Analysis on Sierpinski triangle and similar fractals

The spectral analysis on some symmetric fractals can be completely described in terms of the complex dynamics of a polynomial or a rational function. Examples of such fractals include one dimensional fractals and the familiar Sierpinski gasket. I will discuss the spectral zeta function, complex spectral dimensions, and the spectral type of the lattice Laplacian. The spectral zeta function has a product structure that involves a geometric part and a certain new zeta function of a polynomial. A similar product structure of the spectral zeta function was studied for fractal strings by Lapidus et al. Although on one dimensional fractals the spectrum is singularly continuous, on the infinite Sierpinski gasket the Neumann Laplacian has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpinski gasket has no boundary, but is false for the Dirichlet Laplacian. In all these cases I will describe the spectrum of the Laplacian (in terms of the complex dynamics of a polynomial) and all the eigenfunctions with compact support. I will also discuss harmonic coordinates on the Sierpinski gasket and vector fields.