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Two central themes in conformal dynamics are the iteration of rational maps on the Riemann sphere and the action of Kleinian groups. Although these theories developed along different paths, they share striking conceptual parallels and often exhibit similar forms of chaotic behavior. As early as the 1920s, Fatou anticipated that such systems could be studied within a unified framework of iterated algebraic correspondences.
We provide concrete evidence to Fatou's vision by constructing algebraic correspondences as combinations/matings of complex polynomials and Fuchsian groups, allowing features of both theories to coexist in a single dynamical setting. At the level of parameter spaces, this leads naturally to products of Teichmüller spaces and Mandelbrot sets inside appropriate moduli spaces of ramified covers of the Riemann sphere. We will outline the main analytic and algebraic ideas of this program, discuss connections with other areas of mathematics, and highlight several open questions. |