Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-27T23:50:21.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological conditions for the existence of invariant measures for unimodal maps

Published online by Cambridge University Press:  19 September 2008

H. Bruin
H. Bruin
Affiliation:
Delft University of Technology, Department of Pure Mathematics, PO Box 5031, 2600 GA Delft, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a class of S-unimodal maps having an invariant measure which is absolutely continuous with respect to Lebesgue measure. This measure can often be proved to be finite. We give an example of a map which has such a finite measure and for which the lim inf of the derivatives of the iterates of the map in the critical value is finite. It will be shown that all topologically conjugate non-flat S-unimodal maps have the same properties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 433 - 451
Copyright
Copyright © Cambridge University Press 1994

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

[BL1]Blokh, A. M. and Lyubich, M. Ju.. Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. II.the smooth case Ergod. Th. & Dynam. Sys. 9 (1989), 751–758.CrossRefGoogle Scholar
[BL2]Blokh, A. M. and Lyubich, M. Ju.. Measurable dynamics of S-unimodal maps of the interval. Ann. Scient. Éc. Norm. Sup. 24 (1991), 545–573.CrossRefGoogle Scholar
[CE]Collet, P. and Eckmann, J-P.. Positive Lyapunov exponents and absolute continuity of maps of the interval. J. Stat. Phys. 25 (1983), 17–25.Google Scholar
[D]Devaney, R. L.. An Introduction to Chaotic Dynamical Systems. Benjamin/Cummings: New York, 1986.Google Scholar
[G]Guckenheimer, J.. Sensitive dependence on initial conditions for one dimensional maps. Commun. Math. Phys. 70 (1979), 133–160.CrossRefGoogle Scholar
[GJ]Guckenheimer, J. and Johnson, S. D.. Distortion of S-unimodal maps. Ann. Math. 132 (1990), 73–130.CrossRefGoogle Scholar
[He]Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. IHES 49 (1979), 5–233.CrossRefGoogle Scholar
[Ho]Hofbauer, F.. The topological entropy of the transformation x → ax(l − x). Monath. Math. 90 (1980), 117–141.CrossRefGoogle 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 on recent results about S-unimodal maps. Ann. Inst. Henri Poincaré, Physique Théorique. 53 (1990), 413–425.Google Scholar
[J]Johnson, S. D.. Singular measures without restrictive intervals. Commun. Math. Phys. 110 (1987), 185–190.CrossRefGoogle Scholar
[JS]Jakobson, M. and Światek, G.. Metric properties of non-renormalizable S-unimodal maps, I. Induced expansion and invariant measures. Preprint IHES. (1991).Google Scholar
[K]Keller, G.. Exponents, attractors and Hopf decompositions of interval maps. Ergod. Th. & Dynam. Sys. 10 (1990), 717–744.CrossRefGoogle Scholar
[KN]Keller, G. and Nowicki, T.. Fibonacci maps re(al)-visited. Preprint. University of Erlangen. (1992).Google Scholar
[Le]Ledrappier, F.. Some properties of absolutely continuous invariant measures of an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 77–93.CrossRefGoogle Scholar
[Ly1]Lyubich, M. Ju.. Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. I. The case of negative Schwarzian derivative. Ergod. Th. & Dynam. Sys. 9 (1989) 737–750.CrossRefGoogle Scholar
[Ly2]Lyubich, M. Ju.. Combinatorics, geometry and attractors of quadratic-like maps. Preprint SUNY, Stony Brook (1992).Google Scholar
[LM]Lyubich, M. Ju. and Milnor, J.. The Fibonacci unimodal map. J. Am. Math. Soc. 6 (1993), 425–457.CrossRefGoogle Scholar
[M]Mañé, R.. Ergodic theory and differentiable dynamics. Ergebnisse der Mathematic und ihrer Grenzgebiete. Springer: Berlin and New York, 1987.Google Scholar
[Ma1]Martens, M.. Interval Dynamics Thesis. (1990).Google Scholar
[Ma2]Martens, M.. The existence of σ-finite invariant measures, applications to real 1-dimensional dynamics. Preprint IMPA. (1991) and Preprint StonyBrook (1992/1).Google Scholar
[MMS]Martens, M., de Melo, W. and Strien, S. van. Julia-Fatou-Sullivan theory for real 1-dimensional dynamics. Acta Math. 168 (1992), 273–318.CrossRefGoogle Scholar
[MS1]Melo, W. de and Strien, S. van. One-dimensional dynamics. Ergebnisse der Mathematic und ihrer Grenzgebiete. Springer: Berlin and New York, 1993.Google Scholar
[MS2]Melo, W. de and Strien, S. van. A structure theorem in one-dimensional dynamics. Ann. Math. 129 (1989), 519–546.CrossRefGoogle Scholar
[N]Nowicki, T.. Symmetric S-unimodal mappings and positive Lyapunov exponents. Ergod. Th. & Dynam. Sys. 8 (1988), 425–435.CrossRefGoogle Scholar
[NS]Nowicki, T. and Strien, S. van. Absolutely continuous invariant measures under a summability condition. Invent. Math. 105 (1991), 123–136.CrossRefGoogle Scholar
[R]Renyi, A.. Representation of real numbers and their ergodic properties. Acta Math. Akad. Sc. Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar
[V]Vargas, E.. Markov partitions in non-hyperbolic interval dynamics. Commun. Math. Phys. 138 (1991), 521–535.CrossRefGoogle Scholar