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arXiv · 2511.04027

The growth of eigenfunction extrema on p.c.f. fractals

Hua Qiu·Haoran Tian
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Abstract

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 $\#\mathrm{Extr}(u_\lambda)\asymp\lambda^{d_S/2}$ 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.

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Hua Qiu, Haoran Tian. 2025-11-06. The growth of eigenfunction extrema on p.c.f. fractals. https://arxiv.org/abs/2511.04027

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