A volume preserving dynamical systems with discrete or continuous time on a compact smooth manifold M is said to exhibit essential coexistence if M can be split into two invariant disjoint Borel subsets A and B of positive volume -- the chaotic and regular regions -- such that the Lyapunov exponents at every point in A are all nonzero (except for the Lyapunov exponents in the flow direction in the case of continuous time) while the Lyapunov exponents at every point in B are all nonzero and the restriction of the system on the set A is ergodic (Bernoulli).

There are two types of essential coexistence: type I, when the set A is open (mod0) and dense and type II, when the set B is open. 
I will discuss some general results, conjectures, and present some examples of systems with both discrete and continuous time which exhibit essential coexistence of type I. Finally, I will outline a construction of a Hamiltonian flow with essential coexistence of type I (based on a recent joint work with J. Chen, H. Hu, and Ke Zhang).