We explore the notion of two-sided limit shadowing property introduced by
Pilyugin \cite{P1}. Indeed, we characterize the C1-interior of the set of
diffeomorphisms with such a property on closed manifolds as the set of
transitive Anosov diffeomorphisms. As a consequence we obtain that all
codimention-one Anosov diffeomorphisms have the two-sided limit shadowing
property. We also prove that every diffeomorphism f with such a property on a
closed manifold has neither sinks nor sources and is transitive Anosov (in the
Axiom A case). In particular, no Morse-Smale diffeomorphism have the two-sided
limit shadowing property. Finally, we prove that C1-generic diffeomorphisms
on closed manifolds with the two-sided limit shadowing property are transitive
Anosov. All these results allow us to reduce the well-known conjecture about
the transitivity of Anosov diffeomorphisms on closed manifolds to prove that
the set of diffeomorphisms with the two-sided limit shadowing property
coincides with the set of Anosov diffeomorphisms.Comment: 10 page