Teichmüller space for iterated function systems

In this paper we investigate families of iterated function systems (IFS) and conformal iterated function systems (CIFS) from a deformation point of view. Namely, we introduce the notion of Teichmüller space for finitely and infinitely generated (C)IFS and study its topological and metric properties. Firstly, we completely classify its boundary. In particular, we prove that this boundary essentially consists of inhomogeneous systems. Secondly, we equip Teichmüller space for (C)IFS with different metrics, an Euclidean, a hyperbolic, and a $\lambda$-metric. We then study continuity of the Hausdorff dimension function and the pressure function with respect to these metrics. We also show that the hyperbolic metric and the $\lambda$-metric induce topologies stronger than the non-metrizable $\lambda$-topology introduced by Roy and Urbanski and, therefore, provide an alternative to the $\lambda$-topology in the study of continuity of the Hausdorff dimension function and the pressure function. Finally, we investigate continuity properties of various limit sets associated with infinitely generated (C)IFS with respect to our metrics.