On singularity of energy measures on self-similar sets

Abstract

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

“…The space of all harmonic functions is denoted by H. For any w ∈ W * and h ∈ H, ψ * w h ∈ H. By using the linear map ι : l(V 0 ) ∋ u → h ∈ H, we can identify H with l(V 0 ). In particular, H is a finite dimensional subspace of F , and the norm · F on H is equivalent to (ι 5) which is also considered as a square matrix of size #V 0 . Let I be the set of all constant functions on K, which is a one-dimensional subspace of H. Let Q be an arbitrary projection operator from H to I.…”

“…existence of derivatives with respect to some self-similar measures. Since self-similar measures and energy measures are mutually singular in many cases ( [5,7]), their result and ours are not comparable. (ii) Even when K is not a self-similar fractal, Theorem 5.4 should hold with suitable modification.…”

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

“…This measure valued Lagrangian takes on the fractal K the role of the Euclidean Lagrangian dL(u, v) = ∇u · ∇v dx. Note, that for most nested fractals, such as the Sierpinski gasket, the Lagrangian L K is not absolutely continuous with respect to the Hausdorff measure µ (see [28], [29], [5], [17] and [18]). Therefore, the Lagrangian approach turns out to be a very powerful tool to define energy forms on these fractals and their deformations.…”

“…Actually, for most nested fractals, one can employ only the Lagrangian methods due to the singularity of the Lagrangian with respect to the Hausdorff measure (see [28], [29], [5], [17] and [18]). Note that the "Lagrangian" is also called "energy measure", in particular in Kusuokas papers [28] and [29].…”

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

“…An interesting observation of Kusuoka ([11], see [2] for another proof) is that these measures are all singular with respect to the standard self-similar measure µ on K (the normalized Hausdorff measure), but they are all absolutely continuous with respect to a single measure ν, called the Kusuoka measure, defined below. See [5] and [6] for more recent singularity results.…”