Abstract. Let M be a closed, symplectic connected Riemannian mani-fold and f a symplectomorphism on M. We prove that if f is C 1 -stablyweak shadowable on M, then the whole manifold M admits a partiallyhyperbolic splitting. 1. Introduction, basic definitions and statement of the results1.1. IntroductionThe language of differential equations (discrete or continuous) is very usefulwhen we want to model phenomena in various applications. In this contextthe stability of our model is essential in order to the modeling be useful andconsequently implemented. Hyperbolicity is widely acknowledged to be a keyingredient for stability. Nevertheless, it is an old problem in smooth dynamicsto perceive how the stability of a certain property implies some hyperbolic-type of behavior on the tangent map of the system. The celebrated structuralstability conjecture is the most important example of that (see e.g. [23]). Thestability of several properties like topological conjugacy, topological stability,shadowing, expansiveness, specification, hyperbolic periodic orbits, etc, hasbeen a debated issue in recent years. Here, we are interested in a shadowing-like property.The notion of shadowing applied to dynamics, introduced by Bowen [10] inthe context of hyperbolic dynamics, is motivated by the numerical computa-tional idea of estimating differences between true and approximate solutionsalong orbits and to understand the influence of the errors that we commit andallow on each iterate. In rough terms, we may ask if it is possible to obtainshadowing of approximate trajectories in a given dynamical system by trueorbits of the system. We refer the reader to the monograph by Pilyugin’s [21]for a fairly complete exposition on the subject.