Martina Zähle: Fractal curvatures and dynamical systems

Classical notions of curvatures from differential geometry, convex geometry, geometric measure theory and algebraic geometry have been extended to fractal sets via approximation by parallel sets of small distances. The fractal curvatures are introduced as limits of the rescaled versions from geometric measure theory for the parallel sets, where the Minkowski content is included as marginal case. We give an introduction to these developments including some historical background for the classical cases. Then we indicate how the interplay between renewal theory and ergodic theory for associated dynamical systems leads to integral representations for the fractal curvature measures and for their densities with respect to the corresponding Hausdorff measures. Furthermore, the combination with martingale theory provides variants for certain random fractals.