On singularity of energy measures on self-similar sets
For the classical Brownian motion in \(\mathbb{R}^d\), the associated Dirichlet form \((\mathcal{E}, \mathcal{F})\) on \(L^2(\mathbb{R}^d, dx)\) is given by \[ \mathcal{E}(f,g)=\frac{1}{2} \int_{\mathbb{R}^d} \langle \nabla f, \nabla g\rangle \,dx, \quad f,g \in \mathcal{F}=H^1(\mathbb{R}^d). \] Hence, the corresponding energy measure \(\mu_{\langle f \rangle}:=| \nabla f| ^2 \,dx\), \(f \in \mathcal{F}\), is absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}^d\). In general, this is no longer true if the underlying space is a fractal. The paper provides a general criterion for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type volume measures including the Hausdorff measure. Roughly spoken, the energy measure is singular to the canonical volume measure (i.e. the Hausdorff measure) if the elliptic Harnack inequality holds and the walk dimension is greater than 2. A special feature of the paper is that the considerations include infinitely ramified fractals. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on two-dimensional Sierpinski carpets.
scientific article; zbMATH DE number 2173777
scientific article