  A general invariant manifold theorem is needed to study the topological
classes of smooth dynamical systems. These classes are often invariant under
renormalization. The classical invariant manifold theorem cannot be applied,
because the renormalization operator for smooth systems is not differentiable
and sometimes does not have an attractor. Examples are the renormalization
operator for general smooth dynamics, such as unimodal dynamics, circle
dynamics, Cherry dynamics, Lorenz dynamics, H\'enon dynamics, etc. A general
method to construct invariant manifolds of non-differentiable non-linear
operators is presented. An application is that the $\mathcal C^{4+\epsilon}$
Fibonacci Cherry maps form a $\mathcal C^1$ codimension one manifold.
