In this talk, we will discuss an interesting curve, so-called the Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especially representations corresponding to Riemann surfaces with cusps. Marc Burger and Richard Sharp showed that, for convex-cocompact Fuchsian representations, one can use the shape of the Manhattan curve to derive dynamical and geometric rigidity results. We will discuss how the Thermodynamic Formalism (for countable Markov shifts) helps us to generalize these rigidity results to non-convex-cocompact Fuchsian representations. Speaker(s): Nyima Kao (UChicago)