Spectral decimation for families of self-similar symmetric Laplacians on the Sierpiński gasket

Abstract

We construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit Interval, and this yields an analogue of the classical Sturm-Liouville theory for the eigenfunctions of these one-dimensional Laplacians.

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

“…Hence, we obtain a one-parameter family of centrosymmetric probabilistic Laplacians and it is not difficult to see that this type of probabilistic Laplacians was already studied in [71] under the terminology of a pq-model, see also [11,23] where we set p 1 = p for some p ∈ (0, 1). Note that this type of probabilistic Laplacians is related to the graph Laplacians studied in [32]. Direct computation gives (5.9)…”

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

“…where we set p 1 D p for some p 2 .0; 1/. Note that this type of probabilistic Laplacians is related to the graph Laplacians studied in [41]. Direct computation gives…”