Dirichlet forms and critical exponents on fractals

Abstract

Let B σ 2,∞ denote the Besov space defined on a compact set K ⊂ R d which is equipped with an α-regular measure µ. The critical exponent σ * is the supremum of the σ such that B σ 2,∞ ∩ C(K) is dense in C(K). It is well-known that for many standard selfsimilar sets K, B σ * 2,∞ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied.2010 Mathematics Subject Classification. Primary 28A80; Secondary 46E30, 46E35.

On a recursive construction of Dirichlet form on the Sierpiński gasket

“…The first one is the twisted SG introduced by Mihai and Strichartz [23], it is a modification of the IFS of the SG that reflecting the three subcells of the SG along the angle bisectors at the three vertices. We show that for a 0 > b 0 = c 0 , the closure of V * under the (effective) resistance metric has interesting topology different from the SG; the second one is from [8], it is called a Sierpinski sickle, which is the attractor of an IFS of 17 similitudes and three boundary points, of which the recursive construction does not yield a compatible sequence for a Dirichlet form.…”

“…The recursive construction can be extended to more general p.c.f. sets (see [8] for some examples), but it also have limitation. In Section 4, we give two other examples that this construction have abnormality.…”

“…It follows that Recall that the spectral dimension d s of a Dirichlet form is defined to be lim if the limit exists. Heuristically, d s /2 = d f /d w where d f is the Hausdorff dimension of K, and d w is the walk dimension of K. The walk dimension is the space-time relation E x (|X t − x| 2 ) ≈ t 2/d w of the associate diffusion process [29], which is also the critical exponent of the Besov space corresponding to the domain of the Dirichlet form [7,8,15] (see Section 4). From Theorem 3.5, we see that for a 0 > b 0 , d s = log 9 log(9/2) , d w = log 9 log 2 − 1.…”

“…In this section, we consider two more examples. The first one is a modification of the SG such that for a 0 > b 0 = c 0 , the closure of V * under the resistance metric is different from the SG; the second one is detailed in [8], it is a p.c.f. set constructed by an IFS of 17 maps with three boundary points, of which the recursive construction does not yield a compatible sequence of Dirichlet forms E n , n ≥ 0.…”

“…Recently two of the authors studied some anomalous p.c.f. fractals in regard to the domains of the Dirichlet forms and the associated Besov spaces [8]. In their investigation, a construction of the non-self-similar energy form was considered, and some interesting properties were found (see Section 4).…”