0
$\begingroup$

Why is the renormalization factor of the Dirichlet form for the graph approximations of the Sierpinski gasket (5/3)^m?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

The renormalization factor of the Sierpinski gasket is derived in Spectral Decimation Functions and Forbidden Eigen Values in the Graph of Level Sierpinski Triangles. It depends on the contraction ratio of the construction, the number $\alpha=5/3$ is for a contraction ratio of $1/2$. I am not aware of a simple intuitive argument for why $\alpha$ has that value. Note that a contraction ratio of $1/3$ has $\alpha=15/7$ and a contraction ratio of $1/4$ has $\alpha=103/41$ --- so it seems unlikely that these numbers can be "guessed" without a calculation.

Improve this answer
$\endgroup$
2
  • $\begingroup$ What calculation is done? $\endgroup$
    – TerryL
    Commented Aug 20, 2018 at 12:30
  • 1
    $\begingroup$ It's explained in the linked paper: you calculate an eigenvalue of the graph Laplacian. $\endgroup$
    – Carlo Beenakker
    Commented Aug 20, 2018 at 15:02

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .