An Analytic Proof of the Shadowing Lemma

The shadowing lemma is one of the fundamental technical results used in the study of axiom A dynamical systems and homoclinic phenomena.The idea first appears in Anosov's [1] studies of geodesic flows and was later formalized by Bowen [2] for axiom A systems.These early results were established using detailed information about the intersection of the stable and unstable manifolds associated to hyperbolic sets.Robinson [7] gave a simple geometric proof of this lemma along with a variety of interesting applications.The reader is referred to Robinson's paper for a more complete introduction and further references.In this paper we shall give a simple analytic proof of this lemma along with some applications.This simple proof facilitates a generalization of the shadowing lemma to skew product flows which is given in section mathrmImathrmV .For discrete dynamical systems the shadowing lemma is as follows.Let $f:Mrightarrow M$ be a C¹ diffeomorphism of a smooth manifold M endowed with some Riemannian metric.An invariant set $Lambdasubset M$ for f is called hyperbolic if there is a continuous splitting TM|Lambda=Eᵘoplus Eˢ and constants $C>0,0<mu<1$ such that (1) ||Dfⁿ(x)(v)||leq Cmuⁿ||v|| and ||Df⁻ⁿ(x)(u)||leq Cmuⁿ||u|| for all x $inLambda$ , vin EXˢ , uin EXᵘ and $ngeq 0$ .For $a>0$ an oc-pseudo-orbit for $ f|Lambda$ is a bisequence xᵢᵢ₌ⁱⁿᶠᵗʸᵢnfty of points XᵢinLambda such that d(f(xᵢ₋₁), xᵢ)<a for all i .One says that a mathrmf -orbit fⁱ(y) , $beta$ -shadows xᵢ if d(fⁱ(y), Xᵢ)<beta for all i .The Shadowing Lemma.If $Lambda$ is an compact, hyperbolic invariant set for $f:Mrightarrow M$ , then for every $beta>0$ there is an $a>0$ such that every $alpha-$ pseudo-orbit for $ f|Lambda$ is/3-shadowed by some f -orbit fⁱ(y) .Moreover, there is a beta₀>0 such that if 0<beta<beta₀ the f -orbit given above is uniquely determined by the $alpha-$ pseudo- orbit.This lemma will be proved in the next section.After a brief discussion of some applications, it is generalized to almost periodic flows in section mathrmImathrmV .This paper is dedicated to the memory of Lamont Cranston and the lovely Margo Lane.