Measurable Dynamics, Theory and Applications

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This is a proposal for a research program over the next five years in an area called measurable dynamical systems or ergodic theory. In this field we use techniques from measure theory and modern probability to understand asymptotic or long range behaviour of mathematical objects (or mathematical models of physical objects) that evolve with time. Ergodic theory brings a range of powerful tools to the analysis of dynamical systems with applications both in and outside the field of mathematics. For example, in ordinary differential equations one constructs a Poincare section in order to study periodic or recurrent structures in the continuous time flow. In the case of Hamiltonian systems, Liouville measure induces an invariant measure for the discrete-time map on the section, and the tools from ergodic theory can be applied. As another example, the sequence of digits in the continued fraction expansion of a real number can be generated by iteration of the Gauss map on the unit interval. There is a finite, invariant, Borel measure (Gauss measure) on the interval preserved by the Gauss map. The orbit of a point under iteration of the Gauss map determines the continued fraction digits and their statistical properties via this measure preserving system. As a third example, the flow of the world's oceans can be modelled as a discrete-time dynamical system, where the evolution of little boxes of ocean are tracked over some fixed positive time step (ranging from hours to months, depending on the problem). While an idealized ocean flow should preserve volume, in practice, observational data only roughly coincides with volume or area preservation. A very basic problem is to find an invariant measure for this observed discrete-time transformation and to use that measure along with the tools from ergodic theory to study this important dynamical system. Our proposal considers both theoretical and computational aspects of the field. We will treat both closed (traditional) and open dynamical systems as well as non-autonomous or time-dependent systems arising as random maps. Our objectives are (1) To develop efficient, rigorous numerical schemes to approximate dynamical structures such as invariant measures, conditionally invariant measures, almost invariant structures and other such barriers to mixing in dynamical systems. The same methods will be used to estimate relaxation parameters such as escape rate and correlation decay. (2) Ergodic theory has an analogue of the transition matrix from Markov chains theory called the Perron-Frobenius operator. A second objective of this program is to extend powerful theoretical techniques for spectral analysis of the Perron-Frobenius operator to the multidimensional setting, particularly for non-uniformly hyperbolic examples. This is an important, modern area of focus in the field of ergodic theory. (3) To investigate stability and bifurcation issues that arise from random perturbation of a single transformation and/or more generally to understand the effects of small-scale parameter variation in non-autonomous systems. This is highly relevant for real-life applications such as the ocean flow model outlined above. In order to achieve these objectives the program will rely on coordinated research between the PI and existing international experts in the field and will integrate training activities for HQP located at the University of Victoria and under direct supervision of the PI.