Complex dynamics - Wikipedia
Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics , where a polynomial or rational function is iterated.
This paper discusses the theory of dynamical systems of a single variable on the complex plane. We begin with a brief overview of the general theory of holomorphic dynamics on complex manifolds, defining the Julia set and exploring its elementary properties.
Complex dynamics is the study of dynamical systems whose state spaces are subsets of the complex plane C (or more generally n. C for n 1). Complex dynamics was one of the original motivations for the study of fractal geom-etry, and it includes some of the most famous fractals in mathematics|the Julia sets and the Mandelbrot set.
DANNY CALEGARI. Abstract. These are notes on complex dynamics, based on a graduate course taught at the University of Chicago in Winter 2022. Contents. et. er. 15. 3. Examples. . . ost-critically fin. ot. 19. 24. 33. References. 37. 1. Fatou and Julia Sets. Lemma . nd Montel’s . em in complex analysis is th. (Schwarz Lemma). Let f : D . Then.
1.1 Complex dynamics This work presents a study of renormalization of quadratic polyno-mials and a rapid introduction to techniques in complex dynamics. Around 1920 Fatou and Julia initiated the theory of iterated ra-tional maps f: Cb→ Cb on the Riemann sphere. More recently methods of geometric func-
In complex dynamics we study the behavior of iterated analytic maps. Typical set-up is an analytic self-map f: X!X, where Xis some Riemann surface, typically the complex plane C or the Riemann sphere C^ = C[f1g. (An analytic map to C^ is just a meromorphic map.) We will denote the n-th iterate of a function fby fn= f f ::: f.
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We now turn to studying the dynamical properties of complex-valued functions f: C !C. We will universally denote z= x+ iy, and write such functions either as f(z), f(x+ iy), or f(re i ).
Complex Dynamics - SpringerLink
Complex dynamics is today very much a focus of interest. Though several fine expository articles were available, by P. Blanchard and by M. Yu. Lyubich in particular, until recently there was no single source where students could find the material with proofs.
Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, …
1 Introduction: why study complex analysis? These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis. First, it is, in my