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Chaos and Fractals
Mark McClure
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Contents
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Front Matter
Preface
1
Introduction
Surprise in Newton's method
The scope of chaos
Exercises
2
The iteration of real functions
Basic notions
Computer experimentation
Graphical analysis
Classification of fixed points
Classification of periodic orbits
Critical orbits
Parameterized families of functions
Conjugacy
The doubling map and chaos
Tent maps and the Cantor set
A few notes on computation
Exercises
3
The complex quadratic family
An illustrative example
The filled Julia set
An algorithm for the filled Julia set
Another look at conjugacy
The critical orbit
The Mandelbrot set
The components of the Mandelbrot set
Exercises
4
Self-similarity
Another look at the Cantor set
The Sierpinski gasket
Iterated function systems
Applying iterated function systems
Similarity transformations
Tools for generating self-similar sets
More examples
5
Fractal Dimension
Quantifying dimension
Similarity dimension of an IFS
Box-counting dimension
Comparing fractal dimensions
6
Generalizing self-similarity
Self-affinity
Digraph Iterated Function Systems
Authored in PreTeXt
Chaos and Fractals
Mark McClure
Department of Mathematics
University of North Carolina at Asheville
mcmcclur@unca.edu
August 21, 2023
Preface