Thermodynamic formalism for Quasimorphisms: Bounded Cohomology and Statistics

For a compact negatively curved space, we develop a thermodynamic formalism framework to study the space of quasimorphisms of its fundamental group modulo bounded functions. We prove that this space is Banach isomorphic to the space of Bowen functions on the associated Gromov geodesic flow, modulo a weak form of Livšic cohomology. We also show that each unbounded quasimorphism is associated with a unique invariant measure for the flow, which uniquely determines the cohomology class. As a consequence, we establish the Central Limit Theorem and the invariance principle for any unbounded quasimorphism with respect to Markov measures, and we prove that the associated equilibrium state has the Bernoulli property.