  A goal of this work is to study the dynamics in the complement of KAM tori
with focus on non-local robust transitivity. We introduce $C^r$ open sets
($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems,
exhibiting "large" robustly transitive sets. We show that the $C^\infty$
closure of such open sets contains a variety of systems, including so-called a
priori unstable integrable systems. In addition, the existence of ergodic
measures with large support is obtained for all those systems. A main
ingredient of the proof is a combination of studying minimal dynamics of
symplectic iterated function systems and a new tool in Hamiltonian dynamics
which we call symplectic blender.
