Two central themes in holomorphic dynamics are the iteration of rational maps and the action of Kleinian groups on the Riemann sphere. Although these theories developed along largely independent paths, they exhibit striking conceptual parallels—a phenomenon often referred to as Sullivan’s Dictionary. As early as the 1920s, Fatou envisioned that these two classes of dynamical systems could be unified within the framework of iterated algebraic correspondences.

 After surveying fundamental results in rational dynamics and Kleinian groups, we will present concrete realizations of Fatou’s vision by constructing algebraic correspondences that unite the behaviors of complex polynomials and Fuchsian groups in a single dynamical plane. We will also discuss examples of rational Julia set realizations of Kleinian limit sets, such as classical Apollonian-like gaskets. Time permitting, we will outline the main analytic and algebraic ideas underlying this program, and highlight several applications to other areas of mathematics.