2019/01/24 by Christian Loring, W. Jacob Ogden, Loring, Christian +5
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1901.08713
openalex publication_date 2019/01/24 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28
We study the analogue of polynomials (solutions to \Δn+1 u =0 for\nsome n) on the Sierpinski gasket (SG) with respect to a family of\nsymmetric, self-similar Laplacians constructed by Fang, King, Lee, and\nStrichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on\nthe polynomials with respect to the standard Kigami Laplacian. We define a\nbasis for the space of polynomials, the monomials, characterized by the\nproperty that a certain "derivative" is 1 at one of the boundary points, while\nall other "derivatives" vanish, and we compute the values of the monomials at\nthe boundary points of SG. We then present some data which suggest surprising\nrelationships between the values of the monomials at the boundary and certain\nNeumann eigenvalues of the family of symmetric self-similar Laplacians.\nSurprisingly, the results for the general case are quite different from the\nresults for the Kigami Laplacian.\n