Spectral asymptotics for Laplacians on self-similar sets

Abstract

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a 'geometric counting function' defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.

Spectrum and Heat Kernel Asymptotics on General Laakso Spaces

“…This paper will be concerned instead with the irregular domain being a fractal itself. Some notable works with this type of domain include [3,6,8,16,17,19] among others. Laakso's spaces were introduced in [11].…”

Energy and Laplacian on Hanoi-type fractal quantum graphs

“…We follow a similar argument as in [Kaj10,Lemma 4.5], which is included for completeness: consider L 0 := { w∈A m a w 1 Xw : a w ∈ R}. This is a 3 m −dimensional subspace of Dom E X A m such that E X A m | L 0 ×L 0 ≡ 0.…”

“…Call u A the corresponding eigenfunction, renormalized so that X A m u 2 A dµ = 1. Since (E , H 1 (X)) is a resistance form on X, the associated resistance metric R is compatible with the original topology of X by Theorem 5.2, and u A is orthogonal to L 0 , a uniform Poincaré inequality (see [Kaj10,Definition 4.2] for the self-similar case) holds for u A . This together with equality (6.2) leads to…”

Weyl asymptotics for Hanoi attractors

“…The self-similar case was first discussed in [7,27,28] on p.c.f. sets and later [26] discussed results for the Sierpiński carpet. Non strictly self-similarity can be obtained by introducing randomnes, as the case of homogeneous random p.c.f.…”

“…The proof of this result is based on the minimax principle for the eigenvalues of non-negative self-adjoint operators and it follows ideas of [26]. Details about the minimax principle can be found in [10, Chapter 4].…”