LECTURE NOTES ON HADAMARD-PERRON THEOREM
In these notes we discuss the stable and unstable manifold theorem at a hyperbolic
fixed point. The treatment follow Katok-Hasselblatt, (denoted K-H from now on),
but the proof of contraction mapping uses the version in Brin-Stuck (B-S), and there
are some variations from both sources.
1. Introduction
Our setting is an m-dimensional manifold M, an open set U ⊂ M, and a diffeomorphism f : U → M. Assume that p ∈ U is a periodic point of f , i.e. f N (p) = p
for some n ∈ N.
Definition. A periodic point p of period N is called hyperbolic if the linear map
D( f N )(p) : Tp M → Tp M has no eigenvalue on the unit circle.
We also list here the related definition for flows to highlight some differences. Let
f t : M → M be a flow with vector field F (x).
Definition.
• An equilibrium point p of the flow f t is called hyperbolic if the
linear map D( f t ) : Tp M → Tp M has no eigenvalue on the unit circle for all
t , 0.
• A periodic point p of f t with period T is called hyperbolic if D( f T )(p) :
Tp M → Tp M has a simple eigenvalue χ = 1, and no other eigenvalues on
the unit circle.
In the second case, F (x) is always an eigenvector of D( f T )(p) with χ = 1. We
also have p is a hyperbolic periodic point for the associated Poincaré return map.
Given an m × m hyperbolic matrix A, let sp( A) denote the set of all eigenvalues
for A, sp− ( A) = { χ ∈ sp( A) : | χ| < 1}, and sp+ ( A) = { χ ∈ sp( A) : | χ| > 1}. We
define
µ( A) = inf{| χ| : χ ∈ sp+ ( A)},
λ( A) = sup{| χ| : χ ∈ sp− ( A)},
and E ± ( A) = ⊕ χ∈sp± ( A) E χ ( A), where
E χ ( A) = {v ∈ Rm : ( A − χI) k v = 0 for some k ∈ N}
is the root space for the eigenvalue χ.
In general, given 0 < λ < µ, we say the spectrum of A admits a (λ, µ)-splitting if
sp( A) = { χ ∈ sp( A) : | χ| ≤ λ} ∪ sp( A) = { χ ∈ sp( A) : | χ| ≥ µ}.
We still denote the splitting of the spectrum sp± ( A) and define E ± ( A) in the same
way.
1
2
LECTURE NOTES ON HADAMARD-PERRON THEOREM
Proposition 1 (K-H, Proposition 1.2.2). Assume that A admits a (λ, µ)-splitting.
Then for every > 0, there exists a norm k · k on Rm such that
k Avk ≤ (λ + )kvk,
v ∈ E − ( A),
k A−1 vk ≤ (µ−1 + )kvk,
v ∈ E + ( A).
Proof. We write P AP −1 = J, where J = diag{ J1 , · · · , Js } is the Jordan normal form.
For each Jordan block Ji of size k, we check that
 λ 1
 1


 

...
 λ
  
..

..


.

. 1 

 
(k−1)

λ 

 λ 


..
.
λ


= 
...




λ 
Let Λ = diag{Λ1 , · · · , Λs }, then the norm kvk = kΛPvk∞ , where k · k∞ is the sup
norm on Rm , works.
1

 −1

−1


Λi Ji Λi := 
..
.


−(k−1)


