Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T04:47:25.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive Liapunov exponents and absolute continuity for maps of the interval

Published online by Cambridge University Press:  19 September 2008

P. Collet  and
J.-P. Eckmann
P. Collet
Affiliation:
Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau, France
J.-P. Eckmann
Affiliation:
Département de Physique Théorique, Universite de Genève, 1211 Genève 4, Switzerland
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.

We give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity assumptions, the condition requires positivity of the forward and backward Liapunov exponent of the critical point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 1 , March 1983 , pp. 13 - 46
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Bowen, R.. Invariant measures for Markov maps of the interval. Comm. Math. Phys. 69 (1979), 1–17.CrossRefGoogle Scholar
[2]Collet, P. & Eckmann, J.-P.. On the abundance of aperiodic behaviour for maps on the interval. Comm. Math. Phys. 73 (1980), 115–160.CrossRefGoogle Scholar
[3]Collet, P. and Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems. Birkhäuser: Basel, Boston, Stuttgart, 1980.Google Scholar
[4]Dunford, M. & Schwartz, J. T.. Linear Operators. Interscience: New York, 1958.Google Scholar
[5]Jakobson, M.. Construction of invariant measures absolutely continuous with respect to dx for some maps of the interval. Comm. Math. Phys. 81 (1981), 39–95.CrossRefGoogle Scholar
[6]Lasota, A. & Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. AMS 183 (1979), 418–485.Google Scholar
[7]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES 53 (1981).CrossRefGoogle Scholar
[8]Ruelle, D.. Applications conservant une mesure absolument continue par rapport à dx sur [0, 1]. Comm. Math. Phys. 55 (1977), 47–51.CrossRefGoogle Scholar
[9]Ruelle, D., Sensitive dependence on initial conditions and turbulent behavior of dynamical systems. In Bifurcation Theory and its Applications in Scientific Disciplines. New York Acad. of Sci. 316 (1979).Google Scholar
[10]Szlenk, W.. Some dynamical properties of certain differentiable mappings of an interval. I and II. Preprint IHES M/80/30, M/80/41 (1980), to appear.Google Scholar
[11]Ulam, S. M. & von Neumann, J.. On combinations of stochastic and deterministic processes. Bull. Amer. Math. Soc. 53 (1947), 1120.Google Scholar
[12]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1975), 121–153.CrossRefGoogle Scholar
[13]Yosida, K.. Functional Analysis. Springer-Verlag: Berlin, Heidelberg, New York.Google Scholar