Local behavior of harmonic functions on the Sierpinski gasket

Abstract

The local behavior of a harmonic function on the Sierpinski gasket in the neighborhood of a periodic point is governed by the eigenvalues of the $3 \times 3$ matrix that corresponds to zooming in to that point. We study the case when the matrix has complex conjugate eigenvalues. We develop a theory of local derivatives in this case. We give numerical evidence for the decay in relative frequency of this case, but we show how to construct infinitely many distinct points that fall into this case.