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Full Description
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.
Contents
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
