Self-Similar Energies on Post-Critically Finite Self-Similar Fractals

Abstract

On a large class of post-critically finite (finitely ramified) self-similar fractals with possibly little symmetry, we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal, we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared with previous results, our technique allows us to replace symmetry by connectivity arguments.

Existence of self-similar energies on finitely ramified fractals

“…Statement (S 2 ) is an immediate consequence of Lemma 6.4 ii). C (1) j is separated from C (22) j,l by (6.4), and moreover it is separated from C (3) l . In fact, as C (3) …”

“…Then a) The set G (1) j,n , is separated from all sets of G but itself and G e) The set G (7) j,l,n is separated from the sets G…”

“…j,l,n , and (ϖ 0 , ...ϖ m ), ϖ s ∈ G (5) n , is a path connecting ϖ to some element ϖ ′ of G (4) l,n , then the first element ϖ s of the path in G (4) l,n in fact belongs to G (1) j,n . To see this, let n 2 ϖ i := Π n,n 2 (ϖ i ).…”

“…It suffices to observe that C (1) j ∪ C (21) j,l is a path, and that, by definition, every point of C (7) j,l is connected to j (n 2 ) ∈ C (1) j by a path in C (7) j,l . Thus, in any of the sets…”

“…self-similar sets see [4]. Conversely, there exists a standard way to associate to a given fractal triple a self-similar fractal K, and also an n-fractal triple corresponding to the previously defined set Ψ n of one-to-one maps and generating the same fractal K. Namely, given a fractal triple F := (V (0) , V (1) , Ψ), Ψ = {ψ 1 , ..., ψ k }, we can define a related fractal, F in the following way: Let…”