proof of Hartman-Grobman theorem

proof of Hartman-Grobman theorem

Lemma 1

.

Let

A

:

E

→

E

be an

hyperbolic isomorphism

, and let

φ

and

ψ

be

ε

-Lipschitz maps from

E

to itself such that

φ

⁢

(

0

)

=

ψ

⁢

(

0

)

=

0

.
If

ε

is sufficiently small, then

A

+

φ

and

A

+

ψ

are

topologically conjugate

.

Since

A

is hyperbolic, we have

E

=

E

s

⊕

E

u

and there is

λ

<

1

(possibly changing the norm of

E

by an

equivalent

box-type one), called the

skewness

of

A

,
such that

∥

A

|

E

s

∥

<

λ

,

∥

A

-

1

|

E

u

∥

<

λ

and

∥

x

∥

=

max

⁡

{

∥

x

s

∥

,

∥

x

u

∥

}

.

Let us denote by

(

E

~

,

∥

⋅

∥

0

)

the

Banach space

of all

bounded

,

continuous maps

from

E

to itself, with the norm of the

supremum

induced by the norm of

E

.
The

operator

A

induces a

linear operator

A

~

:

E

~

→

E

~

defined by

(

A

~

⁢

u

)

⁢

(

x

)

=

A

⁢

(

u

⁢

(

x

)

)

, which is also hyperbolic. In fact, letting

E

~

i

be the set of all maps

u

:

E

~

→

E

~

whose range is

contained

in

E

i

(for

i

=

s

,

u

) we have that

E

~

=

E

~

s

⊕

E

~

u

is a hyperbolic splitting
for

A

~

with the same skewness as

A

.

From now on we denote the projection of

x

to

E

i

by

x

i

, and
the

restriction

A

|

E

i

:

E

i

→

E

i

by

A

i

(

i

=

s

,

u

).

We will try to find a conjugation of the form

I

+

u

where

u

∈

E

~

.

Proposition 1

.

There exists

ε

>

0

such that if

φ

and

ψ

are

ε

-

Lipschitz

,
then there is a unique

u

∈

E

~

such that

(

I

+

u

)

⁢

(

A

+

φ

)

=

(

A

+

ψ

)

⁢

(

I

+

u

)

.

Proof.

We want to find

u

such that

A

+

φ

+

u

⁢

(

A

+

φ

)

=

A

+

A

⁢

u

+

ψ

⁢

(

I

+

u

)

which is the same as

φ

+

u

⁢

(

A

+

φ

)

=

A

⁢

u

+

ψ

⁢

(

I

+

u

)

.

This can be rewriten as

u

u

=

A

u

-

1

⁢

(

u

u

⁢

(

A

+

φ

)

+

φ

u

-

ψ

u

⁢

(

I

+

u

)

)

u

s

=

(

A

s

⁢

u

s

+

ψ

s

⁢

(

I

+

u

)

-

φ

s

)

⁢

(

A

+

φ

)

-

1

,

where we use the fact that by the

Lipschitz inverse mapping theorem

, if

Lip

⁡

(

φ

)

<

1

/

λ

≤

∥

A

-

1

∥

-

1

(where

λ

is the skewness of

A

) then

A

+

φ

is

invertible

with Lipschitz

inverse

.

Now define

Γ

:

E

~

→

E

~

by

Γ

s

⁢

(

u

)

=

(

A

s

⁢

u

s

+

ψ

s

⁢

(

I

+

u

)

-

φ

s

)

⁢

(

A

+

φ

)

-

1

Γ

u

⁢

(

u

)

=

A

u

-

1

⁢

(

u

u

⁢

(

A

+

φ

)

+

φ

u

-

ψ

u

⁢

(

I

+

u

)

)

We assert that, if

ε

is small,

Γ

is a

contraction

.
In fact,

∥

Γ

s

⁢

(

u

)

-

Γ

s

⁢

(

v

)

∥

0

=

∥

(

A

s

⁢

(

u

s

-

v

s

)

+

ψ

s

⁢

(

I

+

u

)

-

ψ

s

⁢

(

I

+

v

)

)

⁢

(

A

+

φ

)

-

1

∥

0

≤

∥

A

~

s

∥

⋅

∥

(

u

s

-

v

s

)

