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Frontiers in complex dynamics. (English) Zbl 0807.30013
A rational function is said to be hyperbolic if all critical points tend to attracting cycles under iteration. Two equivalent definitions are that the function is expanding on its Julia set or that the postcritical set and the Julia set are disjoint. One of the main conjectures in complex dynamics is that hyperbolic rational functions are dense among all rational functions. This paper describes the background of this conjecture, connects it to a number of equivalent or related conjectures, and surveys recent results in this direction.
Among the topics and results discussed are: the density of structurally stable rational functions (Mañé-Sad-Sullivan), the connection with (the non-existence of) invariant line fields, the special case of quadratic polynomials and the Mandelbrot set, real quadratic polynomials, local connectivity of the Mandelbrot set implies density of hyperbolicity among quadratic polynomials (Doaudy-Hubbard), renormalization, the Mandelbrot set is connected at $$c$$ if $$z^ 2+ c$$ is not infinitely renormalizable (Yoccoz).
The author also sketches the proof of his result that the Julia set of $$f(z)= z^ 2+ c$$ carries no invariant line field if $$f$$ is an infinitely renormalizable real quadratic polynomial. Finally, he briefly mentions related recent work of Świątek, Lyubich, and Shishikura.
The paper is well written and highly recommended to anyone interested in this active area of research.

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37F99 Dynamical systems over complex numbers
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