Ann. I. H. Poincaré – AN 23 (2006) 641–661
www.elsevier.com/locate/anihpc

Partial hyperbolicity for symplectic diffeomorphisms ✩

Vanderlei Horita a,∗ , Ali Tahzibi b

a UNESP – Universidade Estadual Paulista, Departamento de Matemática, IBILCE, Rua Cristóvão Colombo 2265,
15054-000 S. J. Rio Preto, SP, Brazil

b Departamento de Matemática e de Computação, ICMC/USP, Av. Trabalhador São Carlense, 400-Cx. Postal 668,
13560-320 São Carlos, SP, Brazil

Received 3 December 2004; accepted 7 June 2005

Available online 6 December 2005

Abstract

We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a
dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.
© 2005 Elsevier SAS. All rights reserved.

Keywords: Partial hyperbolicity; Dominated splitting; Symplectic diffeomorphisms; Robust transitivity; Stable ergodicity

1. Introduction

In the 1960’s, one of the main goals in the study of dynamical systems was to characterize the structurally stable
systems and to verify their genericity.

The theory of uniformly hyperbolic systems has introduced a foundation to approach these subjects. Indeed,
uniform hyperbolicity proved to be the key ingredient to characterize structurally stable systems.

We have a good description of uniformly hyperbolic dynamics. The Smale spectral Theorem asserts that the
non-wandering set of a hyperbolic diffeomorphism splits into basic pieces. The dynamic restricted to each piece is
topologically transitive and the transitivity property persists after small perturbations.

Recall a diffeomorphism is transitive if there exists a point x such that its forward orbit is dense. Transitive
systems do not have neither sinks nor source. We say a diffeomorphism is Cr -robustly transitive if it belongs to the
Cr -interior of the set of transitive diffeomorphisms.

✩ The authors was partially supported by FAPESP-Proj. Tematico 03/03107-9. A. Tahzibi would like to thank the financial support CNPq
(Projeto Universal). V. Horita was also supported by CAPES, FAPESP (02/06531-3), and PRONEX.

* Corresponding author.
E-mail addresses: vhorita@ibilce.unesp.br (V. Horita), tahzibi@icmc.usp.br (A. Tahzibi).
URL: http://www.icmc.usp.br/~tahzibi.
0294-1449/$ – see front matter © 2005 Elsevier SAS. All rights reserved.
doi:10.1016/j.anihpc.2005.06.002



642 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
An interesting question is whether robust transitivity implies hyperbolicity in uniform fashion or not.
It is well known that every C1-robustly transitive diffeomorphism on a closed surface is an Anosov diffeo-

morphism. This is a direct consequence of a C1-generic dichotomy between hyperbolicity and C1-Newhouse
phenomenon (the coexistence of infinitely many sinks and sources in a locally residual subset) proved by Mañé
in [11].

In compact manifolds of dimension greater than two, robust transitivity does not imply uniform hyperbolicity,
see [14,10,4,6,15]. However, all these examples present a weak form of hyperbolicity called dominated splitting.
Let M be a compact manifold and f :M → M a diffeomorphism. A Df -invariant decomposition T M = E ⊕F of
the tangent bundle of M , is dominated if for every positive integer � and any x in M ,∥∥Df �|E(x)

∥∥ · ∥∥Df −�|F (
f �(x)

)∥∥ < Cλ�,

for some constants C > 0 and 0 < λ < 1.
If for the dominated splitting T M = E ⊕ F at least one of the subbundles is uniformly hyperbolic then f is

called partially hyperbolic.
Díaz, Pujals and Ures in [9] show that every C1-robustly transitive diffeomorphism in a 3-dimensional compact

manifold should be partially hyperbolic. This result cannot be generalized for higher dimension: Bonatti and Viana
present in [6] an example in 4-dimensional compact manifold of a non-partially hyperbolic C1-robustly transitive
diffeomorphism. This example can be extended in any dimension, see [15].

Bonatti, Díaz and Pujals in [5] proved that every C1-robustly transitive diffeomorphism on a n-dimensional
compact manifold, n � 1 admits a dominated splitting. Recently Vivier proved similar results for flows in [16].
She proved that robustly transitive C1-vector fields on a compact manifold do not admit singularity and that robust
transitivity implies the existence of dominated structure.

In this paper we address the problem of the existence of dominated splitting for C1-robustly transitive symplectic
diffeomorphisms. In the symplectic setting dominated splitting implies strong partial hyperbolicity. This fact was
first observed by Mañé, see [12]. A proof is given in [2] for dimM = 4 and in [3] for the general case.

Let (M,ω) be a 2N -dimensional symplectic manifold, where ω is a non-degenerated symplectic form. We
denote Diff r

ω(M), r � 1, the set of Cr -symplectic diffeomorphisms. We say f ∈ Diff r
ω(M) is Cr -robustly transi-

tive symplectic diffeomorphism if there exists a neighborhood U ⊂ Diff r
ω(M) of f such that every g ∈ U is also

transitive.

Theorem 1. Every C1-robustly transitive symplectic diffeomorphism on a 2N -dimensional, N � 1, compact man-
ifold is partially hyperbolic.

We emphasize that if f ∈ Diff1
ω and is C1-robustly transitive symplectic diffeomorphism then just symplectic

nearby diffeomorphism are transitive. So, our theorem is not a consequence of results in [5]. In 4-dimensional
setting, N = 2, Theorem 1 is a consequence of [2].

In the context of volume preserving diffeomorphisms, ergodicity of the Lebesgue measure is a basic feature.
Recall that a diffeomorphism in Diff1

ω(M) preserves the 2-form ω and consequently the volume form ω ∧ · · · ∧ ω

is also preserved. This volume form induces in a natural way a Lebesgue measure defined on M .
The stable ergodicity of symplectic diffeomorphism is defined as follows. A symplectic diffeomorphism f ∈

Diff2
ω(M) is C1-stably ergodic if there exists a neighborhood U ⊂ Diff1

ω(M) of f such that any g ∈ U ∩ Diff2
ω(M)

is ergodic.
The theories of stable ergodicity and robust transitivity had very parallel development in last few years. Like as in

the robust transitivity case we know examples of stably ergodic diffeomorphisms with weak form of hyperbolicity.
We propose the reader to see [7] for approaches to prove stable ergodicity in the partially hyperbolic case. However,
there are stably ergodic diffeomorphisms which are not partially hyperbolic, see [15].



