Components of degree two hyperbolic rational maps. (English) Zbl 0712.30022

The author considers rational maps f of degree 2 which are hyperbolic in the sense that the orbits of the critical points are bounded away from the Julia set of f. These maps are classified into four distinct types. The paper describes the components of the set of these hyperbolic maps. Each component consists of maps of the same type. A rather long theorem describes the structure of the components of different types. There is a discussion of the boundary points of some of the components. Rather than using quasiconformal techniques the author relies on a result (described in a preprint of W. P. Thurston) on the equivalence of certain branched coverings of the sphere to a rational map.
Reviewer: I.N.Baker


30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics
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