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DCDS

We study various types of shadowing properties and their
implication for $C^1$ generic vector fields. We show that,
generically, any of the following three hypotheses implies that an
isolated set is topologically transitive and hyperbolic: (i) the
set is chain transitive and satisfies the (classical) shadowing
property, (ii) the set satisfies the limit shadowing property, or
(iii) the set satisfies the (asymptotic) shadowing property with
the additional hypothesis that stable and unstable manifolds of
any pair of critical orbits intersect each other. In our proof we
essentially rely on the property of chain transitivity and, in
particular, show that it is implied by the limit shadowing
property. We also apply our results to divergence-free vector
fields.

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