$C^r$-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms

Abstract

We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by non-uniformly hyperbolic diffeomorphisms.

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• On invariant holonomies between centers
• Homoclinic points of symplectic partially hyperbolic systems with 2D centre