Abstract
In this thesis, we aim to study dynamics on the boundary of hyperbolic spaces and hyperbolic groups induced by isometries of these spaces. In particular, we focus on north-south dynamics induced by loxodromic isometries. Much of this thesis covers background material in the area of hyperbolic spaces and hyperbolic groups with Chapter 2 covering the definition of hyperbolic spaces along with properties of these spaces. We also describe the behavior of quasi-geodesics in hyperbolic space, the important Morse lemma, and the quasi-isometric invariance of hyperbolicity. In Chapter 3, we discuss a number of constructions for the boundary of a hyperbolic space and show their equivalence. We then define a topology on the boundary, noting that the boundary is compact under this topology and that quasi-isometries of hyperbolic space extend to homeomorphisms on its boundary. This boundary is a metrizable space and so we then use methods by Bourbaki as well as Ghys and de la Harpe to construct a well-defined metric on the boundary of hyperbolic space. We finish the chapter with a cursory introduction to hyperbolic groups, their boundaries, and actions of a group on hyperbolic space. Chapter 4 covers isometries of hyperbolic spaces and includes the classification of these isometries. A contribution of this thesis is to combine disparate definitions for each class of isometry and prove their equivalence. In Chapter 5, we introduce the force of an isometry along with a new metric on the boundary of hyperbolic space, allowing us to utilize tools that use points in the boundary as a reference rather than points in the space itself. We then reclassify the isometries of hyperbolic space with respect to the force of the isometry and, in another contribution of the thesis, prove that these new definitions are equivalent to the previous ones. While the phenomenon of north-south dynamics is well-referenced in literature, a complete proof is not readily accessible and this is the main goal of this thesis. So lastly, we reference results from Solomon Leader to offer a complete and rigorous proof of north-south dynamics.