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

TL;DR

This paper investigates how the number of local extrema of Laplacian eigenfunctions on p.c.f. fractals, especially the Sierpinski gasket, grows with eigenvalue size, revealing a spectral dimension-dependent behavior.

Contribution

It establishes a sharp two-sided estimate for the growth of eigenfunction extrema on p.c.f. fractals, highlighting the influence of spectral dimension and fractal symmetry.

Findings

Number of extrema scales as λ^{d_S/2} on the Sierpinski gasket

Eigenfunction complexity is governed by spectral dimension

Contrasts with growth laws on Euclidean domains

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.