An Invitation to Fractal Geometry: Fractal Dimensions, Self-Similarity, and Fractal Curves

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Michel L. Lapidus Goran Radunovic

An Invitation to Fractal Geometry

Fractal Dimensions, Self-Similarity, and Fractal Curves

Michel L. Lapidus, Goran Radunovic

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English

American Mathematical Society
31 January 2025

Calculus & mathematical analysis

Series: Graduate Studies in Mathematics

This book offers a comprehensive exploration of fractal dimensions, self-similarity, and fractal curves. Targeting 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.

By: Michel L. Lapidus, Goran Radunovic
Imprint: American Mathematical Society
Country of Publication: United States
Volume: 247
Dimensions: Height: 254mm, Width: 178mm,
ISBN: 9781470476236
ISBN 10: 1470476231
Series: Graduate Studies in Mathematics
Pages: 611
Publication Date: 31 January 2025
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Preliminary material Introduction to concepts in fractal geometry Metric spaces and fixed point theorem Measure theory and integrals Dimension theory Iterated function systems and self-similarity Introduction to Hausdorff measure and dimension $\delta$-Approximate Hausdorff measures, via Caratheodory's theory Construction and properties of Hausdorff measure Minkowski content and Minkowski dimension Fractal curves and their complex dimensions Epilogue: A primer of fractal curves and their complex dimensions Appendices Upper and lower limits Caratheodory's approach to measure theory Acknowledgments Bibliography Index of symbols Author index Subject index

Michel L. Lapidus, University of California, Riverside, CA, and Goran Radunovic, University of Zagreb, Croatia