Hyperbolic maps are the dynamically simplest among all maps in a certain family, and each connected component consisting of all hyperbolic maps is called a hyperbolic component. For rational maps, a hyperbolic component contains a unique post-critically finite map. That is, hyperbolic components can be enumerated by the solutions of algebraic equations. In this talk, we will extend this enumerating method to a family of meromorphic maps, and we give a combinatorial description of hyperbolic components. Furthermore, we will describe measurable dynamics of the maps corresponding to these solutions. In particular, these maps are non-ergodic on their Julia set. This is a joint work with Linda Keen.

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