Criteria for Spectral Gaps of Laplacians on Fractals

Abstract

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove general criteria for the existence of gaps when the Laplacian admits spectral decimation. The known examples, including the Sierpinski gasket and the level-3 Sierpinski gasket, and the new examples including the fractal-3 tree, the Hexagasket and the infinite family of tree-like fractals satisfy the criteria.

Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals

“…Note that L, the degree of the rational function R(z), does not have to be equal to N , which is the number of contraction maps in Definition 1. Given w = w n ...w 1 , a word of length n = |w| on the letters 0, ..., L, we put [36,46,47] formore detail).…”

“…Investigation of the existence of gaps is important to analysis on fractals because of its many interesting applications, as mentioned in the introduction. Some criteria and examples are given in [46] and [47], although the verification can be tedious. In the next section we will derive a simple and easy to apply criterion based on the total disconnectedness of the Julia set of the spectral decimation function, generalizing the results of [46].…”

“…In [46], one of the authors gave general criteria for the existence of gaps in the spectrum of the Laplacian on fractals that admit spectral decimation and used this to establish the existence of gaps in many examples. All of the examples exhibit a common feature that the Julia set of the spectral decimation function (a rational function associated with the Laplacian) is totally disconnected.…”

“…Under the conditions of the theorem above, there exist gaps in σ(∆) if and only if J R is totally disconnected.This result can be generalized for a larger class of finitely ramified symmetric fractals with weights, and with more elaborate combinatorial structure, but doing so would require dealing with many technical details and auxiliary results which are not available in the existing literature. The main ideas and techniques behind these results come from [3,32,36,46]. One of the newest examples of fractals we consider can be found in [16].…”

Explicit Spectral Decimation for a Class of Self-Similar Fractals

“…The spectral decimation method has also shown to be a very useful tool for the analysis of the structure of the spectra of Laplacians of some fractals (e.g. [7,15,19]).…”