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Fractal Zeta Functions and Fractal Drums

Higher-Dimensional Theory of Complex Dimensions

  • Book
  • © 2017

Overview

Authors:
  1. Michel L. Lapidus

    1. Department of Mathematics, University of California, Riverside, USA

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  2. Goran Radunović

    1. Department of Applied Mathematics, University of Zagreb, Zagreb, Croatia

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    You can also search for this author in PubMed   Google Scholar

  3. Darko Žubrinić

    1. Department of Applied Mathematics, University of Zagreb, Zagreb, Croatia

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    You can also search for this author in PubMed   Google Scholar

  • beginning at the advanced graduate level
  • The exposition is gentle with numerous instructive examples and illustrations
  • The book builds on the one-dimensional theory of complex dimensions (the case of fractal strings) and builds towards as well as achieves a higher-dimensional theory of complex dimensions for arbitrary compact subsets of Euclidean spaces of any dimension
  • The content is self-contained and relatively easily accessible to a wide variety of readers with different levels of mathematical maturity

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xl
    Download chapter PDF

  2. Introduction

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 1-42

  3. Distance and Tube Zeta Functions

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 43-184

  4. Applications of Distance and Tube Zeta Functions

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 185-244

  5. Relative Fractal Drums and Their Complex Dimensions

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 245-405

  6. Fractal Tube Formulas and Complex Dimensions

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 407-538

  7. Classification of Fractal Sets and Concluding Comments

    • Michel L. Lapidus, Goran Radunović, Darko Žubrinić
    Pages 539-576

  8. Back Matter

    Pages 577-655
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Keywords

About this book

This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions.  


The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.

Authors and Affiliations

  • Department of Mathematics, University of California, Riverside, USA

    Michel L. Lapidus

  • Department of Applied Mathematics, University of Zagreb, Zagreb, Croatia

    Goran Radunović, Darko Žubrinić

Bibliographic Information

  • Book Title: Fractal Zeta Functions and Fractal Drums

  • Book Subtitle: Higher-Dimensional Theory of Complex Dimensions

  • Authors: Michel L. Lapidus, Goran Radunović, Darko Žubrinić

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-44706-3

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2017

  • Hardcover ISBN: 978-3-319-44704-9Published: 26 June 2017

  • Softcover ISBN: 978-3-319-83115-2Published: 28 July 2018

  • eBook ISBN: 978-3-319-44706-3Published: 07 June 2017

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XL, 655

  • Number of Illustrations: 45 b/w illustrations, 10 illustrations in colour

  • Topics: Number Theory, Measure and Integration, Mathematical Physics

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