Self-Similarity, Operators and Dynamics

We construct a large class of infinite self-similar (fractal, hierarchical or substitution) graphs and show, under a certain strong symmetry assumption, that the spectrum of the Laplacian can be described in terms of iterations of an associated rational function (so-called 'spectral decimation'). We prove that the spectrum consists of the Julia set of the rational function and a (possibly empty) set of isolated eigenvalues which accumulate to the Julia set. In order to obtain our results, we start with investigation of abstract spectral self-similarity of operators.