Complex Dynamics

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 13 in Complex Analysis and Dynamics in One Variable with Applications, 2025, pp 441-586 from Springer

Abstract: Abstract One variable complex dynamics with emphasis on dynamics of rational functions, conjugacy of functions, linearization at attracting, superattracting or repelling periodic points, general properties of Julia and Fatou sets, including selfsimilarity, Cantor sets, chaos, Siegel disks and Herman rings, Denjoy–Wolff theorem, rotation number and its topological invariance, including for a holomorphic homeomorphism at a neutral fixed point established in 1982 with a simpler proof in 1996, dynamics in a neighborhood of a rational neutral point with the Leau–Fatou conjugacy for parabolic periodic points and the parabolic flower theorem, bounds on the number of periodic orbits, dynamics in a neighborhood of an irrational neutral fixed point when it is linearizable (Siegel point) or not (Cremer point), abundance of the latter in Baire category and scarcity in Lebesgue measure, KAM theory to prove the Arnold theorem for homeomorphisms in a circle with Diophantine rotation number and of the Siegel theorem on fixed points of holomorphic functions with multiplier e i 2 πω $$e^{i2\pi \omega }$$ with ω $$\omega $$ a Diophantine rotation number are Siegel points as proved by C.L. Siegel and J. Moser in 1956, and proof of the analogous but stronger Brjuno theorem following a proof presented by J.-C. Yocozz in 2014, the proof of Yoccoz in 1987 of abundance in Lebesgue measure for ω ∈ [ 0 , 1 [ $$\omega \!\in \![0,1[$$ of existence of Siegel disks around 0 for quadratic polynomials z ( z + e i 2 πω ) $$z(z{+}e^{i2\pi \omega })\,$$ , Sullivan classification theorem of dynamics of Fatou components, boundaries of Siegel disks or Herman rings as closures of positive orbits of critical points, notion and general dynamical properties of hyperbolic functions, and of subhyperbolic functions using orbifolds as conceived by W.Thurston in 1982 and explained in detail only in 1993 by A. Douady and J. Hubbard, dynamics of polynomials including results with external rays obtained with the Riemann mapping theorem, properties of the Mandelbrot set of quadratic complex polynomials and relationship with dynamics of quadratic functions in a real interval.

Date: 2025
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DOI: 10.1007/978-3-031-64999-8_13

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