In the talk, we will use the proper mapping theory to study the global topology of hyperbolic components.
This will include the following topics:

1. Proper holomorphic mapping from annulus to disk and its formula. This generalizes the well-known Blaschke products.

We also characterize the topology of model spaces.

2. Proper holomorphic mapping from multiconnected domain to disk. We gives a necessary and sufficient condition for the existence of the map, using

Abel's Theorem for principle divisors.

3. Characterize the global topology of hyperbolic components in the Cantor circle locus.

4. We also posed some interesting problems.