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 643
A natural question is whether stable ergodicity of volume preserving or even symplectic diffeomorphisms im-
plies the existence of a dominated splitting of the tangent bundle. For the symplectic case we give an affirmative
answer.

Theorem 2. Every C1-stably ergodic symplectic diffeomorphism on a 2N -dimensional, d � 1, compact manifold
is partially hyperbolic.

To prove our theorems we follow the arguments in [5] where they obtain a dichotomy between dominated
splitting and the Newhouse phenomenon. This is done by showing that the lack of dominated splitting leads to
creation of sinks or sources by a convenient perturbation. Of course a symplectic diffeomorphism does not admit
sink or source. The idea in the symplectic case is to make a perturbation and create a totally elliptic periodic point.

In Section 2 we state this dichotomy for symplectic diffeomorphisms in Theorem 2.1.
We use this dichotomy to finish the proof of our main results in Section 3. The idea is to eliminate the possibility

of creation of totally elliptic periodic points for robustly transitive or stably ergodic symplectic diffeomorphisms.
For this purpose in Section 3 we use generating functions for symplectic diffeomorphisms to prove that “stably
ergodic and robustly transitive symplectic diffeomorphisms are C1 far from having totally elliptic points”.

To prove the dichotomy for symplectic diffeomorphisms we prove a similar result for linear symplectic systems.
Symplectic linear systems are introduced in Section 4 and some perturbation results are proved there which will be
used in the rest of the paper.

The difficulty to get the dichotomy in the symplectic case is that we have much less space to perform symplectic
perturbations. The perturbations have to preserve the symplectic form ω.

In Section 5 we state the precise result of dichotomy for linear symplectic systems and in Sections 6 and 7 we
prove this statements. In these two final sections we prove new results for symplectic linear systems which enable
us to adapt the approach of [5] for symplectic linear systems.

We mention that for robustly transitive volume preserving diffeomorphisms the dichotomy as mentioned above
is straightforward from the result in [5]. Arbieto and Matheus [1] use this dichotomy and prove that robustly
transitive conservative diffeomorphisms have dominated splitting. So, the main difficulty in the proof of similar
to our results in the volume preserving case is to show that the robustly transitive conservative diffeomorphisms
cannot have totally elliptic points. They overcome this difficulty with a new “Pasting Lemma” which uses a theorem
of Dacorogna and Moser [8].

2. A dichotomy for symplectic diffeomorphisms

An invariant decomposition T M = E ⊕ F is called �-dominated if for any n � � and every x in M ,∥∥Df n|E(x)
∥∥ · ∥∥Df −n|F (

f n(x)
)∥∥ <

1

2
.

From now on we use the notation E ≺� F for �-dominated splitting and E ≺ F for dominated splitting without
specifying the strength of the splitting.

It is easy to verify from definition that �-dominated splitting has the following properties:

1. If Λ ⊂ M is an invariant subset that admits an �-dominated splitting then the same is true for the closure of Λ.
2. If a sequence (fn)n of maps admitting an �-dominated splitting converges to f in C1-topology then f also

admits an �-dominated splitting.

Theorem 2.1. Let f ∈ Diff1
ω(M) be a symplectic diffeomorphism of a 2N -dimensional manifold M . Then there is

� ∈ N such that,



644 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
(a) either there is a symplectic ε-C1-perturbation g of f having a periodic point x of period n ∈ N such that
Dgn(x) = Id,

(b) or for any symplectic diffeomorphism g ε-C1-close to f and every periodic saddle x of g the homoclinic class
H(x,g) admits an �-dominated splitting.

To prove this theorem we introduce the concept of symplectic linear systems in Sections 4. Then, in Section 5,
we reduce the proof of Theorem 2.1 to proof a similar result for that symplectic linear systems.

By the following lemma we are able to extend dominated splittings over homoclinic classes to the whole
manifold for a generic symplectic diffeomorphism. Before stating this result let us recall a symplectic version
of Connecting lemma due to Xia in [17].

Theorem 2.2 ((Xia)). Let M be a compact n-dimensional manifold with a symplectic or volume form ω. Then there
is a residual subset R2 ⊂ Diff1

ω(M) such that if g ∈ R2 and p ∈ M are such that p is a hyperbolic periodic point
of g, then Ws(p) ∩ Wu(p) is dense in both Ws(p) and Wu(p).

Using this result we are able to prove the following result.

Lemma 2.3. There is a residual subset R ⊂ Diff1
ω(M) of diffeomorphisms f such that the non-trivial homoclinic

classes of hyperbolic periodic points of f are dense in M .

Proof. The version of this lemma for conservative diffeomorphisms is given in [5, Lemma 7.8]. We can use the
same arguments adapted for the symplectic case by means of a result of Newhouse and connecting lemma of Xia.

First of all, we recall that for a symplectic diffeomorphism the set of recurrent points is dense in M . Indeed
a symplectic diffeomorphism preserves a Lebesgue measure. Moreover, using C1-Closing lemma of Pugh and
Robinson and the Birkhoff fixed point theorem, Newhouse [13, Corollary 3.2] proved that there is a C1-residual
subset R1 ∈ Diff1

ω(M) such that for any g ∈R1 the set of hyperbolic periodic points of g is dense in M .
By using Theorem 2.2 we take R = R1 ∩R2. Thus, for any g ∈ R the union of non-trivial homoclinic classes

is dense. This completes the proof of lemma. �
Remark 2.4. Using the above lemma and the continuity of dominated splitting the item (b) of Theorem 2.1 can be
rewritten as follows.

(b1) the manifold M is the union of finitely many invariant (by f ) compact sets having a dominated splitting.

We remark that the invariant compact sets mentioned above are Λi , i < 2N , where Λi is closure of the union of
non-trivial homoclinic classes with an �-dominated splitting E ⊕ F with dimension i (i.e. dim(E) = i). Of course
for a transitive diffeomorphism the above item is equivalent to have a dominated splitting on the whole manifold.

3. Proof of Theorems 1 and 2

In this section we prove Theorems 1 and 2 using Theorem 2.1.

3.1. Elliptic points vs. robust transitivity

Let us prove that the item (a) of the dichotomy given in Theorem 2.1 does not occur for C1-robustly transitive
diffeomorphisms.



