proof of Hartman-Grobman theorem
Let us denote by the Banach space of all bounded, continuous maps from to itself, with the norm of the supremum induced by the norm of . The operator induces a linear operator defined by , which is also hyperbolic. In fact, letting be the set of all maps whose range is contained in (for ) we have that is a hyperbolic splitting for with the same skewness as .
From now on we denote the projection of to by , and the restriction by ().
We will try to find a conjugation of the form where .
There exists such that if and are -Lipschitz, then there is a unique such that
We want to find such that
which is the same as
This can be rewriten as
Now define by
We assert that, if is small, is a contraction. In fact,
Thus, if , has Lipschitz constant smaller than , so it is a contraction. Hence exists and is unique. ∎
The map from the previous proposition is a homeomorphism.
Using the previous proposition with and switched, we get a unique such that
It follows that
Also, the previous proposition with implies that that there is a unique such that
The two previous propositions prove the lemma.
If is an open neighborhood of and is a map with , then for every there is such that is -Lipschitz in the ball .
There is a constant such that if is an -Lipschitz map, then there is a -Lipschitz map which coincides with in .
Let be a bump function: an infinitely differentiable map such that for and for , with derivative bounded by and for all . Now define (when is not defined, we assume that it is zero). If and are both in then we have
if is in and is not, then
where is defined as with
This is true because . Also, ; hence
Finally, if both and are outside , then . Letting we get the desired result. ∎
Proof of the theorem. Taking the particular in the lemma, we observe that there is such that for any -Lipschitz map , is conjugate to . Choose such that is -Lipschitz in . Let be the -Lipschitz extension of to obtained from the previous proposition. We have that is conjugate to . But for we have , so that is locally conjugate to .
|Title||proof of Hartman-Grobman theorem|
|Date of creation||2013-03-22 14:25:29|
|Last modified on||2013-03-22 14:25:29|
|Last modified by||Koro (127)|