Let $T\colon R^n\to R^n$ be a linear map. If we want to study the behavior of $T^kx$ for some $x\in R^n$ as integer $k$ grows, we usually look at the eigen structure of $T$.
Now let $S\colon R^n\to R^n$ be a linear map plus a nonlinear perturbation. And I want to study the behavior of $S^kx$ for some $x\in R^n$ as integer $k$ grows. I am wondering if there exists a theory that discusses this kind of problem.
Yes, there is such a theory, and you already gave yourself the answer in the tag. For particular shape of your map $S$, i.e. a linear hyperbolic map (meaning: the spectrum is disjoint from the unit circle) $T$ plus a small (in a sense to be precised) Lipschitz perturbation , the Hartman-Grobman tells you that $S$ is conjugated with $T$ by a Hölder continuous homeomorphism. The proof is very elementary and follows from the contraction principle; see e.g. M.Shub's book, Global stability of Dynamical Systems.
Suppose there is a linear $T$ with $\| Sx - Tx \| = o(\|x\|)$ as $x \to 0$, and all eigenvalues of $T$ have absolute value $< 1$. Then there exist positive integer $N$ and positive reals $\delta$ and $\epsilon$ such that for $\|x\| < \delta$, $\|S^N x\| \le (1 - \epsilon) \|x\|$. It follows that for $\|x\|$ sufficiently small, $S^k x \to 0$ as $k \to \infty$. Was that the sort of result you were looking for?
