Space of Geometric Gibbs Measures: Uniqueness, Ergodicity, and Teichmüller Complex Manifold Structure

Abstract

Building on our previous work, in which geometric Gibbs measures were introduced to complete the space of classical Gibbs measures under the Teichmüller metric, we establish the uniqueness of geometric Gibbs measures in this paper. This result completes the construction of a Teichmüller complex manifold structure on the space of all geometric Gibbs measures. Using the welding technique, we uncover a connection between geometric Gibbs measures and harmonic measures on dynamical quasicircles. This connection allows us to establish asymptotic conformal rigidity for these dynamical quasicircles. Finally, we establish a foundation for the systematic study of the ergodic properties of geometric Gibbs measures.