Poincaré functional equations, harmonic measures on Ju- lia sets, and fractal zeta functions

Connections between the analysis on fractals and the iteration of rational functions were discovered in the earliest publications on diffusion processes on certain self-similar sets, such as the Sierpiński gasket (see, for instance [3, 38]). The connection stems from the fact that time on the successive approximations of the fractal is modelled by a branching process. The relation of branching processes to the iteration of holomorphic functions is known for a long time (see [19]). More precisely, in order to obtain a diffusion on a fractal, define a sequence of random walks on approximating graphs and synchronise time so that the limiting process is non-constant and continuous. This was the first approach to the diffusion process on the Sierpiński gasket given in [3, 14, 26] and later generalised to other “nested fractals” in [29]. In our description we will follow the lines of definition of self-similar graphs given in [24, 25] and adapt it for our purposes. We consider a graph G = (V (G), E(G)) with vertices V (G) and undirected edges E(G) denoted by {x, y}. We assume throughout that G does not contain multiple edges nor loops. For C ⊂ V (G) we call ∂C the vertex boundary, which is given by the set of vertices in V (G)\C, which are adjacent to a vertex in C. For F ⊂ V (G) we define the reduced graph GF by V (GF ) = F and {x, y} ∈ E(GF ), if x and y are in the boundary of the same component of V (G) \F . This requires that removing the set F disconnects the graph G into different components. The following definition is taken from [25]. It is motivated by the properties of the infinite Sierpiński gasket (see Figure 1). Furthermore, it will turn out that this definition of self-similarity of a graph is reflected by according functional equations for the Green function (the generating function of the transition probabilities) and by rational function relations between the eigenvalues of the transition Laplace operator, which will be exploited later.