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 645
Lemma 3.1. If f ∈ Diff1
ω(M) has a totally elliptic periodic point p of period n (Df n(p) = Id) then there exists

g ∈ Diff1
ω(M) and C1-close to f such that g is not transitive.

Proof. In order to simplify our arguments, let us suppose that p is a fixed point. We use generating functions to
construct g in such a way that g coincide to the identity map in a small neighborhood of p.

Let us introduce generating function as in [17]. We fix a local coordinated system (x1, . . . , xd, y1, . . . , yd)

in a neighborhood U of p such that p correspond to (0,0) in this coordinate system. Suppose f (x, y) =
(ξ(x, y), η(x, y)). The fact f is symplectic implies

d∑
i=1

dxi ∧ dyi =
d∑

i=1

dξi ∧ dηi.

Moreover, we may suppose that partial derivative ∂η/∂y of η with respect to y is non-singular at every point of U .
This enable us to solve η = η(x, y) to obtain y = y(x, η). Hence, we can define a new system of coordinates
(x1, . . . , xd, η1, . . . , ηd). Let γ : (x, y) → (x, η(x, y)) be the map of change of coordinates.

Since the 1-form

α :=
d∑

i=1

ξi dηi + yi dxi

is closed, there exists a C2 function Sf (x, η), defined on a neighborhood of (0, η(0,0)) such that dSf = α. The
function Sf is unique up to a constant. Moreover,

∂Sf

∂xi

= yi and
∂Sf

∂ηi

= ξi .

Conversely, for a real C2-function S(x, η) defined on a neighborhood of (0, η(0,0)) such that the second partial
derivative

∂2S

∂x∂η

is non-singular in this neighborhood, we define

ξi(x, η) = ∂S

∂ηi

and yi(x, η) = ∂S

∂xi

.

Then, solving η in terms of x, y we find a symplectic diffeomorphism which maps (x, y) to (ξ, η).
Observe that in the above construction the generating function of f is Ck+1 whenever f is Ck . Moreover, given

a Ck+1 generating function we obtain locally a Ck symplectic diffeomorphism.
We also have that f is C1-close to g if and only if Sf is C2-close to Sg , provided they are defined on the same

domain.
Let ρ be a C∞ bump function such that

ρ(z) =
{

1 if z ∈ γ
(
B(β/2)

)
,

0 if z /∈ γ
(
B(β)

)
where B(r) is the ball of radius r centered in (0,0) in the (x, y)-coordinates.

Define a C2 function given by

Sg := ρ(x, η)Sid + (
1 − ρ(x, η)

)
Sf .

More important is that, if we take β > 0 small enough, Sg is C2-close to Sf . Indeed, f is C1-close to identity in a
neighborhood of the origin and consequently, Sid and Sf are C2-close enough on γ (B(β)).



646 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
Finally, it is easy to see that if Sid and Sf are ε-close on γ (B(β)) in the C2-topology then

ρ(x, η)Sid + (
1 − ρ(x, η)

)
Sf

is Kε-close to Sf in the C2-topology, where K depends on the bump function ρ and β.

Let g ∈ Diff1
ω(M) be the corresponding symplectic diffeomorphism to Sg . As g is C1-close to f and locally it

is equal to the identity, we conclude that g cannot be transitive and this conclude the proof of the lemma. �
The previous lemma proves that a robustly transitive diffeomorphism cannot have a totally elliptic periodic point.

So, if f is robustly transitive as in Theorem 1 then by Remark 2.4 we can conclude that f admits a dominated
splitting. In this way we prove Theorem 1.

3.2. Elliptic point vs. stable ergodicity

In this subsection we prove that any C1-stably ergodic symplectic diffeomorphism admits a dominated splitting.
Let us recall that by C1-stably ergodic diffeomorphism we mean a symplectic C2-diffeomorphism such that all
symplectic C2-diffeomorphisms in a C1-neighborhood are ergodic.

Let f ∈ Diff2
ω(M) be a stably ergodic diffeomorphism and U be a C1-neighborhood of f such that any g ∈

U ∩ Diff2
ω(M) is ergodic. We prove that any diffeomorphism inside R∩ U admits an �-dominated splitting where

R is given in Lemma 2.3. Since the �-dominated splitting property is a closed property in C1-topology we obtain
a dominated splitting for f . The proof is by contradiction.

Suppose that there exists f1 ∈ R∩ U close to f . If f1 does not admit a dominated splitting then, by Theorem 2.1,
there exists g1 ∈ Diff1

ω(M) with a totally elliptic periodic point.
We claim that there exists g ∈ Diff2

ω(M) and C1-close to g1 such that g is not ergodic. This gives a contradiction
with the stable ergodicity of f.

Just to simplify our arguments let us suppose that g1 has a totally elliptic fixed point. Using the same technics
used in the previous subsection we construct g2 ∈ Diff1

ω(M) and C1-close to g1 in such a way that g2 coincides
with the identity map on B(β), for some β > 0. Note that g2 is just C1 and we do not get a contradiction with the
stable ergodicity of f .

However, from a result of Zehnder in [18] there exists g3 ∈ Diff2
ω(M) and C1-close to g2. In order to get a

contradiction, similarly to the previous subsection, let ρ̃ be a C∞ bump function such that

ρ̃(z) =
{

1 if z ∈ γ
(
B(β/3)

)
,

0 if z /∈ γ
(
B(β/2)

)
.

We define

Sg(x, η) := ρ̃(x, η)Sid + (
1 − ρ̃(x, η)

)
Sg3 .

Since g3 is a C2-diffeomorphism we have Sg is a C3-function. Moreover, Sg is C2-close to Sg3 . Therefore, Sg is a
generating function for a symplectic diffeomorphism g ∈ Diff2

ω(M) such that g is C1-close to g3 and consequently
C1-close to f .

By construction, g coincides with the identity map on a neighborhood of zero. Hence g cannot be ergodic. This
gives a contradiction, because we have supposed that f is stably ergodic.

The proof of Theorem 2 is complete.

4. Symplectic linear systems

In this section we introduce the key ingredient in the proof of Theorem 2.1. Following the techniques of Mañé,
we use symplectic linear systems enriched with transition by Bonatti, Díaz, Pujals (see [5, Section 1]) in order



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 647
to prove a dichotomy for linear systems in Proposition 5.3. However, we stress that in our case the symplectic
perturbations we have to perform is much more difficult to realize.

