Measurable dynamics: theory and applications

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This is a proposal for a mathematical 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 study asymptotic or long-range behaviour of mathematical systems (or mathematical models of physical systems) 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. In effect, the invariant measure encodes the equilibrium behaviour of the system's cross section, from which one can compute long term statistics. ******As another 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, presenting a modelling challenge. A more fundamental mathematical challenge is that the parameters in ocean flow change over time; the system is non-autonomous. Equilibria for this system now take the form of equivariant families of measures, evolving along with the changing values of the underlaying parameters. Still, one can use theoretical results from ergodic theory (in this case, the multiplicative ergodic theory) to extract long term statistics of ocean flow. ******Although the examples above are deterministic systems, it is well-known that they can mimic stochastic or probabilistic behaviour. For example, ergodic theory has a version of the law of large numbers (tendency to the mean), the central limit theorem (distributional convergence of observations to normal), large deviations estimates, and so-on. This is one of the most exciting areas of current research in ergodic theory. The work described in this proposal aims to further tease out this important connection between deterministic and random-like behaviour in a range of models that include both autonomous and non-autonomous dynamics. ******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 supervision of the PI. **