$C^{m}$-smoothness of invariant fiber bundles

Abstract

The method of invariant manifolds, now called
the Hadamard-Perron Theorem, was originally developed
by Lyapunov, Hadamard and Perron for time-independent maps and
differential equations at a hyperbolic fixed point. It was
then extended from hyperbolic to non-hyperbolic systems, from
time-independent and finite-dimensional to
time-dependent and infinite-dimensional equations.
The generalization of an invariant manifold for a discrete dynamical
system (mapping) to a time-variant difference equation is called
an invariant fiber bundle.
While in the hyperbolic case the smoothness of the invariant
fiber bundles is easily obtained with the contraction principle, in the
non-hyperbolic situation the smoothness depends on a spectral gap condition,
is subtle to prove and proofs were given under various assumptions
by basically three different approaches, so far:
(1) A lemma of Henry, (2) the fiber-contraction theorem, or
(3) fixed point theorems for scales of embedded Banach spaces.

In this paper we present a new self-contained and basic proof of the
smoothness of invariant fiber bundles which relies only on Banach's fixed point
theorem.
Our result extends previous versions of the Hadamard-Perron Theorem
and generalizes it to the time-dependent, not necessarily hyperbolic, infinite-dimensional,
non-invertible and parameter-dependent case.
Moreover, we show by an example that our gap-condition is sharp.