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Dimension of invariant measures for maps with exponent zero

Published online by Cambridge University Press:  19 September 2008

F. Ledrappier  and
M. Misiurewicz
F. Ledrappier
Affiliation:
Laboratoire de Probabilités, Université Paris VI, T56; 6, Place Jussieu, 75230 Paris Cedex 05, France
M. Misiurewicz
Affiliation:
Instytut Matematyki, Universytet Warszawski, PKiN IXp, 00.901 Warszawa, Poland
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Abstract

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We give examples of maps of the interval with zero entropy for which the continuous invariant measure has no dimension, and we prove a dimension property for maps lying in the stable manifold of Feigenbaum's fixed points.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 4 , December 1985 , pp. 595 - 610
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Billingsley, P.. Ergodic Theory and Information. J. Wiley and Sons, 1965.Google Scholar
[2]Brin, M. & Katok, A.. On local entropy. In Geometric Dynamics. Lect. Notes in Math. 1007 Springer (1983).CrossRefGoogle Scholar
[3]Campanino, M. & Epstein, H.. On the existence of Feigenbaum's fixed point. Commun. Math. Phys. 79 (1981), 261–302.CrossRefGoogle Scholar
[4]Collet, P. & Eckmann, J.-P.. Iterated maps on the interval as dynamical systems. Progress in Physics, Birkhäuser: Boston, 1980.Google Scholar
[5]Collet, P., Eckmann, J.-P. & Lanford, O. E.. Universal properties of maps on an interval. Commun. Math. Phys. 76 1980, 211–254.CrossRefGoogle Scholar
[6]Grassberger, P.. On the Hausdorff dimension of fractal attractors. Journal ofStat. Phys. 26 (1981), 173–179.CrossRefGoogle Scholar
[7]Khanin, K. M., Sinai, J. G. & Feigenbaum, H. B. Vul.universality and thermodynamical formalism. Uspehi 39; 3 (1984).Google Scholar
[8]Lanford, O. E.. A computer-assisted proof of the Feigenbaum conjectures. Bull Amer. Math. Soc. (New Series) 6 (1982), 427–434.CrossRefGoogle Scholar
[9]Ledrappier, F.. Some relations between dimension and Lyapounov exponents. Commun. Math. Phys. 81 (1981), 229–238.CrossRefGoogle Scholar
[10]Misiurewicz, M.. Structure of mappings of an interval with zero entropy. Publ. Math. IHES 53 (1981), 5–16.CrossRefGoogle Scholar
[11]Misiurewicz, M.. Attracting Cantor set of positive measure for a C∞ map of an interval. Ergod. Th & Dynam. Sys. 2 (1982), 405–415.CrossRefGoogle Scholar
[12]Ruelle, D.. An inequality for the entropy of different;able maps. Bol Soc. Bras. Mat. 9 (1978), 83–87.CrossRefGoogle Scholar