Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

Title page......Page 1
Preface......Page 3
Contents......Page 5
1 Basic definitions......Page 9
2.2 The snowflake functor......Page 13
2.3 Cantor sets......Page 14
2.4 Other fractals......Page 15
2.5 A general procedure......Page 17
2.6 Limit sets of discrete groups......Page 20
3.1 Rectifiable sets in R n......Page 21
3.2 Uniform rectifiability......Page 22
4 The Heisenberg group......Page 24
5.3 Ahlfors regular spaces......Page 27
5.4 Assouad's embedding theorem......Page 29
5.5 Dyadic cubes......Page 30
5.6 Semi-regularity......Page 32
6.1 The basic result......Page 34
6.3 Regular subsets......Page 40
6.4 Some remarks about EAC......Page 41
7.1 Basic facts......Page 43
7.2 BPI equivalence and uniform rectifiability......Page 44
7.3 A strengthening of BPI equivalence......Page 48
7.4 Mappings with big bilipschitz pieces......Page 57
8.2 Convergence in Euclidean spaces......Page 60
8.3 Convergence of mappings......Page 61
8.4 Convergence of spaces......Page 62
8.5 Convergence of mappings between spaces......Page 67
8.6 Convergence of measures......Page 69
8.7 Limits of subsets and their measures......Page 71
8.8 Smooth sets......Page 75
9.1 The definition......Page 79
9.2 First facts......Page 80
9.3 Limits of BPI spaces......Page 81
9.4 Weak tangents of subsets......Page 83
9.5 Comparisons with rectifiability......Page 85
9.6 BPI spaces which are not BPI equivalent......Page 88
9.7 Weak tangents of mappings......Page 89
9.8 Weak tangents of measures......Page 90
10 Rest stop......Page 92
11.1 Definitions and basic facts......Page 93
11.2 Mappings defined everywhere......Page 94
11.3 Cantor sets to Euclidean spaces......Page 95
11.4 Looking down from Euclidean spaces......Page 96
11.5 Euclidean and Heisenberg geometries......Page 97
11.6 A Cantor set with sliding......Page 98
11.7 Iterating patterns with cubes......Page 100
11.8 An observation about snowflakes......Page 101
11.9 Looking down between Cantor sets......Page 102
11.10 Remarks......Page 109
12.1 The definition and basic facts......Page 110
12.2 Regular mappings as weak tangents......Page 112
12.3 Looking down from Rn......Page 117
12.4 Measure-preserving weak tangents......Page 118
12.5 Measure-preserving mappings......Page 126
12.6 Spaces not looking down on each other......Page 127
13.1 Preliminary notions......Page 130
13.2 Convergence of families of cubes......Page 132
13.3 Weak tangents of families of cubes......Page 133
13.4 Mappings......Page 135
13.5 Subsets that block connectedness......Page 136
13.6 Going further......Page 139
14.1 Introduction......Page 144
14.2 Some examples......Page 145
14.3 Possibilities for Lipschitz mappings......Page 149
14.4 A stronger example......Page 151
15.1 Introduction......Page 164
15.2 Spaces of dimension :S 1......Page 165
15.3 UItrametrics......Page 169
15.4 Uniformization......Page 170
15.5 Making regular mappings......Page 174
16.1 The definition......Page 180
16.2 Deformations of geometry......Page 181
16.4 Doubling measures on Cantor sets......Page 184
16.5 Reasonably self-similar pairs......Page 188
16.6 Riesz products......Page 190
16.7 Riemann surfaces......Page 192
16.8 Remarks......Page 195
17 Deformations of BPI spaces......Page 197
18 Snapshots......Page 202
19.1 The basic construction......Page 205
19.2 Small variations on the theme......Page 210
20 A few more questions......Page 213
References......Page 215
Index......Page 219