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.