⁢

(

A

+

φ

)

-

1

∥

0

+

∥

(

ψ

s

⁢

(

I

+

u

)

-

ψ

s

⁢

(

I

+

v

)

)

⁢

(

A

+

φ

)

-

1

∥

0

≤

λ

⁢

∥

u

s

-

v

s

∥

0

+

ε

⁢

∥

u

-

v

∥

0

≤

(

λ

+

ε

)

⁢

∥

u

-

v

∥

0

and

∥

Γ

u

⁢

(

u

)

-

Γ

u

⁢

(

v

)

∥

0

=

∥

A

u

-

1

⁢

(

u

u

⁢

(

A

+

φ

)

-

v

u

⁢

(

A

+

φ

)

-

ψ

u

⁢

(

I

+

u

)

+

ψ

u

⁢

(

I

+

v

)

)

∥

0

≤

∥

A

~

u

-

1

∥

⋅

(

∥

u

u

⁢

(

A

+

φ

)

-

v

u

⁢

(

A

+

φ

)

∥

0

+

∥

ψ

u

⁢

(

I

+

u

)

-

ψ

u

⁢

(

I

+

v

)

∥

0

)

≤

λ

⁢

(

∥

u

u

-

v

u

∥

0

+

ε

⁢

∥

u

-

v

∥

0

)

≤

λ

⁢

(

1

+

ε

)

⁢

∥

u

-

v

∥

0

.

Thus, if

ε

<

ε

0

≐

min

⁡

{

λ

,

(

1

-

λ

)

/

λ

}

,

Γ

has Lipschitz constant
smaller than

1

, so it is a contraction. Hence

u

exists and is unique.
∎

Proposition 2

.

The map

u

from the previous

proposition

is a homeomorphism.

Proof.

Using the previous proposition with

φ

and

ψ

switched, we get a unique

v

∈

E

~

such that

(

I

+

v

)

⁢

(

A

+

ψ

)

=

(

A

+

φ

)

⁢

(

I

+

v

)

.

It follows that

(

I

+

v

)

⁢

(

I

+

u

)

⁢

(

A

+

φ

)

=

(

I

+

v

)

⁢

(

A

+

ψ

)

⁢

(

I

+

u

)

=

(

A

+

φ

)

⁢

(

I

+

v

)

⁢

(

I

+

u

)

.

(1)

Also, the previous proposition with

φ

=

ψ

implies that
that there is a unique

w

∈

E

~

such that

(

I

+

w

)

⁢

(

A

+

φ

)

=

(

A

+

φ

)

⁢

(

I

+

w

)

,

which obviously is

w

=

0

. But since

(

I

+

v

)

⁢

(

I

+

u

)

=

I

+

(

u

+

v

+

u

⁢

v

)

and

u

+

v

+

u

⁢

v

∈

E

~

,
(

1

) implies that

w

=

u

+

v

+

u

⁢

v

is a solution of the above equation, so that

u

+

v

+

u

⁢

v

=

0

and

(

I

+

v

)

⁢

(

I

+

u

)

=

I

. In a similar way, we see that

(

I

+

u

)

⁢

(

I

+

v

)

=

I

. Hence

I

+

u

is invertible, with

continuous

inverse.
∎

The two previous propositions prove the lemma.

Proposition 3

.

If

U

is an open neighborhood of

0

and

f

:

U

→

E

is a

C

1

map with

f

⁢

(

0

)

=

0

, then for every

ε

>

0

there is

δ

>

0

such that

φ

≐

f

-

D

⁢

f

⁢

(

0

)

is

ε

-Lipschitz in the ball

B

⁢

(

0

,

δ

)

.

Proof.

This is a direct

consequence

of the

mean value inequality

and the
fact that

D

⁢

φ

is continuous and

D

⁢

φ

⁢

(

0

)

=

0

.
∎

Proposition 4

.

There is a

constant

k

such that if

φ

:

B

¯

⁢

(

0

,

r

)

→

E

is an

ε

-Lipschitz map, then
there is a

k

⁢

ε

-Lipschitz map

φ

~

:

E

→

E

which coincides with

φ

in

B

⁢

(

0

,

r

/

2

)

.

Proof.