Theorem 2 (See K-H, Theorem 6.2.3). Let p be a hyperbolic fixed point of a C l diffeomorphism f : U → M, with l ≥ 1. Then for each δ > 0, there exists
C l -embeded discs W p+ ,W p− ⊂ U with the following properties:
(1) p ∈ W + ,W − , TpW + = E + , TpW − = E − .
(2) W − is forward invariant and W + is backward invariant.
(3) There exist C(δ) > 0 such that for any p1 ∈ W − (p), p2 ∈ W + , n ≥ 0,
dist( f n p1 , p) ≤ C(δ)λ(d f (p)) n d(p1 , p),
dist( f −n p2 , p) ≤ C(δ)(µ−1 (d f (p)) + δ)d(p2 , p).
(4) There exists r 0 > 0 such that if f n p1 ∈ Ur0 (p) for all n ≥ 0, then p1 ∈ W − p;
if f −n (p) ∈ Ur0 (p) for all n ≥ 0, then p2 ∈ W + (p).
2. Local maps
By using a coordinate chart ϕ : Rm → V 3 p, we may consider the map
f 0 = ϕ−1 ◦ f ◦ ϕ : Rm → Rm . Denote k = dimE − . By taking an additional affine
linear coordinate change, we may assume 0 is a hyperbolic fixed point for f 0 , and
E − (D f 0 (0)) = {0} × Rm−k , E + (D f 0 (0)) = R k × {0}. Then the local map f 0 takes
the form
f 0 (x, y) = ( Ax + α(x, y), Bx + β(x, y)),
x ∈ R k , y ∈ Rm−k ,
LECTURE NOTES ON HADAMARD-PERRON THEOREM
3
k A−1 k ≤ µ−1 (D f 0 (0)) + , kBk ≥ λ(D f 0 (0)) + , α, β vanishes up to order one at
(0, 0). In particular, for any σ > 0, there exists r > 0, such that kαkC 1 (Br ) , k βkC 1 (Br ) <
σ.
It suffices to prove Theorem 2 for the map f 0 | Br . We will, however, prove a
general theorem that also covers the case of “uniformly hyperbolic orbits”. Let
f : M → M be a C 1 diffeomorphism of a compact manifold M.
Definition. Given 0 < λ < µ, we say the full orbit { f n (p)}n∈Z of f admits a uniform
(λ, µ)-splitting if there exist a Riemannian metric on M with the following properties.
• For each n ∈ Z, there exists a splitting T f n (p) M = En+ ⊕ En− of constant
dimensions k and m − k, that is invariant under d f . This means
±
d f ( f n p)(En± ) = En+1
,
n ∈ Z.
• Let k · kn denote the norm on tangent vectors given by the Riemannian metric
at the nth spot. Then
kd f ( f n p)vk ≤ λ kvk,
v ∈ En− , n ∈ Z
k(d f ( f n−1 )) −1 v)k ≤ µ−1 kvk, v ∈ En+ , n ∈ Z.
When λ < 1 < µ the orbit is called uniformly hyperbolic.
In the neighborhood of each f n (p), we consider local coordinate ϕn with ϕn (0) =
f n (p), and obtain the local maps f n = ϕn+1 ◦ f ◦ ϕn . We may further change
coordinates such that
dϕn (0)({0} × Rm−k ) = En− ,
dϕn (0)(R k × {0}) = En+ ,
The spaces Rm for each ϕn inherits its own norm k · kn , however, they are uniformly
equivalent to the standard norm. We will omit the subscript in the norm, as which
norm to apply will be clear from context. Furthermore, given any σ > 0, there exists
r > 0 such that
f n (x, y) = ( An x + α n (x, y), Bn y + βn (x, y)),
with k An−1 k ≤ µ−1 , kBn k ≤ λ, and kα n kC 1 (Br ) , k βn kC 1 (Br ) < σ. We now state the
theorem in terms of local maps.
Theorem 3 (Hadamard-Perron, see K-H Theorem 6.2.8, B-S Proposition 5.6.1).
Let 0 < λ < µ with µ > 1. Given C 1 -diffeomorphisms f n : Rm → Rm such that
f n (x, y) = ( An x + α n (x, y), Bn y + βn (x, y)) with k An−1 k ≤ µ−1 , kBn k ≤ λ and
