoogle Scholar
[Br]Bruin, H.. A topological condition for the existence of invariant measures for unimodal maps. Preprint. Delft, 1992.Google Scholar
[GSTV]Graczyk, J., Światek, G., Tangerman, F.M. and Veerman, J.J.P.. Scalings in circle maps III. Preprint. Stony Brook, 1992.Google Scholar
[GJ]Guckenheimer, J. and Johnson, S.. Distortion of S-unimodal maps. Ann. Math. 132 (1990), 71–130.CrossRefGoogle Scholar
[He]Henrici, P.. Discrete Variable Methods in Ordinary Differential Equations. J. Wiley & Sons: New York-London, 1962.Google Scholar
[HK1]Hofbauer, F. and Keller, G.. Quadratic maps without asymptotic measure. Commun. Math. Phys. 127 (1990), 319–337.CrossRefGoogle Scholar
[HK2]Hofbauer, F. and Keller, G.. Some remarks about recent results on S-unimodal maps. Annales de l' Institut Henri Poincarée, Physique Théorique 53 (1990), 413–425.Google Scholar
[JS]Jakobson, M. and Swiatek, G.. Metric properties of nonrenormalizable S-unimodal maps. Preprint. IHES, 1991, revised 1993.Google Scholar
[Ke]Keller, G.. Exponents, attractors, and Hopf decompositions for interval maps. Ergod. Th. & Dynam. Sys. 10 (1990), 717–744.CrossRefGoogle Scholar
[Lu]Lyubich, M.. Combinatorics, geometry and attractors of quasi-quadratic maps. Preprint. Stony Brook, 1992.Google Scholar
[LM]Lyubich, M. and Milnor, J.. The Fibonacci unimodal map. J. Amer. Math. Soc. 6 (1993), 425–457.CrossRefGoogle Scholar
[Ma]Martens, M.. Interval Dynamics. Thesis, University of Delft (1990).Google Scholar
[MS]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer: Berlin, 1993.CrossRefGoogle Scholar
[Mil]Milnor, J.. On the concept of attractor. Commun. Math. Phys. 99 (1985), 177–195.CrossRefGoogle Scholar
[Mis]Misiurewicz, M.. Absolutely continuous invariant measures for certain maps of an interval. Publ. Math. IHES 53 (1981), 17–51.CrossRefGoogle Scholar
[No]Nowicki, T.. Some dynamical properties of S-unimodal maps. Fund. Math. 142 (1993), 45–57.CrossRefGoogle Scholar
[NvS]Nowicki, T. and van Strien, S.. Invariant measures exist under a summability condition for unimodal maps. Inv. Math. 105 (1991), 123–136.CrossRefGoogle Scholar
[TV]Tangerman, F.M. and Veerman, J.J.P.. Scalings in circle maps I. Commun. Math. Phys. 144 (1990), 89–107.Google Scholar