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Letters in Mathematical Physics

, Volume 88, Issue 1–3, pp 101–129 | Cite as

Fractal Strings and Multifractal Zeta Functions

  • Michel L. LapidusEmail author
  • Jacques Lévy-Véhel
  • John A. Rock
Open Access
Article

Abstract

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

Mathematics Subject Classification (2000)

Primary 11M41 28A12 28A80 Secondary 28A75 28A78 28C15 

Keywords

fractal string geometric zeta function complex dimension multifractal measure multifractal zeta functions perfect sets Cantor set 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Michel L. Lapidus
    • 1
    Email author
  • Jacques Lévy-Véhel
    • 2
  • John A. Rock
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA
  2. 2.Projet FractalesINRIA RocquencourtLe Chesnay CedexFrance
  3. 3.Department of MathematicsCalifornia State UniversityStanislaus, TurlockUSA

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