Matings and the other side of the dictionary (Q1950987)

In the theory of rational maps an important role is played by matings; in the theory of Kleinian groups the construction of double limits is a crucial concept. In this article the author looks at the `entries' (a) mating (in dynamical systems) and (b) double limit theorem (in hyperbolic geometry) in Sullivan's dictionary. He points out that the known proofs of (a) and (b) are quite different but maybe they do not need to be. Section 2 addresses matings of polynomials and a suitable setting for hyperbolic manifolds. In Section 3 the author gives some important theorems related to matings of polynomials and the double limit theorem. Section 4 attempts to spell out the similarities and differences between (a) and (b).