Fractal theory of Saturn's ring

The true reason, in our opinion, for the partition of Saturn's ring as well as the rings of other planets into a large number of small subrings is found. This reason is clarified by the Zelikin–Lokutsievskiy–Hildebrand theorem about the fractal structure of solutions to generic piecewise smooth Hamiltonian systems. The instability of the two-dimensional model of the ring with continuous surface density of the distribution of particles is proved both for the Newton and Boltzmann equations. We do not claim that we have solved the problem of stability of Saturn's ring. We rather put questions and suggest some ideas and means for future research.