There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let M be a closed Riemannian manifold. Dieomorphism f : M \to M is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. A similar denition can be given for flows.

Most results about this property prove that it is present in certain hyperbolic situations. Quite surprisingly, recently it has been proven that a quantitative version of it is in face equivalent to hyperbolicity (structural stability).

There is also a notion of inverse shadowing that is a kind of a converse to the notion of classical shadowing. Dynamical system is said to have inverse shadowing property if for any (exact) trajectory there exists a pseudotrajectory from a special class that is uniformly close to the original exact one.

I will describe a quantitative (Lipschitz) version of this property and why it is equivalent to structural stability both for dieomorphisms and for flows.