Measure, topology, and fractal geometry / Gerald A. Edgar.

Format:

Book

Author/Creator:

Edgar, Gerald A., 1949-

Contributor:

Sabin W. Colton, Jr., Memorial Fund.

Series:

Undergraduate texts in mathematics

Language:

English

Subjects (All):

Fractals.

Measure theory.

Topology.

Physical Description:

xv, 268 pages, 8 unnumbered pages of plates : illustrations (some color) ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer-Verlag, [2008]

Summary:

For the Second Edition of this highly regarded textbook, Gerald Edgar has made numerous additions and changes, in an attempt to provide a clearer and more focused exposition. The most important addition is an increased emphasis on the packing measure, so that now it is often treated on a par with the Hausdorff measure. The topological dimensions were rearranged for Chapter 3, so that the covering dimension is the major one, and the inductive dimensions are the variants. A "reduced cover class" notion was introduced to help in proofs for Method I or Method II measures. Research results since 1990 that affect these elementary topics have been taken into account. Some examples have been added, including Barnsley leaf and Julia set, and most of the figures have been re-drawn.

Contents:

1 Fractal Examples 1

1.1 The Triadic Cantor Dust 1

1.2 The Sierpinski Gasket 7

1.3 A Space of Strings 11

1.4 Turtle Graphics 14

1.5 Sets Defined Recursively 18

1.6 Number Systems 31

1.7 Remarks 35

2 Metric Topology 41

2.1 Metric Space 41

2.2 Metric Structures 48

2.3 Separable and Compact Spaces 57

2.4 Uniform Convergence 65

2.5 The Hausdorff Metric 71

2.6 Metrics for Strings 75

2.7 Remarks 81

3 Topological Dimension 85

3.1 Zero-Dimensional Spaces 85

3.2 Covering Dimension 91

3.3 Two-Dimensional Euclidean Space 99

3.4 Inductive Dimension 104

3.5 Remarks 113

4 Self-Similarity 117

4.1 Ratio Lists 117

4.2 String Models 122

4.3 Graph Self-Similarity 125

4.4 Remarks 133

5 Measure Theory 137

5.1 Lebesgue Measure 137

5.2 Method I 146

5.3 Two-Dimensional Lebesgue Measure 152

5.4 Metric Outer Measure 155

5.5 Measures for Strings 159

5.6 Remarks 162

6 Fractal Dimension 165

6.1 Hausdorff Measure 165

6.2 Packing Measure 169

6.4 Self-Similarity 185

6.5 The Open Set Condition 190

6.6 Graph Self-Similarity 199

6.7 Graph Open Set Condition 205

6.8 Other Fractal Dimensions 210

6.9 Remarks 216

7.1 Deconstruction 225

7.2 Self-Affine Sets 229

7.3 Self-Conformal 234

7.4 A Multifractal 238

7.5 A Superfractal 242

7.6 Remarks 247.

Notes:

Includes bibliographical references (pages 257-259) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Sabin W. Colton, Jr., Memorial Fund.

ISBN:

9780387747484

0387747486

OCLC:

190580786

Online:

Publisher description