The essential coexistence phenomenon in Hamiltonian dynamics

I will discuss a phenomenon of the essential coexistence of regular (zero Lyapunov exponents and hence, zero entropy) dynamics and chaotic (non-zero Lyapunov exponents and positive entropy) dynamics in the setting of smooth dynamical systems. I will review some recent results in this direction, discuss some open problems and describe some new examples of essential coexistence. In particular, I will outline a construction of a Hamiltonian system which demonstrate a "complete? KAM-type picture of coexistence of invariant KAM tori and surrounding "chaotic sea".