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Monatshefte für Mathematik

, Volume 121, Issue 1–2, pp 11–40 | Cite as

On the magnification of cantor sets and their limit models

  • Tim Bedford
  • Albert M. Fisher
Article

Abstract

For aC1+γ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC1+γ diffeomorphic withC (for some γ>0, γ∈(0,1)), but not all suchC1+γ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's “scaling function”. An alternative definition of a fractal is discussed.

1991 Mathematics Subject Classification

28A80 58F12 

Key words

Fractals Cantor sets scenery scaling 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Tim Bedford
    • 1
  • Albert M. Fisher
    • 2
  1. 1.Department of MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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