Let Σ be a topological space and f a homeomorphism defined on Σ . Consider a locally trivial vector bundle E
over Σ such that E(x) is a symplectic vector space with anti-symmetric non-degenerated 2-form ω. Furthermore,
we require the dim(E(x)) does not depend on x.

We denote by S(Σ,f,E) the set of maps A :E → E such that for every x ∈ Σ the induced map A(x, ·) is a
linear symplectic isomorphism from E(x) → E(f (x)), that is,

ω(u, v) = ω
(
A(u),A(v)

)
.

Thus, A(x, ·) belongs to Lω(E(x),E(f (x))).
For each x ∈ E the Euclidean metric on E(x) and E(f (x)) induces in a natural way a norm on Lω(E(x),

E(f (x))):∣∣B(x, ·)∣∣ = sup
{∣∣B(x, v)

∣∣, v ∈ E(x), |v| = 1
}
.

Furthermore, for A ∈ S(Σ,f,E) we define |A| = supx∈Σ |A(x, ·)|. Note that, if A belongs to S(Σ,f,E) its inverse
A−1 belongs to S(Σ,f −1,E). The norm of A ∈ S(Σ,f,E) is defined by ‖A‖ = sup{|A|, |A−1|}.

Let (Σ,f,E,A) be a linear symplectic system (or linear symplectic cocycle over f ), that is, a 4-tuple where Σ

is a topological space, f is a homeomorphism of Σ , E is an Euclidean bundle over Σ , A belongs to S(Σ,f,E)

and ‖A‖ < ∞. We say that (Σ,f,E,A) is periodic if any p ∈ Σ is a periodic point of f .
Let (V ,ω) be a symplectic vector space and W ⊆ V a vector subspace of V . Then the symplectic complement

of W is given by

Wω = {
x ∈ V :ω(x,w) = 0, for all w ∈ W

}
.

It is easy to see that Wω is also a vector space and, by definition,

(1) If W ⊂ Wω, then W is an isotropic subspace.
(2) If W ∩ Wω = 0 , then W is a symplectic subspace.
(3) If W = Wω, then W is Lagrangian subspace.

Let (Σ,f,E,A) be a diagonalizable periodic linear symplectic system such that E(x) = R2N . In what follows,
we indicate by λ1(x) < λ2(x) < · · · < λ2N(x) their eigenvalues and we denote by E1(x) ≺ E2(x) ≺ · · · ≺ E2N(x)

their respective eigenspaces.
We know that if λ is an eigenvalue of a symplectic transformation then λ−1 is also an eigenvalue. So, if

λ1 < λ2 < · · · < λ2N are 2N distinct eigenvalues of a symplectic transformation then λi∗ := λ2N−i+1 = λ−1
i .

We say that B = {e1, e2, . . . , e2N } is a symplectic basis for a symplectic vector space (V ,ω) if B is a basis of V

and

ω(ei, ej ) =
{

0 for j 
= i∗,

1 for j = i∗ > i.

The fact that ω is an anti-symmetric 2-form implies that ω(ei∗ , ei) = −1 for i∗ > i.

Let (Σ,f,E,A) be a diagonalizable symplectic linear system. Suppose that {e1(p), e2(p), . . . , e2N(p)} is a
symplectic basis of E(p) then {A(e1(p)),A(e2(p)), . . . ,A(e2N(p))} is a symplectic basis of E(f (p)). For sim-
plicity of notations, we omit the dependence on the point of vectors ei .

Lemma 4.1. Let (Σ,f,E,A) be diagonalizable periodic linear symplectic system with distinct eigenvalues
λ1 < · · · < λ2N and E1 ≺ · · · ≺ E2N be the corresponding eigenspaces. There exists a symplectic basis
{e1, . . . , e2N } constituted by eigenvectors. Moreover, for j 
= i∗, Ei ⊕ Ej is an isotropic subspace and Ei ⊕ Ei∗ is
a symplectic subspace.



648 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
Proof. Let n = n(p) be the period of p ∈ Σ . Let ei ∈ Ei and ej ∈ Ej be eigenvectors of An with respective
eigenvalues λi and λj . Then,

ω
(
An(ei),A

n(ej )
) = λiλjω(ei, ej ).

On the other hand, since A is a linear symplectic system, we have

ω
(
An(ei),A

n(ej )
) = ω(ei, ej ).

Thus, if j 
= i∗ then λiλj 
= 1 and consequently ω(ei, ej ) = 0. Moreover, as ω is non-degenerate we have
ω(ei, ei∗) 
= 0. So, normalizing the vectors ei , 1 � i � N , we can choose a new basis constituted by eigenvectors
of A, which we still denote by ei , such that ω(ei, ei∗) = 1 for all 1 � i � N .

Other claims in the statement are direct consequence of definitions of symplectic and isotropic subspaces. �
Let Ej ⊕ Ek be a vector subspace of a diagonalizable symplectic vector space. Any small symplectic perturba-

tion of A|Ej ⊕Ek
is called a symplectic perturbation along Ej ⊕ Ek . The next lemma asserts that any symplectic

perturbation along Ej ⊕ Ek can be realized as the restriction of a symplectic perturbation of A. More precisely,

Lemma 4.2 ((Symplectic realization)). Let (Σ,f,E,A) be a diagonalizable periodic linear system as above. Given
any ε > 0 and 1 � j < k � 2N , every ε-symplectic perturbation B of

A|Ej ⊕Ek
:Ej ⊕ Ek → Ej ⊕ Ek

along the orbit of x ∈ Σ is the restriction of a symplectic ε-perturbation à of A such that Ã|Ei
= A|Ei

for
i 
= j, k, j∗, k∗.

Proof. If Ej ⊕ Ek is a symplectic subspace, that is k = j∗, then we define Ã|Ej ⊕Ej∗ = B , Ã|Ei
= A, for i 
= j, j∗

and we extend it linearly. In that way, we have

ω
(
Ã(ej ), Ã(ej∗)

) = ω
(
B(ej ),B(ej∗)

) = ω(ej , ej∗) = 1,

and for i 
= j, j∗, we get

ω
(
Ã(ei), Ã(ej )

) = ω
(
A(ei), αej + βej∗

) = 0.

