### $C^{m}$-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.

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.

#### Keywords

Hadamard-Perron Theorem; difference equations; nonautonomous; invariant fiber bundles; smoothness

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