Sabrina Kombrink

Parallel volumes and spectral counting functions for fractals

We will discuss two applications of dynamical renewal theory in geometry. Firstly, we will look at asymptotic expansions of parallel volumes of fractal sets – aiming at introducing notions of “fractal volume”, “fractal surface area”, “fractal curvatures”. Secondly, by finding higher order asymptotic terms of eigenvalue counting functions of the Dirichlet Laplacian on domains with a fractal boundary, we will address the question which geometric features of a drum with a fractal boundary one can hear when listening to its sound. The proofs rely on spectral properties of Ruelle-Perron-Frobenius operators.