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 attracting periodic point automorphism Baker domain basin biholomorphic bounded called choose closed curve compact set completely invariant condition conjugate connected component constant contains contradicts converges uniformly Corollary critical points cycle define denote dense element entire functions equation escaping points Fatou component Fatou set fundamental theorem Hausdorff dimension Hence Henon map Herman rings holomorphic automorphism holomorphic function holomorphic map implies Int K+ invariant component iteration Julia set Kleinian group Lemma Let f map F Math metric neighborhood non-empty normal family open set orbit parabolic periodic point polynomial automorphisms polynomial of degree positive number Proof Let proof of Theorem Proposition rational function Remark Riemann satisfies sequence Siegel disk simply connected singular values space subset sufficiently large superattracting Suppose topology transcendental entire functions univalent wandering domains WS(a zero