Complex dynamics and renormalization.

*(English)*Zbl 0822.30002
Annals of Mathematics Studies. 135. Princeton, NJ: Univ. Press,. vii, 214 p. (1995).

A rational function is called hyperbolic if all critical points tend to attracting cycles under iteration. One of the main conjectures in complex dynamics is that hyperbolic rational functions are dense among all rational functions of a fixed degree. If specialized to quadratic polynomials, the conjecture says the set of all \(c\) for which \(z^ 2 + c\) is hyperbolic is a dense subset of the complex plane. Equivalently, every component of the interior of the Mandelbrot set is hyperbolic, i.e., consists only of \(c\)-values for which \(z^ 2 + c\) is hyperbolic. By work of Mañé, Sad, and Sullivan the above conjecture is also known to be equivalent to the conjecture that the Julia set of a quadratic polynomial carries no invariant line field. Yoccoz has shown that this is true for quadratic polynomials which are not infinitely renormalizable.

In this book it is shown that this is also true for a certain class of infinitely renormalizable polynomials. The hypothesis needed for this is called robustness and, roughly speaking, says that there are infinitely many renormalizations with definite space around the small postcritical sets. It is then shown that infinitely renormalizable real quadratic renormalizable polynomials are robust. This implies that every component of the interior of the Mandelbrot set that intersects the real axis is hyperbolic.

The book gives a good introduction to complex dynamics, discusses the background of the above conjectures, and establishes their equivalence. It includes a good exposition of the various results from conformal geometry and other fields that are needed and thus is as self-contained as possible. The chapters are: introduction, background in conformal geometry, dynamics of rational maps, holomorphic motions and the Mandelbrot set, compactness in holomorphic dynamics, polynomials and external rays, renormalization, puzzles and infinite renormalization, robustness, limits of renormalization, real quadratic polynomials, and two appendices on orbifolds and a closing lemma for rational maps. The book will be useful for everyone interested in this active field of research.

In this book it is shown that this is also true for a certain class of infinitely renormalizable polynomials. The hypothesis needed for this is called robustness and, roughly speaking, says that there are infinitely many renormalizations with definite space around the small postcritical sets. It is then shown that infinitely renormalizable real quadratic renormalizable polynomials are robust. This implies that every component of the interior of the Mandelbrot set that intersects the real axis is hyperbolic.

The book gives a good introduction to complex dynamics, discusses the background of the above conjectures, and establishes their equivalence. It includes a good exposition of the various results from conformal geometry and other fields that are needed and thus is as self-contained as possible. The chapters are: introduction, background in conformal geometry, dynamics of rational maps, holomorphic motions and the Mandelbrot set, compactness in holomorphic dynamics, polynomials and external rays, renormalization, puzzles and infinite renormalization, robustness, limits of renormalization, real quadratic polynomials, and two appendices on orbifolds and a closing lemma for rational maps. The book will be useful for everyone interested in this active field of research.

Reviewer: W.Bergweiler (Berlin)

##### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

37F99 | Dynamical systems over complex numbers |