Riemannian and Finslerian spheres with fractal cut loci

The present paper shows that for a given integer k≥2k≥2 it is possible to construct an at least k-differentiable Riemannian metric on the sphere of a certain dimension such that the cut locus of a point of it becomes a fractal. Moreover, we show that this construction can be extended to the case of Finslerian spheres as well.