Smooth stable invariant manifolds and arbitrary growth rates

For nonautonomous linear equations v′=A(t)vv′=A(t)v with a generalized exponential dichotomy, we show that there is a smooth stable invariant manifold for the perturbed equation v′=A(t)v+f(t,v)v′=A(t)v+f(t,v) provided that ff is sufficiently small. The generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t)ρ(t). We consider the general case of nonuniform exponential dichotomies, and the result is obtained in Banach spaces. Moreover, we show that for an equivariant system, the dynamics on the stable manifold in a certain class of graphs is also equivariant. We emphasize that this result cannot be obtained by averaging over the symmetry.