Moreover, for r, s 
= j, j∗, we have

ω
(
Ã(es), Ã(er )

) = ω
(
A(es),A(er )

) = ω(es, er ).

Therefore, in this case, Ã is symplectic.
Now, we suppose k 
= j∗. An important feature we have to take account is the way we extend B to Ej∗ ⊕ Ek∗ .

Once we have made it, we define à equal to A when restricted to the others subspace Ei , i 
= j, k, j∗, k∗. Finally,
we extend linearly this operator to other vectors.

There are constants α,β, γ, δ ∈ R such that

B(ej ) = αej + βek,

B(ek) = γ ej + δek.

Let us suppose 1 � j < k � N . We construct a symplectic linear system à in this case. In the other cases are
completely analogous, by changing conveniently the sign of the constant in B̃ bellow. We denote Δ = αδ − βγ

and we define

B̃(ej∗) = δ

Δ
ej∗ − γ

Δ
ek∗ ,

B̃(ek∗) = − β
ej∗ + α

ek∗ ,

Δ Δ



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 649
and extend B̃ linearly to Ej∗ ⊕ Ek∗ .
Then, we define à as follows:

• Ã|Ei
= A|Ei

, i /∈ {j, k, j∗, k∗},
• Ã|Ej ⊕Ek

= B ,

• Ã|Ej∗⊕Ek∗ = B̃ ,

and extend à linearly.
Note that, if B in Ej ⊕ Ek is a rotation then the perturbation B̃ is the same rotation in Ej∗ ⊕ Ek∗.
To verify that à is a symplectic linear system, it is enough to show that

ω
(
Ã(er ), Ã(es)

) = ω(er , es) for any 1 � r, s � 2N.

Let us begin by the case when r = j, s = j∗:

ω
(
Ã(ej ), Ã(ej∗)

) = ω

(
αej + βek,

δ

Δ
ej∗ − γ

Δ
ek∗

)
= αδ

Δ
ω(ej , ej∗) − βγ

Δ
ω(ek, ek∗) − αγ

Δ
ω(ej , ek∗) + βδ

Δ
ω(ek, ej∗)

= αδ

Δ
− βγ

Δ
= 1 = ω(ej , ej∗).

Similarly, we obtain

ω
(
Ã(ej∗), Ã(ej )

) = ω(ej∗ , ej ),

ω
(
Ã(ek), Ã(ek∗)

) = ω(ek, ek∗),

ω
(
Ã(ek∗), Ã(ek)

) = ω(ek∗ , ek).

If r = j, s = k∗, then

ω
(
Ã(ej ), Ã(ek∗)

) = ω

(
αej + βek,− β

Δ
ej∗ + α

Δ
ek∗

)

= −αβ

Δ
ω(ej , ej∗) + αβ

Δ
ω(ek, ek∗) + α2

Δ
ω(ej , ek∗) − β2

Δ
ω(ek, ej∗)

= 0 = ω(ej , ek∗).

Analogously, we have

ω
(
Ã(ek), Ã(ej∗)

) = 0 = ω(ek, ej∗).

Hence,

ω
(
Ã(ej∗), Ã(ek)

) = 0 = ω(ej∗ , ek),

ω
(
Ã(ek∗), Ã(ej )

) = 0 = ω(ek∗ , ej ).

The remaining cases are direct consequence of the fact that Ã(ei) = A(ei) belongs to Ei , if i 
= j, j∗, k, k∗. This
completes the proof. �

The following lemma is used in the next sections.



650 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
Lemma 4.3. Let (Σ,f,E,A) be a periodic diagonalizable linear system. If Ẽj (p) ∈ Ei(p) ⊕ Ej(p) is close
to Ej(p) then there exists a symplectic perturbation p of the identity map such that p(Ei(p)) = Ei(p) and
p(Ẽj (p)) = Ej(p).

Proof. By Lemma 4.2 it is enough to define p symplectic on Ei(p) ⊕ Ej(p) close to the identity map.
Let us suppose Ei(p)⊕Ej(p) is an isotropic subspace (j 
= i∗). Given ei ∈ Ei and ẽj ∈ Ẽj , we define p(ei) = ei

and p(ẽj ) the projection of ẽj over Ej and extend p to Ei(p) ⊕ Ej(p) linearly. Since Ẽj is close to Ej , p is close
to the identity map. Moreover, there exists ej ∈ Ej such that ẽj = αei + βej , for some constants α,β . Then

ω
(
p(ei),p(ẽj )

) = 0 = αω(ei, ei) + βω(ei, ej ) = ω(ei, ẽj ).

On the other hand, let Ei(p) ⊕ Ej(p) be a symplectic subspace. We take ei ∈ Ei , ej ∈ Ej such that
ω(ei, ej ) = 1. Let ẽj ∈ Ẽj be close to ej Then, there exists constants r close to zero and s close to 1 such that
ẽj = rei + sej . Hence, {ei, ẽj } is a basis of Ei(p) ⊕ Ej(p) and s = ω(ei, ẽj ). We define p on this base as follows:

p(ẽj ) = ej and p(ei) = sei .

Therefore,

ω
(
p(ei),p(ẽj )

) = ω
(
ω(ei, ẽj )ei, ej

) = ω(ei, ẽj ).

So, in both cases p is symplectic perturbation of the identity map. �
4.1. Symplectic transitions

Here we recall an important notion introduced in [5]: the concept of transitions. In this work we are dealing with
the systems which admit transitions. In order to introduce this important notion let us begin with an example, see
[5, Section 1.4]. Suppose P and Q are saddles of the same index linked by transverse intersection of their invariant
manifolds. The existence of a Markov partition shows that for any fixed finite sequence of times there is a periodic
point expending alternately the times of the sequence close to P and Q, respectively. Moreover, the transition time
between a neighborhood of P and a neighborhood of Q can be chosen bounded. This property allow us to scatter
in the whole homoclinic class of P some properties of the periodic points of this class.

Now, we introduce the concept of linear systems with transitions as in [5]. Given a set A, a word with letters in
A is a finite sequence of elements of A. The product of the word [a] = [a1, . . . , an] by [b] = [b1, . . . , bm] is the
word [a1, . . . , an, b1, . . . , bm]. We say a word is not a power if [a] 
= [b]k for every word [b] and k > 1.

Let (Σ,f,E,A) be a periodic linear system of dimension 2N , that is, all x ∈ Σ is a periodic point for f with
period n = n(x). We denote MA the product An(x) of A along the orbit of x.

