Quantum ergodicity in mixed and KAM Hamiltonian systems

In this thesis, we investigate quantum ergodicity for two classes of Hamiltonian systems satisfying intermediate dynamical hypotheses between the well understood extremes of ergodic flow and quantum completely integrable flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian systems. Hamiltonian systems with mixed phase space decompose into finitely many invariant subsets, only some of which are of ergodic character. It has been conjectured by Percival that the eigenfunctions of the quantisation of this system decompose into associated families of analogous character. The first project in this thesis proves a weak form of this conjecture for a class of dynamical billiards, namely the mushroom billiards of Bunimovich for a full measure subset of a shape parameter $t\in (0,2]$. KAM Hamiltonian systems arise as perturbations of completely integrable Hamiltonian systems. The dynamics of these systems are well understood and have near-integrable character. The classical-quantum correspondence suggests that the quantisation of KAM systems will not have quantum ergodic character. The second project in this thesis proves an initial negative quantum ergodicity result for a class of positive Gevrey perturbations of a Gevrey Hamiltonian that satisfy a mild "slow torus" condition.