Linear-Like Distributions of Energy on the Sierpinski Gasket

Let K denote the Sierpinski gasket. For a harmonic function h, we define an energy measure by

where, since h is harmonic,

where the are the boundary values of h on K. The measure of all of K is the energy of h and we compute the measure of general sets S by summing over the measure of cells contained S.

For a general measure on K, we say that is self-similar if there exist constants so that

where . Clearly, an energy measure is not self-similar in this sense, but there is a linear algebraic analog of self-similarity where, instead of constants determining measures of subcells in terms of the cells containing it, we have matrices expressing the measure of the three m-cells in a (m-1) cell in terms of the (m-1)-cell and the two (m-1) cells adjacent to it, much like the harmonic extension matrices give the values of a harmonic function on the boundary of subcells. More specifically,

,

or more generally,

.

The matrices are

.

Using these matrices have simplified MATLAB calculations considerably, and are helpful for more theoretical work as well. For example, the energy matrices are used in proving the following theorem;

Theorem: Let m be an integer. Then for all harmonic functions h,

where the m-cell of maximum measure will be contained in the 1-cell of maximum measure.

The proof, though rather arduous, involves an induction argument on the level of cells: when one considers the m-cells, those are (m-1)-cells of , which therefore reduces the induction step to looking at the nine corner m-cells of , and by symmetry of K and similar considerations, one need only look at comparing three cells, and diagonalizing the energy matrices to extract a lower bound for the difference.

The existence of such matrices is due to the fact that the energies of the three zero cells, as functions of their boundary values, form a basis spanning all polynomials of the form . Can we do this for other fractals, or particularly, for (the level zero cells are shown above, and energies for a harmonic function h are defined as before). The answer is yes, sort of. It turns out that the cells 0,1,2 and 3,4,5 are two sets of linearly independent energy polynomials. In fact, the energies in the middle are fixed by the ones in the corners. In particular,

More specifically,

So it suffices to find energy matrices that compute the corner energy cells of each subcell. If we let the above matrix be C and e the three-dimensional vector of corner energies 0,1, and 2 (ordered as above), we find that

Where

and can be computed from these using the symmetry of . The matrix maps the three corner energies of to the three corner energies of the j cell, and composing with C gives the center energies 3,4,5 of that cell.

We dealt with the fact that there were more than three cells in this case by using the fact that half the cells were fixed by the others in a convenient way, exploiting an interesting inner symmetry. However, not all cases are as convenient as this. The tetrahedral Sierpinski Gasket has four level zero cells, each with energies whose polynomials are now six dimensional, and so there aren’t enough energies in the first cells to span the rest. However, we can go to the next level, which has 16 cells, more than enough to find a basis. However, if we have to pick six cells for a basis from a sixteen cell structure, we can no longer use the symmetry of the fractal to compute the rest of the matrices from one or two, but they can (with long hours of toying with MATLAB) be computed. As before with , we find matrices that send the measure of our six cells to the measure of the six cells respectively in the j cell. However, this only accounts for six cells, but the other 10 (sigh…) can be found since the six cells form a basis and hence will span the others.

The moral of the story is that, for particular kinds of fractals, we can find matrices that generate a linear extension algorithm of the energy measures on them by replicating the process above and adapting them according to the fractal. The simplest and most elegant example of these is the Sierpinski gasket, and after some study, it is clear that the simplicity is inherent and unique.