Robust Transitivity in Hamiltonian Dynamics

Details Robust Transitivity in Hamiltonian Dynamics Robust Transitivity in Hamiltonian Dynamics

2011-08-30

arxiv.org/abs/1108.6012v2

Mathematics

Dynamical Systems

52 pages, 3 figures

Scientific paper

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.

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