On hyperbolic rational maps with finitely connected Fatou sets

In this paper, we study hyperbolic rational maps where all Fatou components are finitely connected. We show a hyperbolic component of such maps can be described by a finite combinatorial model which we call hyperbolic post-critically finite tree mapping schemes. We also show every such combinatorial model is realizable by some hyperbolic rational maps. This generalizes hyperbolic post-critically finite rational maps when the Julia set is connected, and is interpreted as `center' at infinity for such hyperbolic components. The construction uses the idea of `stretch' degeneration when the Julia set is disconnected. A sequence of degenerating rational maps gives a limiting dynamics on the R-tree, and the tree mapping scheme arises as the `core map' for the limiting dynamics. We establish a semiconjugacy between the Julia set for maps in the hyperbolic component and Julia set in the the R-tree (known as the Berkovich Julia set). As an application, this allows us to get bounds on the complexities of the Julia components and establish a dichotomy on the growth of the multipliers. The tree mapping schemes give flexibility in constructing lots of example of hyperbolic rational maps of this type. In particular, we use it to construct the first example of a sequence of rational maps of a fixed degree with infinitely many non-monomial rescaling limits.