α(0, 0) = β(0, 0) = 0.
Then for any δ > 0, there exists σ > 0 such that if kα n kC 1 (Br ) , k βn kC 1 (Br ) < σ,
for some r > 0, then there exists a unique family of C 1 -manifolds
Wn+ = {(x, φ+n (x)) : x ∈ R k } = graph(φ+n )
verifying the following:
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LECTURE NOTES ON HADAMARD-PERRON THEOREM
+ .
(1) (backward invariance) ( f n−1 ) −1Wn+ ⊂ Wn−1
(2) (backward contraction) k( f n−1 ) −1 (z1 ) − ( f n−1 ) −1 (z2 )k ≤ (µ−1 + δ)kz1 − z2 k
for z1 , z2 ∈ Wn+ .
(3) (criterion for unstable manifold) If for any λ < λ 0 < µ, and C > 0 we have
k( f n− j ) −1 ◦ · · · ( f n ) −1 (z)k ≤ min{C(λ 0 ) −n ,r }
for all j ≥ 0, then z ∈ Wn+ .
The manifolds Wn+ are C l -manifolds if the maps are C l , l ≥ 1.
Remark.
• Suppose r = −∞, i.e. the norm estimates kα n kC 1 , k βn kC 1 < σ
holds globally, then the theorem holds without the assumption µ > 1, except
item (3) need to be modified. See K-H Theorem 6.2.8 for details.
• Suppse λ < 1 < µ we can apply the theorem for the map family g−n = ( f n ) −1
to obtain the stable manifolds. Theorem 2 follows from the Hadamard-Perron
Theorem.
3. Cone families
We follow the presentation of K-H. Given a splitting R k ⊕ Rm−k of Rm and
0 < γ ≤ 1, we can define the standard horizontal and vertical cones in the following
way:
γ
Hp = {(u, v) ∈ Tp Rm : kvk ≤ γkuk},
γ
Vp = {(u, v) ∈ Tp Rm : kuk ≤ γkvk}.
A cone field K is a collection of cones K p defined at every point p of a given set. Let
K be a cone field on Rm , we say a submanifold N is tangent to K if Tp N ⊂ K p for
every p ∈ N.
For the map sequence f n , a cone family is a cone field defined at each copy of Rm .
A map family that admits a (λ, µ)-splitting preserves the standard cone families.
Lemma 4. For any 0 < γ < 1, suppose (γ + γ1 )σ <
z1 ∈ Br , ( f m ) −1 (z2 ) ∈ Br ,
γ
γ
D f m (z)Hz ⊂ intH f m (z) ,
γ
γ
1+γ (µ
(D f m (z)) −1Vz ⊂ intV
− λ), then for any
γ
−1 (z) .
fm
Proof. Assume kvk ≤ γkuk, we write
#" #
" 0#
" # "
u
u
A + Du α
Dv α
u
= d f m (z1 )
=
.
0
v
v
Du β
B + Dv β v
Note that all the derivative terms are bounded by σ. Write σ0 = (γ + γ −1 )σ, we
have
LECTURE NOTES ON HADAMARD-PERRON THEOREM
5
kv 0 k ≤ kBvk + σkvk + σkuk ≤ (λ + σ)kvk + σkuk ≤ (γλ + σγ + σ)kuk
< (γλ + σ0 )kuk
γku0 k ≥ γk Auk − γσkuk − γσkvk ≥ γ(µ − σ)kuk − γ 2 σkuk
> (γ µ − γσ0 )kuk.
We need λγ + σ0 < γ µ − γσ0, which is where we obtained the condition.
1/γ
For the backward invariance, it is equilvalent to prove forward invariance of Hp .
1
This is equivalent to σ0 < 1+γ
(µ − λ).
Lemma 5. Write (u0, v 0 ) = D f m (z)(u, v), then
γ
ku0 k > (µ − 2σ)kuk,
(u, v) ∈ Hz ,
kv 0 k < (λ + 2σ)kvk,
(u, v) ∈ Vz .
γ
The proof can be read directly from the calculations in Lemma 4, so we omit it.
The conclusions of the last two lemmas are called the cone properties. In fact,
these two properties are equivalent to the existence of invariant splittings.
Proposition 6. Suppose for 0 < λ < µ, the map family satisfies the conclusions of
