Dynamics Of Holomorphic Functions In The Hyperbolic Plane

Christodoulou, Argyrios (2020). Dynamics Of Holomorphic Functions In The Hyperbolic Plane. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.000111bb

Abstract

This thesis investigates the interactions between hyperbolic geometry and the dynamcal behaviour of compositions of holomorphic self-maps of the hyperbolic plane. Our analysis draws inspiration from iteration theory and the theory of discrete groups.

First, we prove an inequality that quantifies how close a holomorphic function is to being a conformal self-map of the hyperbolic plane. This can be thought of as a rigidity result for conformal functions that involves the hyperbolic metric.

We then use our rigidity result in order to study the dynamics of holomorphic self-maps of the hyperbolic plane. In particular, we investigate the behaviour of compositions of a sequence of functions that itself converges to some limit function. Our goal is to examine under what conditions the dynamics of the composition sequence is similar to the dynamics of the iterates of the limit function. Intuitively, this question is about whether the Denjoy–Wolff theorem is stable under perturbations in the space of holomorphic functions.

Next, we focus on compositions of Möbius transformations. Due to a result of Jacques and Short, the dynamical behaviour of any composition sequence generated by finitely many Möbius transformations can be inferred from the topological properties, in the Möbius group, of the semigroup that these transformations generate. We introduce geometric conditions on these semigroups that allow us to interpret their topological behaviour.

Finally, we use our analysis of semigroups of Möbius transformations in order to study the parameter space of uniformly hyperbolic PSL(2,R)-cocycles. This topic was previously investigated by Avila, Bochi and Yoccoz, who proved that the uniform hyperbolicity of cocycles is equivalent to certain geometric properties of Möbius transformations in PSL(2,R). The three authors pose several questions about the structure of this parameter space and we provide answers to two of their questions.

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