Conformal geometry of plane domains and holomorphic iterated function systems

We select a sequence of holomorphic functions from a hyperbolic domain O into a subdomain X. Consider the backward iterated function system corresponding to this sequence. By Montel's theorem, this system is a normal family. Therefore, it does have a set of limit functions, which we call the accumulation points of this system. The accumulation points are either open maps from O into X or constants. The constants can be inside X or on its boundary.;Suppose O is the unit disk Delta. Lorentzen and Gill showed that if X is relatively compact in Delta then every iterated function system has a unique accumulation point which is a constant inside X. In other words, they showed that relative non-compactness of X is necessary in order to have a boundary point as the accumulation point of an iterated function system. Beardon, Carne, Minda and Ng (see [2]) defined the notion of hyperbolic Bloch domain. These domains can be non-compact but satisfy a certain condition (see section 2.1). Keen and Lakic showed that if X does not have this property and c is a boundary point of X, we can find an iterated function system with the constant c as a limit function.;Our main result is that if c is a boundary point of a non-relatively compact subdomain of Delta, there always exists an iterated function system with the constant c as a limit function. In other words, we show that relative non-compactness of X in Delta is a sufficient condition to have c as a limit function.;In [10], Keen and Lakic defined new densities that generalize the hyperbolic density for a domain. One is a generalization of the Kobayashi density and the other is a generalization of Caratheodory density. We show that for a large class of domains O, with certain property that we define in chapter 5, the hyperbolic density on a hyperbolic domain X is equal to the generalized Kobayashi density. As a result, if X is a Kobayashi-Lipschitz subdomain of O it is a Caratheodory-Lipschitz subdomain as well.