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The classification of conformal dynamical systems. (English) Zbl 0908.30028
Bott, Raoul (ed.) et al., Current developments in mathematics, 1995. Lectures of a seminar, held in Boston, MA, USA, May 7-8, 1995. Cambridge, MA: International Press. 191-230 (323-360, preliminary version 1994) (1995).
In the early 1980s Sullivan discovered analogies between the theory of Kleinian groups and the Fatou-Julia iteration theory of rational maps. In fact he gave a “dictionary” between these two subjects. This interesting survey further illustrates the connections between the two fields. The contents is best described by the following part of the author’s introdution: “We beginn the Teichm├╝ller space of a dynamical system, which classifies deformations and leads to the finiteness theorems of Ahlfors and Sullivan. A more detailed picture is available for expanding dynamical systems, and we examine the conjecture that such systems are dense. Then we present the program for a topological classification of Kleinian groups via 3-manifolds, along with hints of a parallel theory for rational maps. The Hausdorff dimension, ergodic theory and local connectivity of limit sets and Julia sets are also discussed. Finally we describe the role of renormalization in recent progress on the classification of quadratic polynomials.
For the entire collection see [Zbl 0833.00016].

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F99 Dynamical systems over complex numbers
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)