[Paper Review] Dirichlet forms and critical exponents on fractals

Qingsong Gu, Ka‐Sing Lau|arXiv (Cornell University)|Mar 21, 2017

Mathematical Dynamics and Fractals16 references8 citations

TL;DR

This paper introduces a novel framework for analyzing critical exponents in Besov spaces on inhomogeneous p.c.f. fractals using quotient networks, revealing two distinct critical exponents σ* and σ# for asymmetric self-similar sets. It demonstrates that σ* < σ# can occur, challenging the conventional equality in standard fractals, and establishes the non-denseness of B²,∞σ* in C(K) for constructed asymmetric sets.

ABSTRACT

Let $B^σ_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $α$-regular measure $μ$. The {\it critical exponent} $σ^*$ is the supremum of the $σ$ such that $B^σ_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{σ^*}_{2, \infty}$ 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.

Motivation & Objective

• To investigate the behavior of critical exponents σ* and σ# in Besov spaces B²,∞σ on inhomogeneous p.c.f. fractals with non-uniform resistance scaling.
• To develop a quotient network technique for analyzing energy forms and resistance scaling on p.c.f. fractals with asymmetric structures.
• To construct explicit examples of asymmetric p.c.f. sets where σ* < σ# holds, challenging the standard assumption that σ* = σ#.
• To examine the functional properties of Besov spaces at critical exponents and the existence of local regular Dirichlet forms on such fractals.

Proposed method

• Introduce a quotient network method to model resistance scaling on p.c.f. fractals by collapsing substructures and analyzing induced resistances on boundary points.
• Use the discrete energy form Ej[u] on Vj to represent primal energy, with Ej[u] ∼ ρ^(−(2σ−α)j) [u]²_{B²,∞σ} for 2σ > α.
• Define the critical exponent σ* as sup{σ : B²,∞σ ∩ C(K) is dense in C(K)} and σ# as sup{σ : B²,∞σ contains non-constant functions}.
• Construct two asymmetric p.c.f. fractal sets with distinct resistance scalings to demonstrate cases where σ* < σ#.
• Analyze the trace of energy forms on boundary points and use recursive resistance scaling to derive asymptotic behavior of energy sequences.
• Apply graph-directed iterated function systems with non-strongly connected graphs to model fractals with mixed scaling behaviors, deriving box dimension via Perron-Frobenius eigenvalues.

Experimental results

Research questions

• RQ1 Can critical exponents σ* and σ# differ on p.c.f. fractals with inhomogeneous resistance scaling?
• RQ2 What functional properties do Besov spaces B²,∞σ* exhibit on asymmetric p.c.f. fractals, particularly regarding density in C(K)?
• RQ3 How does the quotient network technique enable analysis of energy and resistance scaling on non-uniform fractals?
• RQ4 Under what conditions does B²,∞σ contain non-constant functions when σ > σ*?
• RQ5 What is the role of graph-directed IFS with non-strongly connected components in generating fractals with different Hausdorff dimensions and infinite measures?

Key findings

• For the constructed asymmetric p.c.f. fractal, σ* < σ# holds, showing that the critical exponents can differ, contradicting the standard equality observed in symmetric fractals.
• The Besov space B²,∞σ* is not dense in C(K), as demonstrated by showing that any function in B²,∞σ* must be constant along certain directions (e.g., along p₁p₃), violating density.
• The energy sequence {7^n E_n[u]} is bounded for u ∈ B²,∞σ*, implying that u must be constant along the direction of p₁p₃, which contradicts density in C(K).
• For a non-strongly connected graph-directed system with λ₁ = λ₂, the box dimension remains α = log λ / |log ρ|, but H^α(E_i) = ∞, indicating infinite Hausdorff measure.
• The maximal eigenvalue λ of the adjacency matrix T determines the box dimension α = log λ / |log ρ|, and N_i(n) ∼ λ^n for i in a strongly connected component.
• When λ₁ ≠ λ₂, N_i(n) ∼ λ^n for i in the component with larger eigenvalue, and the box dimension is still α = log λ / |log ρ|, even if the measure is infinite in the case λ₁ = λ₂.