lemma 4 and 5. We write {z n } (i.e. f n (z n ) = z n+1 ), we have the following statement.
(1) Suppose z n−i are defined (i.e. z n−i ∈ Br ) for all i ≥ 0, then
−∞
\
γ
+
En (z) =
D f n−1 (z n−1 ) ◦ · · · ◦ D f n−i (z n−i )Hm
i=0
is a k-dimensional subspace.
(2) Suppose z n+i are defined (i.e. z n+i ∈ Br ) for all i ≥ 0, then
∞
\
γ
−
En (z) =
D f n (z n−1 ) −1 ◦ · · · ◦ D f n−i (z n+i ) −1Vm
i=0
is a (m − k)-dimensional subspace.
This proposition can be proved directly using the cone estimates, and we leave it
for exercise.
4. Graph transform
Let Br+ denote the closed ball in R k × {0}. Fix a 0 < γ ≤ 1, let L denote the set
of all γ-Lipshitz functions ψ : Br+ → Rm−k and let L0 = L ∪ {ψ(0) = 0}. We define
a special metric
kψ1 (x) − ψ2 (x)k
d(ψ1 , ψ2 ) = sup
,
k xk
x∈Br+ \{0}
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LECTURE NOTES ON HADAMARD-PERRON THEOREM
and verifies that L0 is a complete metric space under this metric.
The following statement under the same assumptions as Lemma 4, f n defines a
mapping on L0 .
γ
(µ − λ), and µ − 2σ ≥ 1.
Lemma 7. For any 0 < γ < 1, suppose (γ + γ1 )σ < 1+γ
Then for any n, ψ ∈ L0 , there exists G f n (ψ) ∈ L0 , such that
f n (graph(ψ)) ∩ Br+ × Rm−k = graph(G f n (ψ)).
We use the notation H γ and V γ to denote the standard cone set in R k × Rm−k . Note
that for any zi = (x i , yi ), i = 1, 2, z2 − z1 ∈ H γ if and only if k y2 − y1 k ≤ γk x 2 − x 1 k.
We have the following analog of Lemma 4 and 5.
Lemma 8. Under the same assumptions as Lemma 7, the following hold.
(1) If z1 , z2 ∈ Br satisfies z2 − z1 ∈ H γ , then f n (z2 ) − f n (z1 ) ∈ H γ . Moreover,
k x 02 − x 01 k > (µ − 2σ)k x 2 − x 1 k.
(1)
(2) If z1 , z2 ∈ Br , and f n (z1 ), f n (z2 ) ∈ Br , then f n (z2 ) − f n (z1 ) ∈ V γ implies
z2 − z1 ∈ V γ . Moreover,
(2)
k y20 − y10 k < (λ + 2σ)k y2 − y1 k.
Proof. Denote zi = (x i , yi ) and zi0 = f n (x i , yi ), i = 1, 2. Note that
" 0
# "
#
x 2 − x 01
A(x 2 − x 1 ) + α(x 2 , y2 ) − α(x 1 , y1 )
.
=
y20 − y10
B(y2 − y1 ) + β(x 2 , y2 ) − β(x 1 , y1 )
Since kα(x 2 , y2 ) − α(x 1 , y1 )k, k β(x 2 , y2 ) − β(x 1 , y1 )k ≤ σk x 2 − x 1 k + σk y2 − y1 k,
by exactly the same calculations as in Lemma 4 and 5, we obtain the conclusions. Proof of Lemma 7. Using the cone invariance part of the last lemma, we obtain
f n (graph(ψ)) must be the graph of a γ-Lipshitz function. We only need to show
that its domain contains Br+ . But this follows from the fact that f n (0, 0) = (0, 0) and
(1).
This map on the space of graphs is usually called the graph transform. Let
(
)
Λ = {ψn }n∈Z : ψn ∈ L0 ,
equipped the metric
d({φn }, {ψn }) = sup d(φn , ψn ),
n
and let f n be the sequence of maps in the assumption of Theorem 3. We now define
the graph transform Φ on Λ by
Φ({ψn })n+1 = G f n (ψn ).
Note that if {ψn+ } is a fixed point of Φ, then
+
f n (graph(ψn+ )) ⊃ ψn+1
,
LECTURE NOTES ON HADAMARD-PERRON THEOREM
7
in other words, the sequence of manifolds {graph(ψn+ )} is backward invariant under
{ f n }. We now show that the fixed point exists and is unique.
Proposition 9. Suppose
