The first part studies the accumulation of the zeros of successive derivatives of a Fuchsian automorphic function, thus generalizing Pólya's “Shire Theorem” and solving a problem of classical complex analysis with the tools of modern hyperbolic geometry. It is shown how the orbital approach toward limit points influences the topology of the function's final set and how to generalize the Shire Theorem in situations which render it invalid in its original form.; Solving a problem posed by Dennis Sullivan, the main part of the thesis is dedicated to the construction of a map on the limit set of a discrete Möbius group with the property that it identifies group-equivalent limit points. First, the presentation of the group is used to model its end set as compact metric space in which each point is identified by an infinite symbolic expansion in the generators. The expansions, in turn, induce a, partition of the end set on which the map is defined piecewise continuously, expanding, and orbit-equivalent to the group. Under this map, the partition is then shown to be a Markov partition in the sense of Smale-space theory.; To round off the picture, a measure is constructed on the end set, and the flow induced by the orbit-equivalent map is shown to be ergodic under this measure. An estimate of the Hausdorff dimension of the end set completes the investigation on the level of geometric group theory.; The group's action on the ball model permits the transferal of the construction from the end set to the limit set. By defining a suitable projection from the space of symbolic expansions to hyperbolic space, the orbit-equivalent map is pushed into the ball model—which completes the solution of the originally posed problem.
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