Shadowing lemma
In the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit stays uniformly close to some true trajectory —in other words, a pseudo-trajectory is "shadowed" by a true one.Given a map f : X → X of a metric space to itself, define a ε-pseudo-orbit as a sequence of points such that belongs to a ε-neighborhood of.
Then, near a hyperbolic invariant set, the following statement holds:
Let Λ be a hyperbolic invariant set of a diffeomorphism f. There exists a neighborhood U of Λ with the following property: for any δ > 0 there exists ε > 0, such that any ε-pseudo-orbit that stays in U also stays in a δ-neighborhood of some true orbit.