λ + 2σ
< 1,
(µ − 2σ)(1 − γ 2 )
then Φ : Λ → Λ is a contraction mapping.
Proof. It suffice to show that for each n ∈ Z,
d(G f n φn , G f n ψn ) < d(φn , ψn ).
We fix x 0 ∈ Br+ , and denote y10 = G f n φn (x 0 ) and y20 = G f n ψn (x 0 ). By definition,
there exists x 1 , x 2 ∈ Br+ and y1 = φ(x 1 ), y2 = ψ(x 2 ), such that f n (x i , yi ) = (x 0i , yi0 ),
i = 1, 2. We further denote y˜2 = ψ(x 1 ).
Applying (1) to (0, 0) and (x 1 , y1 ), we have k x 0 − 0k ≥ (µ − 2σ)k x 1 k; applying
(2) to (x 1 , y1 ) and (x 2 , y2 ), we have k y20 − y10 k ≤ (λ + 2σ)k y2 − y2 k. Hence
kG f n φ(x 0 ) − G f n ψ(x 0 )k k y20 − y10 k
λ + 2σ k y2 − y1 k
=
≤
.
0
0
kx k
kx k
µ − 2σ k x 1 k
On the other hand, since (y20 , x 0 ) − (y10 , x 0 ) ∈ V γ , we know (y2 , x 2 ) − (y1 , x 1 ) ∈ V γ ,
i.e. k x 2 − x 1 k ≤ γk y2 − y1 k. Using the γ-Lipshitz property of ψ, we also get
k ỹ2 − y2 k ≤ γk x 2 − x 1 k. We get
k ỹ2 − y1 k ≥ k y2 − y1 k − k ỹ2 − y1 k ≥ (1 − γ 2 )k y2 − y1 k,
hence
k y1 − ỹ2 k
k y2 − y1 k
≥ (1 − γ 2 )
.
k x1 k
k x1 k
Combine the estimates obtained we have
λ + 2σ
kφ(x 0 ) − ψ(x 0 )k
≤
kG f n φ − G f n ψk = sup
kφ − ψk.
0
kx k
(µ − 2σ)(1 − γ 2 )
x 0 ∈Br+ \{0}
kφ − ψk ≥
We now prove the statements (1)-(3) of the Hadamard-Perron theorem. Indeed,
statements (1) follows immediately, and statement (2) is a easy consequence of
Lemma 5.
To show statement (3), we choose σ sufficiently small such that Lemmas 4, 5, 8
holds with γ = 1. Denote z n = z and z n− j = ( f n− j ) −1 ◦ · · · ( f n ) −1 (z n ). We claim
that if kz n− j k ≤ C(λ 0 ) − j for all j ≥ 0, then all z n− j ∈ H 1 . Suppose for some i > 0,
z n−i ∈ V 1 , then due to backward invariance of the vertical cones, z n− j ∈ V 1 for all
j ≥ i. Then Lemma 8 implies that there exists C 0 > 0 such that kz n− j k ≥ C 0 λ − j for
all j ≥ 0. This is a contradiction. Now all z n− j ∈ H 1 implies that for any j > 0,
there exists a Lipshitz graph ψn− j : R k → Rm−k such that z n− j ∈ graph(ψ j ), which
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LECTURE NOTES ON HADAMARD-PERRON THEOREM
implies z ∈ graph(Gn−1 ◦ · · · Gn− j ψn− j ). Since the latter graph converges to Wn+ as
j → ∞, this implies z n ∈ Wn+ .
Now let {ψn+ } be the unique fixed point of Φ, we write Wn+ = graphψn+ , these
Lipshitz invariant manifolds.
Corollary 10. The fixed point {ψn+ }n∈Z depends continuously on the map family
{ f n }n∈Z in the uniform C 1 norm.
Proof. We use the following observation: the fixed point x ∗ of a contraction mapping
Φ depends continuously on Φ in terms of uniform topology among contraction
mappings.
Now by Proposition 6, the invariant bundles En+ (z) is well defined on z ∈ Wn+ .
We give a different proof of this statement which at the same time proves that En+
depends continuously on the base point z.
Lemma 11. For each z n ∈ Wn+ , and z n− j = ( f n− j ) −1 ◦ · · · ◦ ( f n−1 ) −1 , there exists a
+ ⊂ H γ , namely
unique invariant family of k−dimensional subspaces En−
j
+
d f n−1 (z n− j ) ◦ · · · ◦ d f n− j (z n− j )En−
