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 (which one can think of as a numerically computed trajectory with rounding errors on every step) stays uniformly close to some true trajectory (with slightly altered initial position)—in other words, a pseudo-trajectory is 'shadowed' by a true one. 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 (which one can think of as a numerically computed trajectory with rounding errors on every step) stays uniformly close to some true trajectory (with slightly altered initial position)—in other words, a pseudo-trajectory is 'shadowed' by a true one. Given a map f : X → X of a metric space (X, d) to itself, define a ε-pseudo-orbit (or ε-orbit) as a sequence ( x n ) {displaystyle (x_{n})} of points such that x n + 1 {displaystyle x_{n+1}} belongs to a ε-neighborhood of f ( x n ) {displaystyle f(x_{n})} . 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 (finite or infinite) ε-pseudo-orbit that stays in U also stays in a δ-neighborhood of some true orbit.