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Advanced MechanicsFrom Euler's Determinism to Arnold's Chaos$
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S. G. Rajeev

Print publication date: 2013

Print ISBN-13: 9780199670857

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199670857.001.0001

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Dynamics on the complex plane

Dynamics on the complex plane

Chapter:
(p.137) 17 Dynamics on the complex plane
Source:
Advanced Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199670857.003.0017

Iterations of an analytic function of a complex variable is a wonderful laboratory for chaos. After classifying Mobius transformations by the nature of their fixed points, this chapter turns to iterations of rational functions. Mathematical notions include the Riemann sphere, metric spaces, and compactness. The Julia set of a map is the closure of its unstable periodic point. In essence, that is where the map is chaotic. The Fatou set is its complement. Some examples of Julia sets are plotted, as is the Mandelbrot set.

Keywords:   chaos, Mobius transformations, unstable periodic point, compactness, Julia set, Fatou set

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