Statistical properties of chaotic systems: from 1D maps to high dimensions

Abstract

Many important physical systems, such as the Earth's climate, are chaotic: as such we would like (probabilistic) predictions of these systems into the far future. These systems' long-term statistical behaviour is mathematically encoded by various objects, which can be studied functional-analytically using the so-called transfer operator. While rigorous study of many simple, usually one-dimensional, chaotic systems is theoretically tractable, for more complex, high-dimensional systems it is not: as a result the statistical properties of simple chaotic systems are often used as models for those of more complex systems, an idea that has been partially formalised as the Gallavotti-Cohen chaotic hypothesis. To study these simple dynamical systems better, we will in the first part of this thesis rigorously develop efficient, powerful numerics for two classes of one-dimensional maps: uniformly expanding Markov maps, and intermittent maps. To do this, we harness the smooth structure of these problems, in particular by discretising transfer operators using spectral basis functions. We obtain highly accurate numerical estimates of statistical properties of these maps: later, we apply the methods profitably to a numerical continuation problem associated with more complex systems. In the second part of the thesis we investigate high-dimensional systems, focusing on the differentiability of the response of statistical properties to dynamical perturbations (linear response theory). Although there are rigorous examples of one-dimensional maps that do not have differentiable responses, it is commonly believed that complex, high-dimensional chaotic systems generally do. We examine this belief through a study of model classes of ``high-dimensional systems''. We provide a comprehensive picture of the response properties of these systems and give broadly-applicable criteria governing the response's differentiability. In particular, we find classes of maps that, despite being composed of microscopic subsystems with non-differentiable responses, obey linear response theory, and vice versa.