j (z n− j ) = En (z n ).
j
The family En+ (z n ) depends smoothly on z n .
Proof. Consider the linear maps {D f n (z n− j )} : Rm → Rm , j ≤ 0 as a map family.
We verify directly that Proposition 9 applies in the sense that the graph transform on
the one-sided sequence is well defined an a contraction mapping.
Furthermore, consider the set of all linear functions l : R k → Rm−k with kl k ≤ γ.
This is a subset of L0 , and note that the graph transform G D f n (z n ) now preserves
the set of linear maps. It now follows that the unique fixed point for the graph
transform under D f n is a linear map l n+ , and En+ = graph(l n+ ). Continuity follows
from Corollary 10.
We complete the C 1 part of the discussion by the following statement.
Proposition 12. The manifolds Wn+ are C 1 .
Proof. Fix z n = (x n , yn ) ∈ Wn+ , and let z n− j , j ≥ 0 be the backward orbit of z n . We
then claim that for any N ∈ N, there exists > 0 such that for any k x 0n − x n k < ,
we have
−N
\
γ+σ
0
+ 0
(x n , ψn (x n )) − (x n , yn ) ∈
D f n−1 (z n−1 ) ◦ · · · ◦ D f n−i (z n−N )Hm .
i=0
0 ) the backward orbit of
Denote yn0 = ψn+ (yn0 ), and (x 0n− j , yn−
j
converges uniformly to (x n− j , yn− j ), as x 0n → x n . Since
0 )
(x 0n , yn0 ), then (x 0n− j , yn−
j
0
(x 0n−N , yn−N
) − (x n−N , yn−N ) ∈ H γ
LECTURE NOTES ON HADAMARD-PERRON THEOREM
9
0
and f n1 ◦· · ·◦ f n−N converges to D f n−1 (z n−1 )◦· · ·◦D f n−i (z n−i ) locally as (x 0n−N , yn−N
)
converges
to
(x
,
y
),
the
claim
follows.
T−N n−N n−N
γ+σ
Since i=0
D f n−1 (z n−1 )◦· · ·◦D f n−i (z n−N )Hm converges to E + (z n ) as N → ∞,
we conclude that Wn+ is tangent to En+ at z n . Since En+ is continuous, Wn+ is C 1 . 5. Regularity
We now discuss the statement that Wn+ are C l when the map is C l . Given the map
family f n , we define the linear extension
Fn : T (Rm ) ' Rm × Rm → Rm × Rm ,
Fn (z, v) = ( f n (z), d f n (z)v).
We note that
Fn (z, v) = (d f n (0)z, d f n (0)v) + ( f n (z) − d f n (0)z, (d f n (z) − d f n (0))v)
and the second term vanishes at order 1 at (z, v) = (0, 0). Note that the linear
part (d f n (0), d f n (0)v admits an invariant (λ, µ)-splitting, and the Hadamard-Perron
theorem applies. Write z = (x, y) and v = (v x , v y ), then there exists r > 0 and
Lipshitz functions Φ+n : Br+ × Br+ ⊂ R k × R k → Rm−k × Rm−k , such that
Vn+ (z, v) = {(x, Π y Φ+n (x, v x ), v x , Πvy Φ+n (x, v x )),
(x, v x ) ∈ Br+ },
are the unstable manifolds for Fn . Here Π y , Πvy are the standard projections.
Lemma 13. The function Φ+n described above coincides with the function
(x, v x ) :7→ (φ+n (x), l n+ (x, φ+n (x))v x ),
where l n+ (x, φ+n (x)) : R k → Rm−k is such that En+ (x, φ+n (x)) = graph(l n+ (x), φ+n (x)).
Proof. The proof uses the fact that φ+n (x) and En+ (x) are uniquely characterized by
their backward contraction properties. We leave the details as an exercise.
We remark the fact that Φ+n is linear in v x implies Φ+n is well defined on x ∈
Br+ , v x ∈ R k .
The Hadarmard-Perron theorem shows that Φ+n are C 1 −functions in x, v x . This
