Springer Monographs in Mathematics

Fractal Zeta Functions and Fractal Drums

Higher-Dimensional Theory of Complex Dimensions

Authors: Lapidus, Michel, Radunović, Goran, Žubrinić, Darko

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  • 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, beginning at the advanced graduate level
  • The exposition is gentle with numerous instructive examples and illustrations
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eBook 107,09 €
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  • ISBN 978-3-319-44706-3
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Hardcover 135,19 €
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  • ISBN 978-3-319-44704-9
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  • Usually dispatched within 3 to 5 business days.
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Softcover 135,19 €
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  • ISBN 978-3-319-83115-2
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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.

Table of contents (6 chapters)

Table of contents (6 chapters)

Buy this book

eBook 107,09 €
price for Spain (gross)
  • ISBN 978-3-319-44706-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 135,19 €
price for Spain (gross)
  • ISBN 978-3-319-44704-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 135,19 €
price for Spain (gross)
  • ISBN 978-3-319-83115-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Fractal Zeta Functions and Fractal Drums
Book Subtitle
Higher-Dimensional Theory of Complex Dimensions
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-44706-3
DOI
10.1007/978-3-319-44706-3
Hardcover ISBN
978-3-319-44704-9
Softcover ISBN
978-3-319-83115-2
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XL, 655
Number of Illustrations
45 b/w illustrations, 10 illustrations in colour
Topics