Cambridge University Press, Jan 13, 2000 - Mathematics - 338 pages
Here is a comprehensive introduction to holomorphic dynamics, that is, the dynamics induced by the iteration of various analytic maps in complex number spaces. This has been the focus of much attention in recent years, for example, with the discovery of the Mandelbrot set, and work on chaotic behavior of quadratic maps. The mathematically unified treatment emphasizes the substantial role of classical complex analysis in understanding holomorphic dynamics and offers up-to-date coverage of the modern theory. The authors cover entire functions, Kleinian groups and polynomial automorphisms of several complex variables such as complex Hénon maps, as well as the case of rational functions.
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assertion assume attracting fixed point automorphism Baker domain basin biholomorphic called closed curve compact set completely invariant complex condition conjugate connected component constant contains contradicts Corollary critical points cycle define denote dense dynamics element entire function equation escaping points Fatou component Fatou set function f fundamental theorem Hence Hénon map Herman rings holomorphic automorphism holomorphic function holomorphic map implies Int Kt integer invariant component iteration Jacobian Julia set Kleinian group Lemma Let f map F Möbius transformations neighborhood normal family open set orbit parabolic periodic point point of f polynomial automorphism polynomial of degree positive number Proof Let proof of Theorem Proposition rational function Remark Riemann sequence Siegel disk simply connected singular values subset sufficiently large superattracting Suppose topologically transcendental entire functions univalent wandering domains