Fractured fractals and broken dreams: self-similar geometry through metric and measure
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1997
|
| Schriftenreihe: | Oxford lecture series in mathematics and its applications
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IX, 212 S. Ill., graph. Darst. |
| ISBN: | 0198501668 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804126248922775552 |
|---|---|
| adam_text | CONTENTS
1 Basic definitions 1
2 Examples 5
2.1 Euclidean spaces 5
2.2 The snowflake functor 5
2.3 Cantor sets 6
2.4 Other fractals 7
2.5 A general procedure 9
2.6 Limit sets of discrete groups 12
3 Comparison with rectifiability 13
3.1 Rectifiable sets in R 13
3.2 Uniform rectifiability 14
4 The Heisenberg group 16
5 Background information 19
5.1 Extending Lipschitz functions 19
5.2 A covering lemma 19
5.3 Ahlfors regular spaces 19
5.4 Assouad s embedding theorem 21
5.5 Dyadic cubes 22
5.6 Semi regularity 24
6 Stronger self similarity for BPI spaces 26
6.1 The basic result 26
6.2 A corollary 32
6.3 Regular subsets 32
6.4 Some remarks about EAC 33
7 BPI equivalence 35
7.1 Basic facts 35
7.2 BPI equivalence and uniform rectifiability 36
7.3 A strengthening of BPI equivalence 40
7.4 Mappings with big bilipschitz pieces 49
8 Convergence of metric spaces 52
8.1 Hausdorff convergence 52
8.2 Convergence in Euclidean spaces 52
8.3 Convergence of mappings 53
8.4 Convergence of spaces 54
viii CONTENTS
8.5 Convergence of mappings between spaces 59
8.6 Convergence of measures 61
8.7 Limits of subsets and their measures 63
8.8 Smooth sets 67
9 Weak tangents 71
9.1 The definition 71
9.2 First facts 72
9.3 Limits of BPI spaces 73
9.4 Weak tangents of subsets 75
9.5 Comparisons with rectifiability 77
9.6 BPI spaces which are not BPI equivalent 80
9.7 Weak tangents of mappings 81
9.8 Weak tangents of measures 82
10 Rest stop 84
11 Spaces looking down on other spaces 85
11.1 Definitions and basic facts 85
11.2 Mappings defined everywhere 86
11.3 Cantor sets to Euclidean spaces 87
11.4 Looking down from Euclidean spaces 88
11.5 Euclidean and Heisenberg geometries 89
11.6 A Cantor set with sliding 90
11.7 Iterating patterns with cubes 92
11.8 An observation about snowflakes 93
11.9 Looking down between Cantor sets 94
ll.lORemarks 101
12 Regular mappings 102
12.1 The definition and basic facts 102
12.2 Regular mappings as weak tangents 104
12.3 Looking down from Rn 109
12.4 Measure preserving weak tangents 110
12.5 Measure preserving mappings 118
12.6 Spaces not looking down on each other 119
13 Sets made from nested cubes 122
13.1 Preliminary notions 122
13.2 Convergence of families of cubes 124
13.3 Weak tangents of families of cubes 125
13.4 Mappings 127
13.5 Subsets that block connectedness 128
13.6 Going further 131
14 Big pieces of bilipschitz mappings 136
14.1 Introduction 136
CONTENTS ix
14.2 Some examples 137
14.3 Possibilities for Lipschitz mappings 141
14.4 A stronger example 143
15 Uniformly disconnected spaces 156
15.1 Introduction 156
15.2 Spaces of dimension 1 157
15.3 Ultrametrics 161
15.4 Uniformization 162
15.5 Making regular mappings 166
16 Doubling measures and geometry 172
16.1 The definition 172
16.2 Deformations of geometry 173
16.3 Uniformization revisited 176
16.4 Doubling measures on Cantor sets 176
16.5 Reasonably self similar pairs 180
16.6 Riesz products 182
16.7 Riemann surfaces 184
16.8 Remarks 187
17 Deformations of BPI spaces 189
18 Snapshots 194
19 Some sets that are far from BPI 197
19.1 The basic construction 197
19.2 Small variations on the theme 202
20 A few more questions 205
References 207
Index 211
|
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| author | David, Guy 1957- Semmes, Stephen |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:14:27Z |
