[Paper Review] Approximation of fractals by manifolds and other graph-like spaces

Olaf Post, Jan Simmer|arXiv (Cornell University)|Feb 8, 2018

Spectral Theory in Mathematical Physics23 references3 citations

TL;DR

This paper introduces a quasi-unitary equivalence framework to approximate energy forms on fractals—specifically pcf fractals—using sequences of energy forms on metric graphs and graph-like manifolds. By quantifying the distance between these energy forms via geometric and spectral quantities, the authors prove convergence of resolvents, spectra, and eigenfunctions, enabling rigorous approximation of fractal Laplacians by smooth geometric objects.

ABSTRACT

We define a distance between energy forms on a graph-like metric measure space and on a discrete weighted graph using the concept of quasi-unitary equivalence. We apply this result to metric graphs and graph-like manifolds (e.g. a small neighbourhood of an embedded metric graph) as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary) and express the distance of the associated energy forms in terms of geometric quantities. We showed in arXiv:1704.00064 that the approximating sequence of energy forms on weighted graphs used in the definition of an energy form on a pcf fractal converge in the sense that the distance in the quasi-unitary equivalence tends to 0. By transitivity of quasi-unitary equivalence, we conclude that we can approximate the energy form on a pcf fractal by a sequence of energy forms on metric graphs and graph-like manifolds. In particular, we show that there is a sequence of domains converging to a pcf fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. Quasi-unitary equivalence of energy forms implies a norm estimate for the difference of the resolvents of the associated Laplace operators. As a consequence, suitable functions of the Laplacians are close resp. converge as well in operator norm, e.g. the corresponding heat operators and spectral projections. The same is true for the spectra and the eigenfunctions in all above examples.

Motivation & Objective

• To develop a quantitative framework for approximating energy forms on fractals using discrete and continuous graph-like spaces.
• To establish a notion of distance between energy forms on different Hilbert spaces via quasi-unitary equivalence.
• To demonstrate that energy forms on pcf fractals can be approximated by sequences of energy forms on metric graphs and graph-like manifolds.
• To show convergence of spectral properties—resolvents, spectra, eigenfunctions, and heat operators—under this approximation.
• To express the approximation error in terms of geometric parameters such as edge lengths, volumes, and spectral gaps.

Proposed method

• Define a distance between energy forms on discrete weighted graphs and metric measure spaces using quasi-unitary equivalence, generalizing unitary equivalence.
• Construct identification operators $ J: \ell_2(V,\mu) \to L_2(X,\nu) $ and $ J': L_2(X,\nu) \to \ell_2(V,\mu) $ via partition of unity $ \psi_v $ with compact, connected supports.
• Use energy rescaling factor $ \tau $ and isometric rescaling factor $ c $ to align energy scales between graph and metric space.
• Bound the deviation $ \|f - J'Jf\|_{\ell_2} $ and $ \|u - JJ'u\|_{L_2} $ using graph weights $ \gamma_e $, vertex weights $ \mu(v) $, and measure $ \nu(v) $.
• Apply Poincaré-type inequalities and spectral estimates to control the second eigenvalue $ \lambda_2 $ of graph-like manifolds, ensuring uniform lower bounds independent of shrinking parameter $ \varepsilon $.
• Use scaling arguments and convergence results from [EP05] to derive uniform bounds on $ \lambda_2(X_{v,\varepsilon}) $ in terms of $ \ell_0, \ell_\infty, \varepsilon $, and geometric invariants.

Experimental results

Research questions

• RQ1 Can energy forms on pcf fractals be approximated by energy forms on metric graphs and graph-like manifolds?
• RQ2 What is the quantitative distance between energy forms on discrete graphs and continuous metric spaces?
• RQ3 How do the spectra and eigenfunctions of the associated Laplacians behave under such approximation?
• RQ4 What geometric conditions ensure uniform convergence of spectral properties in the approximation process?
• RQ5 Can the convergence of resolvents and heat operators be quantified via the quasi-unitary equivalence framework?

Key findings

• The energy form on a pcf fractal is approximated by sequences of energy forms on metric graphs and graph-like manifolds in the sense of $ \delta $-quasi-unitary equivalence, with $ \delta \to 0 $ as the approximation refines.
• The difference of resolvents of the associated Laplacians is bounded in operator norm by $ \delta $, implying convergence of all spectral functions such as heat operators and spectral projections.
• The second eigenvalue $ \lambda_2(X_{v,\varepsilon}) $ of a graph-like manifold $ X_{v,\varepsilon} $ is uniformly bounded below by $ \ell_0^{-2} $ for $ \varepsilon \leq \varepsilon_0 = C_v^{-2} \ell_0 $, independent of $ \varepsilon $.
• The approximation error $ \delta $ depends explicitly on graph weights $ \gamma_e $, vertex weights $ \mu(v) $, and geometric quantities like $ \nu(v) $, $ \operatorname{vol}Y_e $, and $ \lambda_2({\check{X}_v}) $.
• The convergence of eigenfunctions and spectra is guaranteed under quasi-unitary equivalence, with spectral gaps preserved uniformly in the limit.
• The framework applies to both Kirchhoff Laplacians on metric graphs and Neumann Laplacians on manifolds with boundary, enabling a unified approximation theory.