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.


37A99 Ergodic theory
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