To define the Sierpinski Gasket more precisely we use three fixed point contractions F0, F1, F2 which merely translate points half the distance toward the three boundary points of the SG, q0, q1, and q0, respectively. All the repeated iterates of the three contractions produce the SG in the limit after taking the closure.
"Periodic" refers to the address of a point. If, for example, you have a point F0F1(q0) its address is "01". A cell is a subset of the SG produced by permutations of the contractions on the SG. So the point I just referred to is in the "01" cell. A periodic point has an address with a repeated string of digits in its address. I concentrated on the periodic point with address "01010101..." This case is actually six cases due to the symmetry of the SG.
This case is the simplest to first analyze. In computing the harmonic functions we'll need matrices M0,M1,M2 corresponding to the 3 contractions. What is important are the eigenvalues. The eigenvalues of M0*M1 are all real: {1., 14.729932, 4.7145120}. But the periodic point 012 uses eigenvalues of M0*M1*M2 which are complex {1., 23.148148 + 6.5472851 i, 23.148148 - 6.5472851 i}. This would be a good case to study later. Other cases have real eigenvalues