Robustly transitive behavior in symplectic dynamics

TL;DR

This paper demonstrates that certain symplectic systems, under specific conditions, can be deformed to produce large, robustly transitive sets, including examples not exhibiting uniform hyperbolicity.

Contribution

It introduces new perturbation techniques to create blender horseshoes in real-analytic symplectic systems, leading to robust transitivity without uniform hyperbolicity.

Findings

Existence of large robustly transitive sets in symplectic systems

Construction of real-analytic robustly transitive symplectomorphisms not uniformly hyperbolic

Development of perturbation methods for creating blender horseshoes

Abstract

We consider the direct product of two symplectomorphisms, one of which exhibits a basic set and the other one a non-degenerate elliptic equilibrium. Under a domination condition we show that a broad class of real-analytic deformations of this system display large robustly transitive sets. As a corollary of our construction we also obtain new examples of real-analytic robustly transitive symplectomorphisms which are not uniformly hyperbolic. To establish these results we develop perturbation techniques to create blender horseshoes in the real-analytic setting and import ideas from control theory which show that, typically, these objects have a large domain of influence.