Spectral decimation of a self-similar version of almost Mathieu-type operators

Abstract

We introduce and study self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians {Δp}p∈(0,1) instead of the standard discrete Laplacian and includes the classical almost Mathieu operators as a particular case, namely, when the Laplacian’s parameter is p=12. Our main result establishes that the spectra of these self-similar almost Mathieu operators can be described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. The spectral-type of the self-similar Laplacians used in our model is singularly continuous when p≠12. In these cases, the self-similar almost Mathieu operators also have singularly continuous spectra despite the periodicity of the potentials. In addition, we derive an explicit formula of the integrated density of states of the self-similar almost Mathieu operators as the weighted pre-images of the balanced invariant measure on a specific Julia set.