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

Hua Qiu, Haoran Tian

Abstract

This paper studies the growth of local extrema of Laplacian eigenfunctions on post-critically finite (p.c.f.) fractals. We establish the precise two-sided estimate $N(u)\asympλ^{d_S/2}$ for the Sierpinski gasket, demonstrating that the complexity of eigenfunctions is governed by the spectral exponent $d_S$. This stands in sharp contrast to the general $λ^{(n-1)/2}$ law on smooth manifolds, with the attainment of the exponent $d_S/2$ reflecting the high symmetry of the underlying fractal. Our result reveals a distinct spectral-geometric phenomenon on singular spaces.

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

Abstract

This paper studies the growth of local extrema of Laplacian eigenfunctions on post-critically finite (p.c.f.) fractals. We establish the precise two-sided estimate for the Sierpinski gasket, demonstrating that the complexity of eigenfunctions is governed by the spectral exponent . This stands in sharp contrast to the general law on smooth manifolds, with the attainment of the exponent reflecting the high symmetry of the underlying fractal. Our result reveals a distinct spectral-geometric phenomenon on singular spaces.

Paper Structure

This paper contains 14 sections, 25 theorems, 154 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Propositions \ref{['prothm1']} and \ref{['prothm2']}
  3. asymptotic estimate of the extremum counting function
  4. Preminaries
  5. Proofs of Propositions \ref{['prothm1']} and \ref{['prothm2']}
  6. An equivalent condition of (A)
  7. Eigenfunctions on the Sierpinski Gasket
  8. The spectral decimation method
  9. Statements of Theorems \ref{['thm4']} and \ref{['thm5']}
  10. Proof of Theorem \ref{['thm4']}
  11. The first step
  12. The second step
  13. Proof of Theorem \ref{['thm5']}
  14. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

There exists a constant $C>1$ such that holds for any global Dirichlet or Neumann eigenfunction $u$ on $\mathcal{SG}$ with eigenvalue $\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.

Figures (5)

  • Figure 1: The Sierpinski gasket $\mathcal{SG}$.
  • Figure 2: The modified Koch curve.
  • Figure 3: The Sierpinski gasket $\mathcal{SG}$ and the set $V_1$.
  • Figure 4: $\mathcal{A}_\lambda$ (the shaded open equilateral triangle), $\mathcal{B}_\lambda^1$ (the thickened Y-shaped line segments, with endpoints removed), and $\mathcal{G}_{i}:=\mathcal{G}_{\psi^{-1}(\lambda),i}=(\mathcal{T}_\lambda^i)^{-1}\mathcal{A}_{5^{-1}\lambda}$, $i\in S$ (the three small open triangles).
  • Figure 5: $(\mathcal{P}_\alpha^1)^{-1}(\mathcal{D}_\alpha)$ (the shaded region), and the outer circle represents the line at infinity.

Theorems & Definitions (50)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['prothm1']}
  • proof : Proof of Proposition \ref{['prothm2']}
  • ...and 40 more