Thermodynamic Formalism and Its Applications to Deformation Spaces

In this dissertation, we discuss applications of Thermodynamic Formalism to various deformation spaces, namely, deformation spaces of metric graphs and immersed surfaces in hyperbolic 3-manifolds.

For spaces of metric graphs, we define two conformal but different pressure metrics and study Riemannian geometry induced by these two metrics. Moreover, we compare the Riemannian geometry features, such as curvature and completeness, of these two pressure metrics with the Weil-Petersson metric on Teichmüller spaces.

For immersed surfaces in hyperbolic 3-manifolds, we find relations between the critical exponent of the surface group (with respect to the ambient hyperbolic distance) and the topological entropy of the geodesics flow on the unit tangent bundle of the immersed surface (with respect to the induced Riemannian metric). These relations lead us to recover the remarkable Bowen rigidity for quasi-Fuchsian manifolds as well as give a Riemannian metric on the Fuchsian space which is bounded below by the classical Weil-Petersson metric.