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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†
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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.

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

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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