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The singularity spectrum f(α) for cookie-cutters

Published online by Cambridge University Press:  19 September 2008

D. A. Rand
D. A. Rand
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, Warwick University, Coventry CV4 7AL, UK
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Abstract

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I use a thermodynamic formalism to study the spectrum f(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, that f(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimension Dq of the Gibbs state by the formula (1 − q)Dq = τ(q).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 527 - 541
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Arneodo, A., Grasseau, G. & Kostelich, E. J.. Fractal dimensions and f(α) spectrum of the Henon attractor. Preprint 1987.Google Scholar
[2]Bohr, T. & Rand, D. A.. The entropy function for characteristic exponents. Physica 25D (1986), 387–398.Google Scholar
[3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Mathematics, No. 470 (Springer-Verlag: New York, 1975).Google Scholar
[4]Collet, P., Lebowitz, J. & Porzio, T.. The dimension spectrum for some dynamical systems. Preprint.CrossRefGoogle Scholar
[5]Cvitanovic, P., In: Zweifel, P., Galivotti, G. & Anile, M., eds. Non-linear Evolution and Chaotic Phenomena (Plenum: New York, 1987).Google Scholar
[6]Grassberger, P., Badii, R. & Politi, A.. Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. Preprint, June 1987.Google Scholar
[7]Gunaratne, G. & Procaccia, I.. Organisation of chaos. Phys. Rev. Lett. 59 (1987), 1377.CrossRefGoogle ScholarPubMed
[8]Gundlach, M.. Warwick M.Sc. Dissertation, August 1986.Google Scholar
[9]Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I. & Shraiman, B.. Fractal measures and their singularities: the characterisation of strange sets. Phys. Rev. A 33 (1986), 1141.CrossRefGoogle ScholarPubMed
[10]Lanford, O.. Entropy and equilibrium states in classical statistical mechanics. In: Lenard, A., ed., Statistical Mechanics and Mathematical Problems. Springer Lecture Notes in Physics, No. 20 (Springer-Verlag: New York, 1973) 1–113.Google Scholar
[11]Ledrappier, F.. Proprietes ergodiques des mesures de Sinai. Publ. Math. IHES 59 (1984), 163–188.CrossRefGoogle Scholar
[12]Ledrappier, F. & Young, L.-S.. The metric entropy of diffeomorphisms. Ann. Math. 122 (1985), 509–574.CrossRefGoogle Scholar
[13]Manning, A.. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–459.CrossRefGoogle Scholar
[14]McCluskey, H. & Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1985), 251–260.CrossRefGoogle Scholar
Errata in Ergod. Th. & Dynam. Sys. 5 (1985), 319.Google Scholar
[15]Parisi, G.. On the singularity structure of fully-developed turbulence. Appendix to U. Frisch, Fully-developed turbulence and intermittency. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (Soc. Italiana di Fisica: Bologna, 1985).Google Scholar
[16]Parry, W.. Private communication.Google Scholar
[17]Ruelle, D.. In: Rota, G.-C., ed. Thermodynamic Formalism. Vol.5 of Encyclopedia of Mathematics and its Applications (Addison-Wesley: Massachusetts, 1978).Google Scholar
[18]Young, L.-S.. Dimension, entropy and Liapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–124.Google Scholar