Measure, topology, and fractal geometry

Gerald Edgar0387747486, 9780387747484

From reviews of the first edition:

“In the world of mathematics, the 1980’s might well be described as the “decade of the fractal”. Starting with Benoit Mandelbrot’s remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar’s book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology…the book also contains many good illustrations of fractals (including 16 color plates).”

Mathematics Teaching

“The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples.”

Christoph Bandt, Mathematical Reviews

“…not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out.”

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2–6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

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Table of contents :
Preface……Page 6
Contents……Page 13
The Triadic Cantor Dust……Page 16
The Sierpinski Gasket……Page 22
A Space of Strings……Page 26
Turtle Graphics……Page 29
Sets Defined Recursively……Page 33
Number Systems……Page 46
*Remarks……Page 50
Metric Space……Page 56
Metric Structures……Page 63
Separable and Compact Spaces……Page 72
Uniform Convergence……Page 80
The Hausdorff Metric……Page 86
Metrics for Strings……Page 90
*Remarks……Page 96
Zero-Dimensional Spaces……Page 100
Covering Dimension……Page 106
*Two-Dimensional Euclidean Space……Page 114
Inductive Dimension……Page 119
*Remarks……Page 128
Ratio Lists……Page 132
String Models……Page 137
Graph Self-Similarity……Page 140
*Remarks……Page 148
Lebesgue Measure……Page 152
Method I……Page 161
Two-Dimensional Lebesgue Measure……Page 167
Metric Outer Measure……Page 170
Measures for Strings……Page 174
*Remarks……Page 177
Hausdorff Measure……Page 180
Packing Measure……Page 184
Examples……Page 192
Self-Similarity……Page 200
The Open Set Condition……Page 205
Graph Self-Similarity……Page 214
Graph Open Set Condition……Page 220
*Other Fractal Dimensions……Page 225
*Remarks……Page 231
*Deconstruction……Page 240
*Self-Affine Sets……Page 244
*Self-Conformal……Page 249
*A Multifractal……Page 253
*A Superfractal……Page 257
*Remarks……Page 262
Terms……Page 266
Notation……Page 269
Reading……Page 270
References……Page 272
Index……Page 276