Lecture Notes in Fractal Geometry

                                                      Natalie Priebe Frank

Contents

Chapter 1. Introduction ……………………………………………................................................… 1                                                            

     1.1. Classic examples ………………………………………………................................................ 1

     1.2. A geometric approach to transformations. …….................................................. 4

     1.3. Collage maps: the building blocks of iterated function systems............ 6

     1.4. Ane transformations in two dimensions: a geometric approach............ 8

     1.5. Collage maps in two dimensions .................................................................................. 11

     1.6. What is a fractal? ....................................................................................................................15

     1.7. Exercises .........................................................................................................................................16

Chapter 2. Hausdor metric and the space of fractals.........................................................19

     2.1. A tiny bit of point-set topology ....................................................................................19

     2.2. H(X), the space of fractals................................................................................................. 21

     2.3. Metric spaces ..............................................................................................................................21

     2.4. Hausdor metric on H(X) .....................................................................................................23

     2.5. Exercises ........................................................................................................................................28

Chapter 3. Iterated Function Systems............................................................................................ 29

     3.1. Collage maps as contractions on the space of fractals ................................29

     3.2. Existence of Attractors for Iterated Function Systems ................................33

     3.3. Two computer algorithms for IFS fractals ............................................................34

     3.4. The Collage Theorem ..............................................................................................................36

     3.5. Exercises ...........................................................................................................................................37

Chapter 4. Dimensions .................................................................................................................................39

     4.1. Motivating examples, or, Fun with length and area ...........................................39

     4.2. The idea of fractal dimension ............................................................................................42

     4.3. Similarity dimension ..................................................................................................................46

     4.4. Box-counting dimension .........................................................................................................51

     4.5. When the dimensions are equivalent ..............................................................................54

     4.6. Exercises ..............................................................................................................................................56

Chapter 5. Julia Sets ...........................................................................................................................................57

     5.1. Basic example: dynamical systems in R ...............................................................................58

     5.2. Classic example: dynamical systems in C ..........................................................................60

     5.3. The Escape Time Algorithm ........................................................................................................62

     5.4. Exploring some more, algebraically and Mathematica-ally ................................62

Chapter 6. Mandelbrot Sets ..........................................................................................................................65

     6.1. Using the escape-time algorithm to plot the Mandelbrot set ..........................66

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