Spectral Operators on Vicsek Sets

Denglin Zhou proved in his thesis that a class of fractals called Vicsek sets admit spectral decimation, an algorithm for recursively constructing eigenfunctions of a Laplacian operator. This page displays work done at the summer 2008 Cornell REU. During a summer 2008 Cornell math REU we implemented spectral decimation in MATLAB and used it to investigate eigenvalues, eigenfunctions, and spectral operators:

• Eigenvalues
  • Data
  • Eigenvalue counting function and Weyl ratios
  • Gaps in ratios
  • Clusters
• Eigenfunctions
  • Data
  • Restriction to the Diagonal
  • Limit as n goes to infinity
• Spectral Operators
  • Spectral Projections
  • Heat Kernel
  • Wave Propagator

Send any comments/suggestions to: mhw33@cornell.edu

Additional acknowledgements: Steven Heilman, Yingying Chan, Taryn Flock, Steve Gaarder, Bob Strichartz, Russ Thompson

Last Update: March 30, 2009