Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

Abstract

Abstract. On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E ϕ and conformal Laplacian ∆ ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n = 2. We derive a differential equation that describes the dependence of the effective resistances of E ϕ on ϕ. We show that the spectrum of ∆ ϕ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function ϕ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the L p dimensions of these measures for integer values of p ≥ 2. We derive analogous linear extension algorithms for energy measures on related fractals.

Energy forms on conformal C1‐diffeomorphic images of the Sierpinski gasket

“…Obviously, another measure on the right-hand side of (4.3) would create another Laplacian on G. The choice of the measure depends on the application one has in mind. If one thinks about a body K made from a certain material, the pull back measureμ is the natural choice in order to measure the "weight" on the deformed set G. However, other measures, leading to different Laplacians, have been considered in the literature (see [1], [25], [43] and [44]). Remark 4.6 From Theorem 4.3 it follows that there exists a strong Markovian process (X t ) t≥0 with continuous paths on G, which can be regarded as the "natural Brownian motion" on G (see e.g.…”

“…Moreover, the authors wish to express their thanks to the referees for pointing out the references [1], [5], [17], [18], [25], [27], [31], [38], [41], [43] and [44].…”

Weyl asymptotics for Hanoi attractors

“…Non strictly self-similarity can be obtained by introducing randomnes, as the case of homogeneous random p.c.f. fractals and carpets treated in [18,20], or constructing deterministic examples like self-conformal IFS's, treated in [5,13], fractafolds [37,39], fractal fields [21], or fractal quantum graphs [4].…”

“…This set is called the Hanoi attractor of parameter α and it is not self-similar because G α,4 , G α, 5 and G α, 6 are not similitudes. The quantity α should be understood as the length of the segments joining the copies G α,1 (K α ), G α,2 (K α ) and G α,3 (K α ).…”

Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket

“…The Kusuoka measure is not self-similar, and in fact it is singular with respect to the self-similar measure defined by equation (1.4) [11,12,3,5]. However, since it is defined in terms of the energy form, it deserves attention, and a number of recent papers have explored the properties of both ν and ν [2,10,18]. In this paper we will study the local behavior of functions in the domain of powers of ν , analogous to the theory of Taylor approximations on the real line.…”