Why is the renormalization factor of the Dirichlet form for the graph approximations of the Sierpinski gasket (5/3)^m?
1 Answer
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.
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1$\begingroup$ It's explained in the linked paper: you calculate an eigenvalue of the graph Laplacian. $\endgroup$ Commented Aug 20, 2018 at 15:02