Analysis on Sierpinski n-Gaskets
PCF-IFS's and the 3-Gasket
The familiar Sierpinski gasket is part of a large class of fractals called post-critically finite iterated
function systems, or pcf-ifs's for short. (Say pick-fiffs.) The `pcf' part roughly means
that the fractal can be
disconnected by the removal of finitely many carefully selected points, i.e. the fractal is finitely
ramified, and was first defined and exploited by Kigami. The `ifs' part indicates that the fractal is
obtained as the limit of iterations of a set of contractive similarities applied to an initial set of
vertices and edges.
Let us examine what this means for the Sierpinski gasket. Recall that the standard construction
of the Sierpinski gasket begins with a triangle with vertices q0, q1, and q2, which we call
generation zero. We then define three
contractive similarities---F0, F1, and F2---which fix the respective vertices of the initial
triangle, and reduce the side-lengths by a factor of 1/2. If we take the union of the images of the
original triangle under these three maps, we get what appears to be the original triangle, with a
smaller upside-down triangle inside cut out of it. Repeating this operation on our new object gives
somthing that looks quite similar, but with smaller copies of our first generation object in place
of the old smaller triangles. We call each step of this process a graph approximation to the
Sierpinski 3-gasket. As one increases the generation, one gets closer and closer to the structure
of the complete gasket. One can gain a lot of information about the gasket from the first two graph
approximations (generations zero and one), and it is primarily these at which we look in this
project.
The n-Gasket
In general, we define an analogue to the Sierpinski gasket by taking the complete graph on n
vertices, containing all possible edges, and use n contractive similarities, each one fixing one
of the n vertices. This can be hard to picture for n greater than 4 (in which case we have a nice
friendly tetrahedron). It is easier to not worry about embedding the n-Gasket into any particular
space, and to just concern oneself with the various vertex and edge relationships. If you really
want an embedding, there is a nice one in R^n. Consider section of a hyperplane defined by
x1+x2+...+xn=1, with all of the xi non-negative. The gasket is then the set of points where the xi
are summed in binary with no carries. For example, a point in SG4 might look like
x1=0.00010101110000...
x2=0.10001000001100...
x3=0.01000010000001...
x4=0.00100000000010...
You may also think of n-Gaskets as Sierpinski simplices. This approach has the advantage of being
alliterative, but not much else, so far as I can tell. Why do we use complete graphs instead of
other graphs to seed the fractal? One reason is that this guarantees that the limit of the ifs will
actually be pcf and a fractal. This is not the case, for instance, if we use n-cycles. Complete
graphs also have a very simple combinatorial structure and are thus quite easy to describe for
computations and proofs.
Differential Equations
We know enough about the behavior of the 3-gasket to make sense of many concepts from continuous
analysis on differential equations. We start by defining a graph energy on each graph
approximation, and take a limit as the approximations approach the gasket. Graph energies are
precisely the energies of the corresponding electrical networks, with resistors at each edge and
voltages given by the values of a function at each point of the graph. Any method of
manipulating resistors, such as combining those in series or in parallel, continues to apply
on the gasket without affecting the energies of harmonic functions (those minimizing energy
for a given set of boundary values). To each energy form we associate a measure which
assigns weights to the different cells of the fractal. A cell is an image of our initial
triangle under some finite list of contractive similarities. There are other measures as
well which assign weights to junction points on the fractal. Once we have an energy form on
the gasket with an associated measure, we can define a second derivative using integration by
parts. Measures allow us to compute integrals, and energy can be thought of as an inner
product taking the integral of the products of derivatives of two functions. If we force the
boundary terms of our function to equal 0, then the energy of a function will be equal to the
inner product taken instead on the Laplacian of the function and the function itself. We can
also define a graph Laplacian on each graph approximation to the gasket, and this can be shown
to be equivalent to the other formulation. One can also define two types of derivatives,
normal and tangential, based on our familiar concepts of differentiation, and there is a
similar concept of a gradient given by approximations of functions by harmonic functions.
Our Problem
Our problem on n-gaskets has been in developing a meaningful energy form with which to
understand differential equations. We can define graph energies with resistances as above, but
it is not clear which such definitions give non-degenerate energy forms as n tends to infinity.
The first stage of the project investigated extensibility of some nice properties derived
from the electrical network model of graph energies. The second stage investigated some
twisting phenomena which have proved useful in understanding energies on the 3-gasket. The
current stage of the project has been to investigate energy forms on small perturbations of the
unit interval and the 3-gasket in an attempt to build new energy forms from old ones.
Problem-within-a-Problem: Renormalization
Given a set of boundary values on an n-gasket with vertices labled 0--n-1 and edge conductances
cij between vertices i and j, we can easily determine a harmonic extension h of the boundary
values onto lower and lower levels of the gasket. That is, we can supply values of h to new
vertices such that the graph energy is minimized over all possible such values. Ideally, we
want the new energy to be the same as the old one, so we must scale our initial conductances so
that this happens. The factor by which we scale them is the solution to the renormalization
problem. On the 3-gasket, there are several simple methods for solving this problem to varying
degrees of abstraction. The basic idea is that the graph energy is the sum over all pairs of
adjacent vertices x and y of cxy(h(x)-h(y))^2, where cxy is the conductance on the edge between
x and y. This is true on every level of the graph, so we have an equivalent expression for
level one. The energy of the harmonic extension, however, depends only on the level zero
vertices. Thus, we can write the level one energy as a sum of kxy(h(x)-h(y))^2 where x and y
are adjacent on level zero and k is some constant. For all x and y we must have kxy/cxy equal
to the same constant, which is the reciprocal of the renormalization factor.
One can quickly see that not all combinations of conductances lead to soluble renormalization
equations. In fact, if we just do the construction we've been describing thusfar, the only solution
is to have the cxy constant, with the renormalization factor equal to n/(n+2) on SGn (the
n-gasket). We thus introduce scaling factors ri corresponding to the contractive similarities
Fi and divide each conductance by the corresponding scaling factor each time the edge is mapped
by a given contractive similarity. Then there is at most one set of conductances for each set
of scaling factors which gives a renormalizable energy form, according to a theorem of Sabot's.
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