Abstract: As a counterpart of the Arnold diffusion, the essential coexistence of regular and hyperbolic
behavior provides an alternative picture, that is, the KAM tori are surrounded by chaotic sea. Although
the coexistence phenomena are numerically observed in many mechanical systems, a general theory is
far out of reach. In this talk, we first review some results on the essential coexistence phenomena in
volume-preserving category, whose constructions are based on the perturbation techniques in partially
hyperbolic systems. Then we focus on a recent result by Huyi Hu, Yakov Pesin, Ke Zhang and myself,
which provides a construction of essential coexistence in a 4-dimensional Hamiltonian systems. The
key ingredient is an explicit embedding the Katok map.