Spectral Decimation for Families of Self-Similar Symmetric Laplacians on the Sierpinski Gasket

2017/09/07 by Sizhen Fang, Dylan King, Fang, Sizhen +5
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #31C45 (Primary) #42C99 (Secondary) #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1709.02031

openalex publication_date 2017/09/07 · openalex created_date 2017/09/15 · openalex updated_date 2026/07/28

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.

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