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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