The main results one sees about the existence of invariant manifolds in dynamical systems (e.g. Theorem III.7 in Shub's book - Global stability of dynamical systems) usually require that the map we consider be either a diffeomorphism or, at least, a local diffeomorphism.
I am also looking in J. Jost's book, Dynamical Systems, where in chapter 2.10 he states theorems 4 and 5, which are similar to the one in Shub's book, but he makes no assumption on the map other than it be continuously differentiable. Also, he only provides a sketch of the proof and I can't find the reference he gives for a "more detailed explanation".
Is it necessary that we have (local-)diffeomorphisms or does the theorem hold for the case on non-invertible mappings too?