×

The entropy function for characteristic exponents. (English) Zbl 0643.58006

Summary: Using a thermodynamic formalism, we define an entropy function S(\(\alpha)\) which measures large deviations of the Lyapunov characteristic exponents of certain hyperbolic dynamical systems. The function S(\(\alpha)\) is a Legendre transform of a free-energy or pressure associated with the dynamical system. We show that S(\(\alpha)\) is the noncompact topological entropy of the set of points \(\Lambda_{\alpha}\) with characteristic exponent \(\alpha\), that S(\(\alpha)\)/\(\alpha\) is the Hausdorff dimension of \(\Lambda_{\alpha}\) and that \(\alpha\)-S(\(\alpha)\) is the escape rate from \(\Lambda_{\alpha}\). We explain how to use the formalism for cookie-cutters to describe the distribution of scales in the universal period-doubled attractor and critical golden circle mapping. We relate S(\(\alpha)\) to the Renyi entropies, prove a conjecture of Kantz and Grassberger relating the escape rate from hyperbolic repellors and saddles to the characteristic exponents and information dimension and study the fluctuations of escape rates. We discuss the escape rate from \(f(x)=(4+\epsilon)x(1-x)\) and the behaviour of the associated S(\(\alpha)\) as \(\epsilon\searrow 0\). Finally we discuss how to apply these ideas to experimental time series and non-hyperbolic attractors.

MSC:

37A99 Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Benzi, R.; Paladin, G.; Parisi, G.; Vulpiani, A., On the multifractal nature of turbulence and chaotic systems, J. phys. A, 17, 3521, (1984)
[2] Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, (), No. 470 · Zbl 0308.28010
[3] Bowen, R., Topological entropy for noncompact sets, Trans. AMS, 184, 125-131, (1973)
[4] Coullet, P.; Tresser, C., Iterations d’endomorphismes et groupe de renormalisation, J. phys. C., 5, 25, (1978) · Zbl 0402.54046
[5] Eckmann, J.-P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. mod. phys., 57, 617, (1985) · Zbl 0989.37516
[6] Feigenbaum, M., Quantitative universality for a class of nonlinear transformations, J. stat. phys., 19, 25, (1978) · Zbl 0509.58037
[7] Feigenbaum, M., The transition to aperiodic behaviour in turbulent systems, Comm. math. phys., 77, 65, (1980) · Zbl 0465.76050
[8] Feigenbaum, M.; Kadanoff, L.; Shenker, S., Quasi-periodicity in dissipative systems: A renormalisation group analysis, Physica, 5d, 370, (1982)
[9] Frisch, U.; Parisi, G., On the singularity structure of fully-developed turbulence, (), Appendix to U. Frisch
[10] Fujisaka, H., Statistical dynamics generated by fluctuations of local Lyapunov exponents, Progr. theoret. phys., 70, 1264, (1983) · Zbl 1161.37352
[11] Grassberger, P., Information flow and maximum entropy measures for 1-D maps, Physica, 14D, 365, (1985) · Zbl 0584.94008
[12] Grassberger, P.; Procaccia, I., Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Physica, 13D, 34, (1984) · Zbl 0587.58031
[13] Halsey, T.; Jensen, M.; Kadanoff, L.; Procaccia, I.; Shraiman, B., Fractal measures and their singularities: the characterisation of strange sets, (1985), preprint
[14] Kadanoff, L.P.; Tang, C., Escape from strange repellors, (), 1276 · Zbl 0548.58023
[15] Kantz, H.; Grassberger, P., Repellors and semi-attractors and long-lived chaotic transients, Physica, 17D, 75, (1985) · Zbl 0597.58017
[16] O. Lanford, Entropy and equilibrium states in classical statistical mechanics, in: Statistical Mechanics and mathematica problems, A. Lenard, Ed., Springer Lecture Notes in Physics, No. 20, p. 1.
[17] Lanford, O., A computer-assisted proof of the Feigenbaum conjectures, Bull. AMS, 6, 427, (1982) · Zbl 0487.58017
[18] McClusky, H.; Manning, A., Hausdorff dimension for horseshoes, Ergodic theory and dynamical systems, Errata in ergodic theory and dynamical systems, 5, 319, (1985) · Zbl 0566.58027
[19] Ostlund, S.; Rand, D.A.; Sethna, J.; Siggia, E., Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica, 8D, 303, (1983) · Zbl 0538.58025
[20] Rand, D.A., The singularity spectrum for hyperbolic Cantor sets and attractors, (1986), preprint
[21] Renyi, A., (), 526
[22] Ruelle, D., Thermodynamic formalism, () · Zbl 0702.58056
[23] Young, L.-S., Dimension, entropy and Liapunov exponents, Ergodic theory and dynamical systems, 2, 109, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.