implies that l n+ (x, φ+n (x)) is C 1 in x. Given that l n+ coincide with dφ+n , we obtain φ+n
is in fact C 2 . This argument can be applied inductively, yielding C l differentiability
of φ+n .
6. Inclination lemma
In this section, we consider a hyperbolic fixed point p of a C 1 map f . It suffices
to consider the map
f : U ⊂ Rm → Rm
and the fixed point is 0.
The inclination lemma, sometimes referred to as the λ lemma, is the following
statement.
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LECTURE NOTES ON HADAMARD-PERRON THEOREM
Lemma 14. Assume that p is a hyperbolic fixed point of f : U → M, with kdimensional unstable direction and m − k-dimensional stable direction. Then for
any embeded disk D intersecting W − transversally at q, and any > 0, there exists
an embedded disk D1 ⊂ D containing q and N > 0 such that f N (D1 ) is close to
W + (p) in the C 1 -distance.
To prepare for the proof of this lemma, we first make a coordinate change such
that W + and W − becomes the coordinate axes. We also choose an r neighborhood
of 0 on which the γ-cones are invariant. We then show that there exists q ∈ D0 D
and n0 ∈ N such that f n0 (D) is contained in Br and tangent to the horizontal cones
H γ . It suffices to show that d f n (q)Tq D is contained in the horizontal cone for some
n. Arguing by contradiction: assume d f n (q)Tq D is contained in the vertical cone
H 1/γ for all n ≥ 0 implies Tq D must be contained in the stable subspace E − (q), this
contradicts our assumption.
Step 2 shows if f n0 D2 is tangent to γ-cone, then further iterates of f takes it close
to W + . This is left as an exercise.
7. Comments on the assumption µ > 1
We have been stating our theorem with the assumption µ > 1. A version without
this assumption exists, but its application requires caution. We have
Theorem 15. Let 0 < λ < µ. Given C 1 -diffeomorphisms f n : Rm → Rm such
that f n (x, y) = ( An x + α n (x, y), Bn y + βn (x, y)) with k An−1 k ≤ µ−1 , kBn k ≤ λ and
α(0, 0) = β(0, 0) = 0.
For any δ > 0, there exists σ > 0 such that if
kα n kC 1 (Rm ) , k βkC 1 (Rm ) < σ,
there exists a unique family of C 1 -manifolds
Wn+ = {(x, φ+n (x)) : x ∈ R k } = graph(φ+n )
verifying the following:
+ .
(1) (backward invariance) ( f n−1 ) −1Wn+ ⊂ Wn−1
(2) (backward contraction) k( f n−1 ) −1 (z1 ) − ( f n−1 ) −1 (z2 )k ≤ (µ−1 + δ)kz1 − z2 k
for z1 , z2 ∈ Wn+ .
(3) (criterion for unstable manifold) If for any λ < λ 0 < µ, and C > 0 we have
k( f n− j ) −1 ◦ · · · ( f n ) −1 (z)k ≤ C(λ 0 ) −n ,
for all j ≥ 0, then z ∈ Wn+ .
The regularity part of the manifolds are trickier and we will not discuss it here.
Given a map family f n : U ⊂ Rn → Rn for which d f n admits a (λ, υ)-splitting.
One can always modify f outside of Br such that the assumption of Theorem 15
hold. One has to be very careful, as in the case of µ ≤ 1, the invariant manifolds
LECTURE NOTES ON HADAMARD-PERRON THEOREM
11
obtained may depend on the modified part of the map, and hence do not reflect
properties of the original map.