Perturbations

Motivation

It is fairly clear that any substantial amount of symmetry in an n-gasket is unlikely to lead to a non-degenerate energy form on the infinite (Hilbert) gasket. It becomes a weighted variant of the averaging problem discussed in the constant case. Instead, we can start with a well-behaved finite gasket, and hope to perturb it using new edges with small conductances in such a way that we stay close to the structure of the smaller gasket but can arbitrarily increase the dimension using non-trivial conductances. In some sense, perturbation is the only option after the constant case, since conductances must go to zero in order for the energy form to be non-degenerate.

Numerical Estimates

We can use Maple to systematically obtain large amounts of numerical data on the behavior of conductances and renormalization factors for different combinations of scaling factors. We are principally concerned with the critical scaling factors near which there is a collapse onto a smaller gasket. Right now we have a lot of data, but don't understand it all that much except in the context of some of these collapses.

Here are some data (OpenOffice Calc). If they make sense to you, you're ahead of me.

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