Thermodynamic formalism for Quasi-Morphisms: Bounded Cohomology and Statistics

TL;DR

The paper develops a thermodynamic formalism for bounded cohomology in negatively curved settings by linking unbounded quasi-morphisms on a hyperbolic group to weak Bowen observables along the Gromov geodesic flow. A Banach isomorphism $\Psi$ identifies $\widetilde{\mathscr{QM}}(\Gamma)$ with $\mathrm{Bow}_{Liou}(X)/\sim$, enabling a robust probabilistic and ergodic analysis of these classes via equilibrium states. It establishes a comprehensive framework: Livsic-type cohomology for weak Bowen functions on SFTs and suspensions; a full variational principle for quasi-morphisms with a unique equilibrium state, and a transfer-operator viewpoint that yields CLT, invariance principle, and Bernoulli properties for the induced stochastic processes. The results generalize classical thermodynamic formalism to non-regular potentials, provide a canonical dynamical representation of $\ker(c_2)$, and offer a unified treatment of statistical limit laws for unbounded quasi-morphisms. Collectively, these contributions illuminate the Banach structure of $H^2_b(M;\mathbb{R})$ and link geometric group theory with symbolic dynamics and probabilistic limit theorems.

Abstract

For a compact negatively curved space, we develop a notion of thermodynamic formalism and apply it to study the space of quasi-morphisms of its fundamental group modulo boundedness. We prove that this space is Banach isomorphic to the space of Bowen functions corresponding to the associated Gromov geodesic flow, modulo a weak notion of Livsic cohomology. The results include that each such unbounded quasi-morphism is associated with a unique invariant measure for the flow, and this measure uniquely characterizes the cohomology class. As a consequence, we establish the Central Limit Theorem for any unbounded quasi-morphism with respect to Markov measures, the invariance principle, and the Bernoulli property of the associated equilibrium state.

Theorems & Definitions (90)

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