Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

This advanced mathematical text offers a comprehensive exploration of equidistribution theory and counting problems for geodesics in negatively curved spaces, extending classical approaches to non-compact CAT(-1) manifolds and simplicial trees. The work systematically develops Patterson-Sullivan and Bowen-Margulis measures within the framework of potential functions, providing rigorous proofs for the equidistribution of hypersurfaces toward Gibbs measures. Researchers will find particularly valuable the extension of Oh-Shah skinning measures to these generalized geometric settings without relying on compactness assumptions.

The distinctive contribution lies in unifying geodesic counting problems with thermodynamic formalism across both continuous and discrete negatively curved spaces. This synthesis creates powerful tools for analyzing common perpendicular geodesics weighted by potentials, bridging geometric and dynamical perspectives. Specialists in ergodic theory, geometric group theory, and hyperbolic dynamics will discover sophisticated techniques for handling equidistribution in settings where traditional compactness arguments fail, advancing the frontier of modern geometric analysis.