Table of Contents
Preface v
Acknowledgments ix
List of Figures xvii
List of Tables xxi
1 Examples of curious curves 1
1.1 Variations of the Koch curve 2
1.1.1 Koch curve 2
1.1.2 Modified Koch curve 2
1.1.3 Basics of complex numbers 4
1.2 More examples of curious curves 6
1.2.1 The unit square is a curve 6
1.2.2 Iterated function systems produce curves 7
1.3 Construction of a family of Cantor sets 10
1.3.1 Middle-thirds Cantor set 11
1.3.2 Construction of generalized Cantor sets 12
1.3.3 The length of the Cantor set C0 13
1.3.4 The sets Ch 15
1.4 What is not a curve? 15
2 The Koch curve and tangent lines 19
2.1 Construction of the Koch curve 19
2.1.1 Representation in base 4 20
2.1.2 Formulas for fk 21
2.1.3 Convergence of the sequence {fk} 24
2.1.4 An equation for f 25
2.1.5 Length of the Koch curve 26
2.2 Tangent lines to simple curves in C 26
2.2.1 Definition of a tangent line to a simple curve 27
2.2.2 Another construction of the Koch curve 28
2.2.3 Tangent lines to graphs of continuous maps from I to R 30
2.2.3.1 Modified Cantor functions 34
2.2.3.2 The graph Gφ of the Cantor function 34
2.3 Problems 35
3 Curves and Cantor sets 37
3.1 A square is a curve! 37
3.2 Simple curves 40
3.2.1 A homeomorphism g with C x C ≈ g(I) 42
3.2.2 Simple curves with positive area 44
3.2.3 Proofs of Propositions 3.1 and 3.2 48
3.2.3.1 Proof of Proposition 3.1 48
3.2.3.2 Proof of Proposition 3.2 49
3.3 Continuous images of the Cantor set 52
3.4 Subsets of C that are not curves 55
3.5 Generalized curves 56
3.6 More examples of curves 57
3.7 Problems 63
4 Generalizations ofthe Koch curve 67
4.1 Construction of generalizations 67
4.1.1 The iteration process 70
4.1.2 A decomposition of Ka,θ 70
4.2 Double points in Ka,θ with θ = π/3 and a = 1/4 71
4.2.1 The pivotal value a (θ) = 1/4 for θ =π/3 75
4.3 Investigation of Ka,π/4 76
4.3.1 Verifying <$$> 78
4.3.2 Double points form Cantor sets 80
4.4 Problems 81
5 Metric spaces and the Hausdorff metric 83
5.1 Metric spaces 83
5.1.1 Equivalent metrics 84
5.1.2 Topological properties of metric spaces 85
5.1.3 Complete metric spaces 89
5.2 The Hausdorff metric 90
5.3 Metrics and norms 92
5.4 Problems 94
6 Contraction maps and iterated function systems 101
6.1 Contraction maps 101
6.2 Iterated function systems 104
6.3 An iterated function system defines a curve 106
6.4 Implementation of iterated function systems 108
6.5 Problems 112
7 Dimension, curves and Cantor sets 115
7.1 Intervals, squares and cubes 115
7.2 Hausdorff dimension of a bounded subset of R2 117
7.2.1 Basic facts about dimension 117
7.3 Tent maps and Cantor sets with prescribed dimension 118
7.3.1 Dimension of Cantor sets 122
7.3.2 Dimension of Cantor sets in the plane 123
7.4 Dimension and simple curves 123
7.4.1 Simple curves with prescribed dimension 123
7.4.2 Dimension of the Koch curve 123
7.4.3 Functions with prescribed dimension of points of non-tangency 124
7.5 Symmetric Cantor sets (optional section) 125
7.5.1 Construction of a symmetric Cantor set 125
7.5.2 Definition of dimension of symmetric Cantor sets 126
7.6 Saw tooth maps 127
7.7 Problems 127
8 Julia sets and the Mandelbrot set 129
8.1 Theory of Julia sets 129
8.1.1 Observations 130
8.1.2 Visual images 131
8.1.3 Two facts about Julia sets 132
8.2 The Mandelbrot set 132
8.2.1 Fixed points of fc 133
8.2.2 The central cardioid 136
8.2.3 The great circle 136
8.2.3.1 Period two points 136
8.2.3.2 Description of the great circle 137
8.2.4 Super-attracting fixed points 138
8.2.5 The two large bulbs adjoining the central cardioid 139
8.3 Generalized curves and Julia sets 140
8.4 Problems 140
Appendix A Points on a line 143
A.1 Labeling points on a line 143
A.l.l base b representations 143
A.2 Convergence 144
A.2.1 The geometric series 145
A.3 The special nested interval property 147
A.4 Bounds on subsets of a line 147
A.4.1 Bounded sequences have convergent subsequences 149
A.5 The real numbers R 150
A.6 Eventually periodic base b representations 150
A.7 Problems 151
Appendix B Length and area 155
B.l Intervals and length 155
B.2 Lengths of subsets of intervals 157
B.3 Intervals and rectangles in the plane 158
B.4 Length of a curve 159
B.5 Areas of subsets of the plane 159
B.5.1 Areas of rectangles 160
B.5.2 Areas of general subsets of the plane 162
B.6 Problems 163
Appendix C Maps and sets in the plane 165
C.l Definition of a map 165
C.2 Properties of points in the plane 166
C.3 Continuity and limits 168
C.4 Topological properties of subsets of R2 169
C.4.1 Closed sets 170
C.4.2 Compact sets 171
C.4.3 Connected sets 174
C.4.4 Fixed points of maps 176
C.4.5 Uniform continuity of maps 177
C.5 Convergence of maps 178
C.6 Linear maps from R2 to R2 182
C.7 Homeomorphisms: Inverse maps on compact subsets of R2 184
C.8 Problems 185
Appendix D Infinite sets 191
D.l Countable and uncountable sets 191
D.l.l The positive rational numbers are countably infinite 192
D.l.2 The Cantor set is not a countable set 192
D.1.3 The continuum question 193
D.2 Problems 193
Bibliography 195
Solutions to selected problems 197
Index 207
