Dynamics on spaces of quasimorphisms and applications to approximate lattice theory

Abstract We study the dynamics of countable groups on their respective spaces of quasimorphisms. For cohomologically non-trivial quasimorphisms, we show that there are no invariant measures and classify stationary measures. Within the equivalence class of any given quasimorphism, we find both uniquely stationary orbit closures that are in fact boundaries and orbit closures with uncountably many ergodic stationary probability measures. We apply these results to study hulls of uniform approximate lattices that arise from twists by quasimorphisms. We show that these hulls do not admit invariant probability measures (extending results by Machado and Hrushovski) and classify stationary probability measures on these hulls.