Shadowing lemma

Question

I am trying to find a resource with a proof of the shadowing lemma (for maps: https://en.wikipedia.org/wiki/Shadowing_lemma) that is not too general. Ideally, a planar (or in $\mathbb{R}^n$) non-linear discrete dynamical system around a hyperbolic fixed point. This, to avoid all the differential geometry formalism, and focus on the main ideas of the proof, for now. Any good reference or ideas on the main structure of the proof? So far I could not even find a book with its proof, the only reference I got leaves it as an exercise... I tried to simplify things, and I got to prove it for contractions for example, and I think for linear maps one could go exploiting the splitting of $\mathbb{R}^n$ into a direct sum of the eigenspaces that are super and sub-unitary. But what about the non-linear case? Surely something concerning splitting into the Jacobian's eigenspaces...

Answer 1

I think you'll find all the details that you need in Chapter III of Eduard Zehnder's excellent book Lectures on Dynamical Systems.