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.