Shape and Spectrum: On the Heat and Volume of Self-Similar Fractals

TL;DR

This paper explores the deep connection between the geometric structure and spectral properties of self-similar fractals, revealing that their volume and heat content asymptotics are governed by shared complex dimensions.

Contribution

It introduces a unified framework linking the geometry and spectrum of fractal drums via complex dimensions and scaling functional equations.

Findings

Complex dimensions determine volume asymptotics.

Heat content asymptotics are governed by the same complex dimensions.

Scaling ratios influence both geometric and spectral properties.

Abstract

In this work, we examine the relationship between geometry and spectrum of regions with fractal boundary. The relationship is well-understood for fractal harps in one dimension, but largely open for fractal drums in larger dimensions. To that end, we study fractals arising as attractors of self-similar iterated function systems with some separation conditions. On the geometric side, we analyze the tube zeta functions and their poles, called complex dimensions, which govern the asymptotics of the volume of tubular neighborhoods of such fractals. On the spectral side, we study a Dirichlet problem for the heat equation, closely related to spectrum of the Laplacian. We show that the asymptotics of the total heat content are controlled by the same set of possible complex dimensions. Our method is to establish scaling functional equations and to solve by means of truncated Mellin transforms, wherefrom the scaling ratios of the underlying dynamics can be seen to govern both the geometry and spectra of these self-similar fractal drums.