An invitation to fractal geometry : fractal dimensions, self-similarity and fractal curves / Michel L. Lapidus and Goran Radunovic.

Format:

Book

Author/Creator:

Lapidus, Michel L. (Michel Laurent), 1956- author.

Radunović, Goran, author.

Series:

Graduate studies in mathematics ; Volume 247.

Graduate studies in mathematics, 1065-7339 ; volume 247

Language:

English

Subjects (All):

Fractals.

Medical Subjects:

Fractals.

Physical Description:

1 online resource (xxvii, 600 pages) : illustrations.

Edition:

First edition.

Other Title:

Fractal dimensions, self-similarity and fractal curves

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2024]

Summary:

This book offers a comprehensive exploration of fractal dimensions, self-similarity, and fractal curves. Aimed at undergraduate and graduate students, postdocs, mathematicians, and scientists across disciplines, this text requires minimal prerequisites beyond a solid foundation in undergraduate mathematics. While fractal geometry may seem esoteric, this book demystifies it by providing a thorough introduction to its mathematical underpinnings and applications. Complete proofs are provided for most of the key results, and exercises of different levels of difficulty are proposed throughout the book. Key topics covered include the Hausdorff metric, Hausdorff measure, and fractal dimensions such as Hausdorff and Minkowski dimensions. The text meticulously constructs and analyzes Hausdorff measure, offering readers a deep understanding of its properties. Through emblematic examples like the Cantor set, the Sierpinski gasket, the Koch snowflake curve, and the Weierstrass curve, readers are introduced to self-similar sets and their construction via the iteration of contraction mappings. The book also sets the stage for the advanced theory of complex dimensions and fractal drums by gently introducing it via a variety of classical examples, including well-known fractal curves. By intertwining historical context with rigorous mathematical exposition, this book serves as both a stand-alone resource and a gateway to deeper explorations in fractal geometry.

Contents:

Part 1. Preliminary Material

Chapter 1. Introduction to Concepts in Fractal Geometry

1.1. Monsters

1.2. A Preliminary Fractal Toolbox

1.3. Notions of Dimension

1.4. Fractal Functions and Mathematical Monsters

1.5. Randomly Constructed Fractals

1.6. Complex Dynamics and Fractals: Julia Sets and the Mandelbrot Set

1.7. Chaotic Dynamics and Fractals: Strange Attractors

1.8. A Sample of Books on Fractal Geometry

Exercises

Chapter 2. Metric Spaces and Fixed Point Theorem

2.1. Definition and Properties

2.2. Analysis and Topology

2.3. Contraction Mapping Principle

Chapter 3. Measure Theory and Integrals

3.1. Elementary Families and Applications

3.2. Measures

3.3. Extension of a Premeasure to a Measure

3.4. Lebesgue Measure

3.5. Content and the Riemann Integral

3.6. Lebesgue Integration

Part 2. Dimension Theory

Chapter 4. Iterated Function Systems and Self-Similarity

4.1. Hausdorff Metric

4.2. Iterated Function Systems

4.3. Self-Similar Sets

4.4. Self-Affine Sets

4.5. Symbolic Dynamics: Words and Addresses

4.6. Lattice and Nonlattice Self-Similar Systems

4.7. Moran's Equation and Similarity Dimension

4.8. Self-Similar Measures

4.9. Perfect, Nowhere Dense Sets

4.10. Additional Examples

Chapter 5. Introduction to Hausdorff Measure and Dimension

5.1. Hausdorff Measure

5.2. Hausdorff Dimension

5.3. Hausdorff Dimension Calculations

5.4. Mass Distribution Principle and Moran's Theorem

5.5. Further Techniques for Calculating Hausdorff Dimension

5.6. Density and Hausdorff Measure of Products

Chapter 6. -Approximate Hausdorff Measures, via Carathéodory's Theory

6.1. A Few Generalities About Metric Spaces.

6.2. Metric Outer Measures and Their Associated Borel Measures

6.3. Construction and Properties of -Approximate Hausdorff Measures

Chapter 7. Construction and Properties of Hausdorff Measure

7.1. Construction of Hausdorff Measure, via Its -Approximate Counterparts

7.2. Comparison of Hausdorff Measures for Various Families

7.3. Key Geometric Properties of Hausdorff Measure

7.4. Lebesgue Measure vs. -Dimensional Hausdorff Measure on ℝ^{ }

7.5. Hausdorff Dimension in General Metric Spaces

Chapter 8. Minkowski Content and Minkowski Dimension

8.1. Minkowski Content and Dimension

8.2. Examples: The -String and the Cantor String (or Set)

8.3. Box-Counting Dimension and Content

8.4. Properties of Minkowski and Box-Counting Dimensions

8.5. Hausdorff versus Minkowski

8.6. Topological versus Hausdorff and Minkowski Dimensions

8.7. Dimensions of Attractors

8.8. Distinct Upper and Lower Box-Counting (or Minkowski) Dimensions

8.9. Generalized Hausdorff, Packing, and Minkowski Dimensions

Part 3. Fractal Curves and Their Complex Dimensions

Chapter 9. Epilogue: A Primer of Fractal Curves and Their Complex Dimensions

9.1. Continuous Curves: Their Lengths and Polygonal Approximations

9.2. Interchange of Lengths and Uniform or Hausdorff Limits

9.3. Interlude: An Introduction to Complex Dimensions and Fractal Zeta Functions

9.4. Devil's Staircase (or Cantor Graph): A Self-Affine and Rectifiable Fractal Curve

9.5. Koch Curve: A Self-Similar and Unrectifiable Curve

9.6. Sierpinski Gasket and Carpet: Universality and Beyond

9.7. Weierstrass Curve: A Continuous but Nowhere Differentiable Curve

Part 4. Appendices

Appendix A. Upper and Lower Limits

A.1. Upper and Lower Limits of Sequences

A.2. Upper and Lower Limits of Functions

Exercises.

Appendix B. Carathéodory's Approach to Measure Theory

B.1. Carathéodory's Measurability and Restriction Theorems

B.2. Completion of a Measure

B.3. Extension Theorems for Algebras and Semialgebras

B.4. Application to Lebesgue-Stieltjes and Cantor Measures

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

ISBN:

9781470478964

147047896X