Gaps in the spectrum of the Laplacian on $3N$-Gaskets
| Autor: | Jason Marsh, Daniel J. Kelleher, Maxwell Margenot, William Oakley, Nikhar Gupta, Alexander Teplyaev |
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| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
Polynomial Geodesic FOS: Physical sciences 01 natural sciences 010305 fluids & plasmas Mathematics - Spectral Theory Fractal Mathematics - Metric Geometry 0103 physical sciences FOS: Mathematics Mathematics::Metric Geometry 0101 mathematics Spectral Theory (math.SP) Mathematical Physics Heat kernel Mathematics Decimation Applied Mathematics Probability (math.PR) 010102 general mathematics Metric Geometry (math.MG) Mathematical Physics (math-ph) Mathematics::Spectral Theory Primary 81Q35 60J35 28A80 Secondary 31C25 31E05 35K08 Functional Analysis (math.FA) Sierpinski triangle Mathematics - Functional Analysis Metric (mathematics) Laplace operator Mathematics - Probability Analysis |
| Zdroj: | Communications on Pure and Applied Analysis. 14:2509-2533 |
| ISSN: | 1534-0392 |
| DOI: | 10.3934/cpaa.2015.14.2509 |
| Popis: | This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum. |
| Databáze: | OpenAIRE |
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