Stably weakly shadowing symplectomorphisms are partially hyperbolic

TL;DR

This paper proves that symplectomorphisms with a stable weak shadowing property on a closed symplectic manifold necessarily exhibit a partially hyperbolic structure, linking shadowing stability to dynamical hyperbolicity.

Contribution

It establishes a new connection between the C1-stable weak shadowing property and partial hyperbolicity for symplectomorphisms on closed manifolds.

Findings

C1-stably weakly shadowing implies partial hyperbolicity

Whole manifold admits a partially hyperbolic splitting

Results apply to symplectomorphisms on closed manifolds

Abstract

Let M be a closed, symplectic connected Riemannian manifold, f a symplectomorphism on M. We prove that if f is C1-stably weakly shadowing on M, then the whole manifold M admits a partially hyperbolic splitting.