### Parallel volumes and spectral counting functions for fractals

Ergodic Theory and Dynamical Systems Seminar

2nd February 2023, 2:00 pm – 3:00 pm

Fry Building, G.07

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.

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