If (Σ,f,A) is a periodic symplectic linear system of matrices in SP(2N,R), then for any x ∈ Σ we write,

[M]A(x) = (
A

(
f n−1(x)

)
, . . . ,A(x)

)
,

where n is period of x. The matrix MA(x) is the product of the words [M]A(x).

Definition 4.4 ((Definition 1.6 of [5])). Given ε > 0, a periodic linear system (Σ,f,E,A) admits ε-transitions if for
every finite family of points x1, . . . , xn = x1 ∈ Σ there is an orthonormal system of coordinates of the linear bundle
E so that (Σ,f,E,A) can now be considered as a system of matrices (Σ,f,A), and for any (i, j) ∈ {1, . . . , n}2

there exist k(i, j) ∈ N and a finite word [t i,j ] = (t
i,j

1 , . . . , t
i,j

k(i,j)) of matrices in SP(2N,R), satisfying the following
properties:

(1) For every m ∈ N, ı = (i1, . . . , im) ∈ {1, . . . , n}m, and α = (α1, . . . , αm) ∈ Nm consider the word[
W(ı,α)

] = [
t i1,im

][
MA(xim)

]αm
[
t im,im−1

][
MA(xim−1)

]αm−1 · · · [t i2,i1][MA(xi1)
]α1,



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 651
where the word w(ı,α) = ((xi1 , α1), . . . , (xim,αm)) with letters in M × N is not a power. Then there is
x(ı, α) ∈ Σ such that
• The length of [W(ı,α)] is the period of x(ı, α).
• The word [M]A(x(ı, α)) is ε-close to [W(ı,α)] and there is an ε-symplectic perturbation à of A such that

the word [M]
Ã
(x(ı, α)) is [W(ı,α)].

(2) One can choose x(ı, α) such that the distance between the orbit of x(ı, α) and any point xik is bounded by
some function of αk which tends to zero as αk goes to infinity.

Given ı, α as above, the word [t i,j ] is an ε-transition from xj to xi . We call ε-transition matrices the matrices
Ti,j which are product of the letters composing [t i,j ]. We say a periodic linear system admits transitions if for any
ε > 0 it admits ε-transitions.

The following lemma gives an example of linear systems with symplectic transitions. It is a symplectic version
of [5, Lemma 1.9] and its proof, based on the existence of Markov Partitions, is analogous the proof of [5].

Lemma 4.5. Let f be a symplectic diffeomorphism and let P be a hyperbolic saddle of index k (dimension of
its stable manifold). The derivative Df induces a continuous periodic symplectic linear system on the set Σ of
hyperbolic saddles in the homoclinic class H(P,f ) of index k and homoclinically related to P .

Remark 4.6. We have some good properties that we use during the next sections. Consider points x1, . . . , xn =
x1 ∈ Σ and ε-transitions [t i,j ] from xj to xi . Then

(1) for every positive α � 0 and β � 0 the word([M]A(xi)
)α[

t i,j
]([M]A(xj )

)β

is also an ε-transition from xj to xi .
(2) for any i, j and k the word [t i,j ][tj,k] is an ε-transition from xk to xi .

The following lemma whose proof is analogous of [5, Lemma 1.10] states that every periodic symplectic system
with transitions can be approximated by a diagonalizable systems defined on a dense subset of Σ . We emphasize
that this lemma is also true for symplectic case.

Lemma 4.7. Let (Σ,f,E,A) be a periodic linear symplectic system with transition. Then for any ε > 0 there is a
diagonalizable symplectic ε-perturbation à of A defined on a dense invariant subset Σ̃ of Σ .

Remark 4.8. We remark that the diagonalizable system near to A as required in the above lemma is not necessarily
continuous, but it does not matter in the way we apply this lemma.

5. A dichotomy for symplectic linear systems

In this section, we reduce the study of the dynamics of symplectic diffeomorphisms in Theorem 2.1 to a prob-
lem on symplectic linear systems in Proposition 5.3. We split the proof of this proposition in two propositions
(Propositions 5.5 and 5.6) whose proofs is given in Sections 6 and 7, see [5, Section 2.1] for more details.

An important tool to make the interplay between a dichotomy for diffeomorphisms and for linear symplectic
systems is a symplectic version of Franks’ lemma.

Lemma 5.1 ((Symplectic Franks’ lemma)). Let f ∈ Diff1
ω(M) and E a finite f -invariant set. Assume that B

is a small symplectic perturbation of Df along E. Then for every neighborhood V of E there is a symplectic



652 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
diffeomorphism g arbitrarily C1-close to f coinciding with f on E and out of V , and such that Dg is equal to B

on E.

Remark 5.2. If the symplectic diffeomorphism in Lemma 5.1 belongs to Diff r
ω(M), 1 � r � ∞, then the cor-

responding perturbed diffeomorphism can be taken in Diff r
ω(M). That is because the perturbation envolves ex-

ponential function and some bump functions which are C∞. However, it is important to note that the perturbed
diffeomorphism is just C1-close to the initial one.

The following proposition is a result that provides a dichotomy for symplectic linear systems.

Proposition 5.3 ((Main proposition)). For any ε > 0, N ∈ N and K > 0 there is � > 0 such that any continuous pe-
riodic 2N -dimensional linear system (Σ,f,E,A) bounded by K (i.e. ‖A‖ < K) and having symplectic transitions
satisfies the following,

• either A admits an �-dominated splitting,
• or there are a symplectic ε-perturbation à of A and a point x ∈ Σ such that M

Ã
(X) is the identity matrix.

In the proof of this theorem we use the following notions.

Definition 5.4 ((Definition 2.2 of [5])). Let M ∈ GL(N,R) be a linear isomorphism of RN such that M has some
complex eigenvalue λ, i.e., λ ∈ C \ R. We say λ has rank (i, i + 1) if there is a M-invariant splitting of RN ,
F ⊕ G ⊕ H , such that:

• every eigenvalue σ of M|F (resp. M|H ) has modulus |σ | < |λ| (resp. |σ | > |λ|),
• dim(F ) = i − 1 and dim(H) = N − i − 1,
• the plane G is the eigenspace of λ.

We say a periodic linear system (Σ,f,E,A) has a complex eingenvalue of rank (i, i + 1) if there is x ∈ Σ such
that the matrix MA(x) has a complex eigenvalue of rank (i, i + 1).

We split the proof of Proposition 5.3 into the following two results, which are symplectic version of [5, Propo-
sitions 2.4 and 2.5]:

Proposition 5.5. For every ε > 0, N ∈ N and K > 0 there is � ∈ N satisfying the following property: Let
(Σ,f,E,A) be a continuous periodic 2N -dimensional linear system with symplectic transitions such that its norm
‖A‖ is bounded by K . Assume that there exists i ∈ {1, . . . ,2N − 1} such that every symplectic ε-perturbation Ã

of A has no complex eigenvalues of rank (i, i + 1). Then (Σ,f,E,A) admits an �-dominated splitting F ⊕ G,
F ≺� G, with dim(F ) = i.

Proposition 5.6. Let (Σ,f,E,A) be a periodic linear system with symplectic transitions. Given ε > ε0 > 0 as-
sume that, for any i ∈ {1, . . . ,2N − 1}, there is a symplectic ε0-perturbation of A having a complex eigenvalue
of rank (i, i + 1). Then there are a symplectic ε-perturbation à of A and x ∈ Σ such that M

Ã
is the identity

matrix.

The proof of Proposition 5.5 follows from 2-dimensional arguments of Mañé [11] and higher dimensional argu-
ments of Bonatti, Díaz and Pujals in [5]. However, when we change a symplectic linear system along a subspace
by a symplectic perturbation, it produces an effect along it conjugated symplectic subspace. So, in many time, we
have to deal with 4-dimensional arguments instead of Mañé’s 2-dimensional arguments.



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 653
Although our arguments are based on [5], all perturbations we have to perform are symplectic. For this reason,
we have to be more careful and introduce new techniques.

6. Proof of Proposition 5.5

The main ideas of the proof of Proposition 5.5 is to use the argument of Mañé in 2-dimensional case and
reduction techniques in [5]. In fact, by symplectic nature of our systems, the problem will not be reduced to a
2-dimensional problem. Roughly speaking, the problem will be reduced to a problem in a 4-dimensional subspace.

So, first of all, let us recall the 2-dimensional version of Proposition 5.5.

Proposition 6.1 ((Mañé)). Given any K and ε > 0 there is � ∈ N such that for every 2-dimensional linear system
(Σ,f,E,A), with norm ‖A‖ bounded by K and such that the matrices MA(x) preserve the orientation,

(1) either A admits an �-dominated splitting,
(2) or there are an ε-perturbation B of A and x ∈ Σ such that MB(x) has a complex (non-real) eigenvalue.

Remark 6.2. If A in the above proposition preserves a non-degenerate form ω defined on E then it is possible to
choose B ε-close to A also preserving ω. Recall that, in 2-dimensional setting, if det(B) = 1 then B preserves ω. In
order to construct such B suppose that det(B̃) 
= 1 where B̃ comes from the above proposition, we just substitute
B̃ by B = B̃/det(B̃). The fact that the determinant map is continuous and det(A) = 1 implies that B is close
to A.

6.1. Dimension reduction

To generalize Proposition 6.1 to higher dimensions, Bonatti, Diaz and Pujals in [5] introduced dominated split-
tings for quotient space. Here we just recall the definitions, for more details we recommend the reader to see
[5, Section 4].

Let (Σ,f,E,A) be a linear system and F an invariant subbundle of E with constant dimension. We denote
by AF the restriction of A to F and by A/F the quotient of A along F endowed with the metric of orthogonal
complement F⊥ of F ; i.e., given a class [v] we let |[v]| = |v⊥

F |.
An important statement proved in [5] is the following.

Lemma 6.3 ((Lemma 4.4 of [5])). For any K > 0 and � ∈ N, there exists L with the following property: Given any
linear system (Σ,f,E,A) such that ‖A‖ is bounded by K with an invariant splitting E ⊕ F ⊕ G, one has

(1) E ≺� F and E/F ≺� G/F ⇒ E ≺L F ⊕ G.
(2) F ≺� G and E/F ≺� G/F ⇒ E ⊕ F ≺L G.

Using the previous lemma, the proof of Proposition 5.5 follows from Lemma 6.4 besides a inductive process
completely analogous to that in [5, Lemmas 5.2 and 5.3]. In fact the next lemma is a symplectic version of [5,
Lemma 5.1] and it can be understood as a 2-dimensional version of Proposition 5.5 and in it proof we have to take
account that we are deal with symplectic systems. Therefore, we give a complete proof of this statement. We omit
the inductive argument necessary to the proof of Proposition 5.5, since it is similar to the mentioned work.

Lemma 6.4. Given K > 0 and ε > 0 there is � ∈ N such that for any diagonalizable linear periodic system
(Σ,f,E,A) of dimension 2N and bounded by K , and any 1 � i � 2N − 1 one has



654 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
• Either there is an ε-perturbation of A having a complex eigenvalue of rank (i, i + 1),
• or for every j � i � k

Ej/(Ej+1 ⊕ · · · ⊕ Ek) ≺� Ek+1/(Ej+1 ⊕ · · · ⊕ Ek).

Proof. Suppose that λ1 < · · · < λN < λN+1 < · · · < λ2N are the eigenvalues of A. As A is symplectic we have
λi = (λ2N−i+1)

−1 = λ−1
i∗ .

Fix ε > 0 and let � be the dominance constant in Proposition 6.1. If Ej/(Ej+1 ⊕ · · · ⊕ Ek) ≺� Ek+1/(Ej+1 ⊕
· · · ⊕ Ek) we are done. Otherwise, by Proposition 6.1, we perturb the quotient to collapse the eigenvalues λj and
λk+1. Moreover, by Lemma 4.2, this perturbation of the quotient gives a perturbation à of A having a pair of
eigenvalues λ̃j = λ̃k+1, which are respectively continuation of λj and λk+1 and preserving the eigenvalues of the
restriction of A to Ej+1 ⊕ · · · ⊕ Ek .

Consider a symplectic isotopy At , 0 � t � 1, such that A0 = A, A1 = Ã and denote λj,t and λk+1,t the contin-
uations of λj and λk+1 at time t . Let us assume that λj,t � λk+1,t for every 0 � t < 1. We analyze the following
cases:

(1) λi � 1 � λi+1,
(2) λi < λi+1 < 1,
(3) 1 < λi < λi+1.

