Fractured Fractals and Broken Dreams: Self-similar Geometry Through Metric and MeasureClarendon Press, 1997 - Broj stranica: 212 Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information. |
Sadržaj
Basic definitions | 1 |
Comparison with rectifiability | 13 |
Background information | 19 |
Stronger selfsimilarity for BPI spaces | 26 |
BPI equivalence | 35 |
Convergence of metric spaces | 52 |
Weak tangents | 71 |
Rest stop | 84 |
Sets made from nested cubes | 122 |
Big pieces of bilipschitz mappings | 136 |
Uniformly disconnected spaces | 156 |
Doubling measures and geometry | 172 |
Deformations of BPI spaces | 189 |
Some sets that are far from BPI | 197 |
A few more questions | 205 |
| 211 | |
Uobičajeni izrazi i fraze
Ahlfors regular assume assumption basepoints bilipschitz constant bilipschitz equivalent Borel measure Borel sets bounded number BPI condition BPI equivalent BPI sets BPI spaces Cantor sets cell closed subset compact subset conformally bilipschitz mapping construction contained convergence define Definition 8.9 denote depend diam doubling measures embeddings Euclidean spaces existence fact family of cubes finite geometry given Hausdorff dimension Hausdorff measure Heisenberg group implies integer isometric isometrically equivalent K-conformally Lebesgue measure Lemma Lipschitz mapping look mapping packages measure-preserving measures on F nonempty normalized family number of elements pointed metric spaces positive measure probability measure properties proves the lemma quasimetric quasisymmetric mappings radii regular mappings regular metric space regular of dimension regularity constants restriction s-blocking scale factor Section self-similarity sense of Definition Sierpinski Sierpinski carpet snowflake space of dimension subword Suppose theorem topological dimension uniform rectifiability uniformly bounded uniformly disconnected weak tangents WT(M