On Complex Dimensions and Heat Content of Self-Similar Fractals

TL;DR

This paper demonstrates that complex fractal dimensions, derived from fractal zeta functions, also govern the heat content asymptotics of self-similar fractals, extending previous results to more general non-arithmetic cases.

Contribution

It establishes a link between complex fractal dimensions and heat content asymptotics for self-similar fractals, including non-arithmetic cases, broadening the understanding of fractal geometry and heat diffusion.

Findings

Complex dimensions determine heat content asymptotics.

Extension of results to non-arithmetic self-similar fractals.

Application to generalized von Koch snowflakes.

Abstract

Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allow for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and in particular extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting.