Spectral analysis of Laplacians on the Vicsek set

Abstract

We find the spectral decimation function for the standard Laplacian on the symmetric Vicsek set, expressed in terms of Chebyshev polynomials. This allows us to determine the order of the eigenvalues of the Laplacian, describe their asymptotic behavior and prove that there exist gaps in the spectrum.

The "hot spots" conjecture on the Vicsek set

“…In order to study eigenvalues and eigenfunctions on V S 2 , we use the process of spectral decimation as described in [9,21] that we review next. First, there is a local extension algorithm which shows a unique way to extend a function u that satisfies the eigenvalue equation…”

“…and h 2 (λ) = 6λ − 5. Zhou proved in [21] (see also [9,22]) that the spectral decimation function R is…”

“…The Vicsek set [18][19][20] is another type of fractal that has been studied heavily. Zhou [21] described the spectral decimation on a family of Vicsek sets, V S n , in terms of Chebyshev polynomials, and the authors of [22] used Zhou's formulas to study the Laplacian and spectral operators on the Vicsek set. We study in this paper the Vicsek set V S 2 that is generated by five contractions; however, its analytic boundary consists of only four of the five fixed points of the iterated function systems.…”

“…with the domain of the energy, dom E, consisting of all continuous functions u on V S 2 such that E(u) < ∞. As detailed in [8,9,21], the choice of the renormalization constant in the definition of E m guarantees that the domain of the energy is non-trivial. Now, knowing that the energy exists and can be written as above, we know from [8,9] that we can determine a fractal's Laplacian.…”

Criteria for Spectral Gaps of Laplacians on Fractals

“…Theorem 13 can also be applied to the infinite family of Vicsek sets. We refer the readers to [20] for more details. Fig.…”