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

Abstract

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E φ \mathcal {E}_\varphi and conformal Laplacian Δ φ \Delta _{\varphi } for a given conformal factor φ \varphi , based on the corresponding notions in Riemannian geometry in dimension n ≠ 2 n\neq 2 . We derive a differential equation that describes the dependence of the effective resistances of E φ \mathcal {E}_\varphi on φ \varphi . We show that the spectrum of Δ φ \Delta _{\varphi } (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function φ \varphi 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 L^{p} dimensions of these measures for integer values of p ≥ 2 p\geq 2 . We derive analogous linear extension algorithms for energy measures on related fractals.