Asymptotics for the spectral and walk dimension as fractals approach Euclidean space

B. M. Hambly*, T. Kumagai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

Original languageEnglish
Pages (from-to)403-412
Number of pages10
JournalFractals
Volume10
Issue number4
DOIs
Publication statusPublished - 2002 Dec
Externally publishedYes

Keywords

  • Sierpinski Carpet
  • Sierpinski Gasket
  • Spectral Dimension
  • Walk Dimension

ASJC Scopus subject areas

  • Modelling and Simulation
  • Geometry and Topology
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Asymptotics for the spectral and walk dimension as fractals approach Euclidean space'. Together they form a unique fingerprint.

Cite this