The growth of eigenfunction extrema on p.c.f. fractals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918510781792256 |
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| author | Qiu, Hua Tian, Haoran |
| author_facet | Qiu, Hua Tian, Haoran |
| contents | This paper studies the growth of local extrema of Laplacian eigenfunctions on post-critically finite (p.c.f.) fractals. We establish the sharp two-sided estimate $\#\mathrm{Extr}(u_λ)\asympλ^{d_S/2}$ for the Sierpinski gasket, demonstrating that the complexity of eigenfunctions is governed by the spectral dimension $d_S$. This behavior stands in sharp contrast to the corresponding growth law on Euclidean $n$-dimensional rectangles or balls. The attainment of the exponent $d_S/2$ reflects the high symmetry of the underlying fractal. Our result reveals a distinct spectral-geometric phenomenon on singular spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The growth of eigenfunction extrema on p.c.f. fractals Qiu, Hua Tian, Haoran Functional Analysis 28A80, 31E05 This paper studies the growth of local extrema of Laplacian eigenfunctions on post-critically finite (p.c.f.) fractals. We establish the sharp two-sided estimate $\#\mathrm{Extr}(u_λ)\asympλ^{d_S/2}$ for the Sierpinski gasket, demonstrating that the complexity of eigenfunctions is governed by the spectral dimension $d_S$. This behavior stands in sharp contrast to the corresponding growth law on Euclidean $n$-dimensional rectangles or balls. The attainment of the exponent $d_S/2$ reflects the high symmetry of the underlying fractal. Our result reveals a distinct spectral-geometric phenomenon on singular spaces. |
| title | The growth of eigenfunction extrema on p.c.f. fractals |
| topic | Functional Analysis 28A80, 31E05 |
| url | https://arxiv.org/abs/2511.04027 |