Density evolution in systems with slow approach to equilibrium

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2004

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Abstract

This dissertation investigates the evolution of probability densities under the Frobenius-Perron operator U in chaotic iterated-map systems that are slow to reach equilibrium. It first concentrates on one-dimensional maps that are slow to reach equilibrium because they feature intermittent chaos due to the presence of a marginal fixed point. Using the method of shift states and coherent states under U, certain results are obtained concerning the spectrum of U in various functional spaces, using as the main example the cusp map f (x) = 1

  • √ |1 - 2x |. Those results are applied to obtain corrections to the well-known leading 1/t form of the x-x auto correlation function C(t). The symbolic dynamics of one-dimensional maps are then investigated, with particular emphasis on the implications of the existence of intermittent chaos and with applications to topological conjugation. Next, the statistics of extreme values in one dimensional maps are investigated. Fn(x) is defined as the probability that a point

chosen from an initial probability distribution and its first n- 1 iterates under a particular map are all less than x; the properties of Fn(x) are derived analytically for a wide variety of one-dimensional maps, and the on conclusions are confirmed numerically. Finally, higher-dimensional area-preserving maps are investigated. The technique of local spectral decomposition for U, in which approximate right and left eigenstates for U are constructed localized on unstable periodic points, is used to study density evolution and correlation of observables over time.

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