Automorphisms of compact quantum groups

This paper initiates the study of non-commutative dynamical systems of the form (G,), where a discrete group acts on a compact quantum group (CQG) G by quantum automorphisms. We obtain combinatorial conditions for such dynamical systems to be ergodic, mixing, compact, etc. and provide a wide variety of examples to illustrate these conditions. We generalize a well-known theorem of Halmos to demonstrate reversal of arrows' in the ergodic hierarchy relevant to the context and make a study of spectral measures for actions of (non-commutative) groups. We investigate the structure of such dynamical systems and under certain restrictions exhibit the existence and uniqueness of the maximal ergodic invariant normal subgroup of such systems. As an application, we study the size of normalizing algebras of masas arising from groups in von Neumann algebraic CQGs and show that the normalizing algebra of such masas are the von Neumann algebras generated by co-commutative CQGs.