Let

η

:

ℝ

→

ℝ

be a

𝒞

∞

bump function

: an infinitely

differentiable map

such that

η

⁢

(

x

)

=

1

for

x

<

1

/

2

and

η

⁢

(

x

)

=

0

for

x

>

1

, with

derivative

bounded by

M

and

|

η

⁢

(

x

)

∥

≤

1

for all

x

∈

ℝ

.
Now define

φ

~

⁢

(

x

)

=

φ

⁢

(

x

)

⁢

η

⁢

(

∥

x

∥

/

r

)

(when

φ

⁢

(

x

)

is not
defined, we assume that it is zero).
If

x

and

y

are both in

B

⁢

(

0

,

r

)

then we have

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

)

∥

=

∥

φ

⁢

(

x

)

⁢

η

⁢

(

∥

x

∥

/

r

)

-

φ

⁢

(

y

)

⁢

η

⁢

(

∥

y

∥

/

r

)

∥

≤

∥

(

φ

⁢

(

x

)

-

φ

⁢

(

y

)

)

⁢

η

⁢

(

∥

x

∥

/

r

)

∥

+

∥

φ

⁢

(

y

)

⁢

(

η

⁢

(

∥

x

∥

/

r

)

-

η

⁢

(

∥

y

∥

/

r

)

)

∥

≤

ε

⁢

∥

x

-

y

∥

+

∥

φ

⁢

(

y

)

-

φ

⁢

(

0

)

∥

⋅

∥

η

⁢

(

∥

x

∥

/

r

)

-

η

⁢

(

∥

y

∥

/

r

)

∥

≤

ε

⁢

∥

x

-

y

∥

+

ε

⁢

∥

y

∥

⁢

(

M

⁢

∥

x

-

y

∥

/

r

)

≤

(

M

+

1

)

⁢

ε

⁢

∥

x

-

y

∥

;

if

x

is in

B

⁢

(

0

,

r

)

and

y

is not, then

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

)

∥

=

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

*

)

∥

,

where

y

*

is defined as

x

+

τ

⁢

(

y

-

x

)

with

τ

=

sup

⁡

{

t

:

x

+

t

⁢

(

y

-

x

)

∈

E

∖

B

⁢

(

0

,

r

)

}

This is true because

φ

~

⁢

(

y

*

)

=

0

. Also,

∥

x

-

y

*

∥

=

τ

⁢

∥

x

-

y

∥

≤

∥

x

-

y

∥

; hence

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

)

∥

=

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

*

)

∥

≤

(

M

+

1

)

⁢

ε

⁢

∥

x

-

y

*

∥

≤

(

M

+

1

)

⁢

ε

⁢

∥

x

-

y

∥

.

Finally, if both

x

and

y

are outside

B

⁢

(

0

,

r

)

, then

∥

φ

~

⁢

(

x

)

-

φ

~

⁢

(

y

)

∥

=

0

≤

(

M

+

1

)

⁢

∥

x

-

y

∥

. Letting

k

=

M

+

1

we
get the desired result.
∎

Proof of the

theorem

.

Taking the particular

ψ

=

0

in the lemma, we observe that
there is

ε

>

0

such that for any

ε

-Lipschitz map

φ

,

D

⁢

f

⁢

(

0

)

is conjugate to

φ

+

D

⁢

f

⁢

(

0

)

.
Choose

δ

such that

f

-

D

⁢

f

⁢

(

0

)

is

ε

/

k

-Lipschitz in

B

⁢

(

0

,

2

⁢

δ

)

.
Let

φ

~

be the

ε

-Lipschitz

extension

of

f

-

D

⁢

f

⁢

(

0

)

to

B

⁢

(

0

,

δ

)

obtained from the previous proposition. We have that

D

⁢

f

⁢

(

0

)

+

φ

~

is conjugate to

D

⁢

f

⁢

(

0

)

. But for

x

∈

B

⁢

(

0

,

δ

)

we have

D

⁢

f

⁢

(

0

)

+

φ

~

=

f

, so that

f

is locally conjugate to

D

⁢

f

⁢

(

0

)

.

Title

proof of Hartman-Grobman theorem

Canonical name

ProofOfHartmanGrobmanTheorem

Date of creation

2013-03-22 14:25:29

Last modified on

2013-03-22 14:25:29

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Proof

Classification

msc 37C25

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