Geometry and dynamics of quadratic rational maps (with an appendix by J. Milnor and Tan Lei). (English) Zbl 0922.58062

In the author’s customarily clear style he presents an exposition of various results concerning quadratic rational maps of the Riemann sphere \(\mathbb{C}\cup \infty\) to itself. The geometric aspect concerns the 5-manifold \(\text{Rat}_2\) of such maps and its subspace \(\mathcal M_2\), the “moduli space”, consisting of all holomorphic conjugacy classes of maps in \(\text{Rat}_2\). Note that \(\mathcal M_2\) is biholomorphic to \(\mathbb{C}^2\). An important part is also played by the locus \(\text{Per}_n(\mu)\) of conjugacy classes in \(\mathcal M_2\) of maps with a periodic point of period \(n\) and multiplier \(\mu\). This locus is an algebraic curve of degree equal to the number of hyperbolic components (i.e., hyperbolic on their Julia set) of period \(n\) in the Mandelbrot set, and is a line when \(n=1\) and \(n=2\). The compactification of the moduli space is also studied, as are the spaces (and moduli spaces) of quadratic maps which have a marking (for example, by ordering) of the critical or fixed points. Following the discussion of geometry and topology he turns to results, due largely to M. Rees and Tan. The author shows the difference between working in the compactified moduli space \(\widehat {\mathcal M}_2\) and working in \(\mathcal M_2\cong \mathbb{C}^2\) and then goes on to explore visually the kinds of behavior for a conjugacy class belonging to a one-dimensional slice through \(\mathcal M_2\). Appendices include background material and details of proofs. Appendix F, written by the author and Tan, describes an example of a hyperbolic map for which no two Fatou components have a common boundary point; the corresponding Julia set is a “Sierpinski carpet” (usually called “Sierpinski curve”). The references include several unpublished theses.


37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
32S70 Other operations on complex singularities
37F99 Dynamical systems over complex numbers
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