| institution | BVB |
| isbn | 0198501668 |
| language | English |
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| spelling | David, Guy 1957- Verfasser (DE-588)112614086 aut Fractured fractals and broken dreams self-similar geometry through metric and measure Guy David and Stephen Semmes Oxford Clarendon Press 1997 IX, 212 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford lecture series in mathematics and its applications 7 Espaces métriques ram Fractales ram Fractals gtt Mesure, Théorie de la ram Patroongeneratie gtt Fractals Measure theory Metric spaces Topologie (DE-588)4060425-1 gnd rswk-swf Fraktalgeometrie (DE-588)4473576-5 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Selbstähnlichkeit (DE-588)4286650-9 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Fraktalgeometrie (DE-588)4473576-5 s DE-604 Selbstähnlichkeit (DE-588)4286650-9 s Fraktal (DE-588)4123220-3 s Analysis (DE-588)4001865-9 s Maßtheorie (DE-588)4074626-4 s Metrischer Raum (DE-588)4169745-5 s Topologie (DE-588)4060425-1 s Semmes, Stephen Verfasser aut Oxford lecture series in mathematics and its applications 7 (DE-604)BV009910017 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007897643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | David, Guy 1957- Semmes, Stephen Fractured fractals and broken dreams self-similar geometry through metric and measure Oxford lecture series in mathematics and its applications Espaces métriques ram Fractales ram Fractals gtt Mesure, Théorie de la ram Patroongeneratie gtt Fractals Measure theory Metric spaces Topologie (DE-588)4060425-1 gnd Fraktalgeometrie (DE-588)4473576-5 gnd Metrischer Raum (DE-588)4169745-5 gnd Maßtheorie (DE-588)4074626-4 gnd Analysis (DE-588)4001865-9 gnd Selbstähnlichkeit (DE-588)4286650-9 gnd Fraktal (DE-588)4123220-3 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4473576-5 (DE-588)4169745-5 (DE-588)4074626-4 (DE-588)4001865-9 (DE-588)4286650-9 (DE-588)4123220-3 |
| title | Fractured fractals and broken dreams self-similar geometry through metric and measure |
| title_auth | Fractured fractals and broken dreams self-similar geometry through metric and measure |
| title_exact_search | Fractured fractals and broken dreams self-similar geometry through metric and measure |
| title_full | Fractured fractals and broken dreams self-similar geometry through metric and measure Guy David and Stephen Semmes |
| title_fullStr | Fractured fractals and broken dreams self-similar geometry through metric and measure Guy David and Stephen Semmes |
| title_full_unstemmed | Fractured fractals and broken dreams self-similar geometry through metric and measure Guy David and Stephen Semmes |
| title_short | Fractured fractals and broken dreams |
| title_sort | fractured fractals and broken dreams self similar geometry through metric and measure |
| title_sub | self-similar geometry through metric and measure |
| topic | Espaces métriques ram Fractales ram Fractals gtt Mesure, Théorie de la ram Patroongeneratie gtt Fractals Measure theory Metric spaces Topologie (DE-588)4060425-1 gnd Fraktalgeometrie (DE-588)4473576-5 gnd Metrischer Raum (DE-588)4169745-5 gnd Maßtheorie (DE-588)4074626-4 gnd Analysis (DE-588)4001865-9 gnd Selbstähnlichkeit (DE-588)4286650-9 gnd Fraktal (DE-588)4123220-3 gnd |
| topic_facet | Espaces métriques Fractales Fractals Mesure, Théorie de la Patroongeneratie Measure theory Metric spaces Topologie Fraktalgeometrie Metrischer Raum Maßtheorie Analysis Selbstähnlichkeit Fraktal |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007897643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009910017 |
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