Description:
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 Cr open sets (r = 1; 2; : : : ;1) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting large robustly transitive sets. We show that the C1 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".