On Aspects of Numerical Ergodic Theory: Stability of Ulam’s Method, Computing Oseledets Subspaces and Optimal Mixing

A dynamical system is a pairing between a set of states X ⊂ Rd and a map T : X which describes how the system evolves from state to state over time. The Perron– Frobenius, or transfer, operator is a natural extension of the point-by-point dynamics defined by T to an ensemble theory which describes the evolution of distributions of points. It features heavily in dynamical systems theory and in a wide array of numerical methods including the approximation of invariant densities, physical measures, almost-invariant partitionings, coherent structures, Lyapunov exponents and topological entropy. This thesis focuses on numerical methods that take advantage of the Perron– Frobenius operator and its statistical representation of dynamical systems. Central to these is Ulam’s method—the most commonly used scheme to produce a finite-rank approximation of the Perron–Frobenius operator. Of interest are the circumstances in which Ulam’s method fails to produce an approximation whose spectral values converge to the spectral values of the Perron–Frobenius operator. We discuss a simple advection–diffusion dynamical system whose eigenvalues exhibit unusual behaviour. The eigenvalues of the Perron–Frobenius operator acting on L2 functions depend on the diffusion parameters, while the eigenfunctions do not. However, if the eigenfunctions are ordered by the magnitudes of their corresponding eigenvalues, the same eigenfunctions appear in a different order depending on the diffusion parameters. Finally, as Ulam’s method has its own diffusive effect, the same effect is observed by keeping the diffusion parameters constant, but altering the shape of the partition elements used in Ulam’s method.