Invariant Manifolds for Non-differentiable Operators

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.