Observe that, as this perturbation is symplectic, the eigenvalues λ̃j∗ and λ̃(k+1)∗ also will collapse. So, we get Ã

having the same eigenvalues λs , for s /∈ {j, j∗, k + 1, (k + 1)∗} of A and λ̃j = λ̃k+1, λ̃(k+1)∗ = λ̃j∗ . Furthermore,
when we get a complex eigenvalue of rank (i, i + 1), we also get a complex eigenvalue of rank ((i + 1)∗, i∗).
Hence, the proof of item (2) yields to the proof of item (3).

In order to proof item (1), note that we must have i = N . It is because λi � 1 and λi+1 = λ∗
i � 1 are consecutive

eigenvalues of a symplectic system. Moreover, since A is diagonalizable then λN and λN+1 = λ∗
N cannot assume

the value 1. For the proof of this item, we have the following alternatives:

(1.a) λ̃j = λ̃k+1 < λN . So, there exists 0 � t � 1 such that λk+1,t = λN and λj,t < λN . Hence, λ(k+1)∗,t = λN+1 >

1. Therefore, there exists t ′ < t such that λk+1,t ′ = λ(k+1)∗,t ′ = 1. Then, we perturb At ′ to get a complex
eigenvalue of rank (i, i + 1).

(1.b) λ̃j = λ̃k+1 > λN+1. This case is similar to the previous one.
(1.c) λN < λ̃j = λ̃k+1 < λN+1. Recall that λN < 1 < λN+1. So, there exists t such that either λj,t = λj∗,t = 1

or λk+1,t = λ(k+1)∗,t = 1. In both cases we obtain a complex eigenvalue of rank (i, i + 1) after a small
perturbation of At .

Now, we consider the second case where λi < λi+1 < 1. Again we take account the following alternatives:

(2.a) λ̃j = λ̃k+1 < λi . If λk+1 < λ(i+1)∗ then there exists 0 � t � 1 such that λk+1,t = λi . After a small perturbation
of At , we get a complex eigenvalue of rank (i, i + 1). Otherwise, λk+1 > λ(i+1)∗ implies λ(k+1)∗ < λi . Then,
there exists 0 � t ′ < 1 such that either λj,t ′ < λ(k+1)∗,t ′ = λi+1 (recall that λ̃(k+1)∗ > 1) or λ(k+1)∗,t ′ < λj,t ′ =
λi+1 (there are no reason to λj,t remains less than λi+1 during all the isotopy). In both cases we perturb
slightly At ′ to produce a complex eigenvalue of rank (i, i + 1).

(2.b) λ̃j = λ̃k+1 > λi+1. If λk+1 < λ(i+1)∗ , let 0 � t ′ � 1 be the smallest t such that or λi < λj,t = λ(k+1)∗,t < λi+1
or λj,t = λi+1 < λ(k+1)∗,t or λj,t < λi = λ(k+1)∗,t . After a small perturbation of At ′ , we get a complex
eigenvalue of rank (i, i + 1). Otherwise, λk+1 > λ(i+1)∗ implies λ(k+1)∗ < λi . Then, there exists 0 � t ′ � 1
such that either λ(k+1)∗,t ′ < λj,t ′ = λi+1 or λj,t ′ < λ(k+1)∗,t ′ = λi+1. In both cases we perturb slightly At ′ to
produce a complex eigenvalue of rank (i, i + 1).



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 655
(2.c) λi < λ̃j = λ̃k+1 < λi+1. If λk+1 > λ(i+1)∗ then λ∗
k+1 < λi . Then, there exists 0 � t ′ � 1 such that either

λj,t ′ < λ(k+1)∗,t ′ = λi or λ(k+1)∗,t ′ < λj,t ′ = λi . In both cases we perturb slightly At ′ to produce a complex
eigenvalue of rank (i, i + 1). Otherwise, λk+1 < λ(i+1)∗ implies that a small perturbation of à gives us a
complex eigenvalue of rank (i, i + 1).

This completes the proof. �
End of the proof of Proposition 5.5. Observe that Lemma 6.4 is for diagonalizable systems. But, now we use
Lemma 4.7 to end the proof of Proposition 5.5.

Assume that there are ε > 0 and i ∈ {1, . . . ,2N − 1} such that every ε-perturbation of A has no complex eigen-
value of rank (i, i + 1). Choose a sequence εn < ε/2 converging to zero. As the system (Σ,f,A) has transition,
using Lemma 4.7, we get a dense subset Σn and diagonalizable εn-perturbation Bn of A defined on Σn. Then, we
apply Lemma 6.4 for Bn: there is an integer L > 0 such that every Bn admits an L-dominated splitting En ⊕Fn with
dim(En) = i. Finally, as Σn are dense and ‖Bn − A‖ → 0, we conclude that A admits an L-dominated splitting
E ⊕ F with dim(E) = i. �

7. Proof of Proposition 5.6

After getting periodic points with complex eigenvalues with rank (i, i+1) the idea in the proof of Proposition 5.6
is to use the symplectic transitions to multiply matrices corresponding to different points of Σ having complex
eigenvalues of different ranks. Observe that as f is symplectic a periodic point with complex eigenvalue of rank
(i, i + 1) is also of rank ((i + 1)∗, i∗).

The next proposition is a symplectic version of Lemma 5.4 of [5].

Proposition 7.1. Let (Σ,f,E,A) be a continuous periodic symplectic linear system with symplectic transitions.
Fix ε0 > 0 and assume that a symplectic ε0-perturbation of A has a complex eigenvalue of rank (i, i + 1) for some
i ∈ {1, . . . ,2N − 1}. Then for every 0 < ε1 < ε0 there is a point p ∈ Σ such that for every 1 � i < 2N there is a
symplectic ε1-transition [t i] from p to itself with the following properties:

• There exists a symplectic ε1-perturbation [M]
Ã
(p) of the word [M]A(p) such that the corresponding matrix

[M]
Ã
(p) has only real positive eigenvalues with multiplicity 1. Denote by λ̃1 < · · · < λ̃2N such eigenvalues

and by Ei(p) their respective (1-dimensional) eigenspaces.
• There is a symplectic (ε0 + ε1)-perturbation [t̃ i] of the transition [t i] such that the corresponding matrix T̃ i

satisfies

– T̃ i (Ej (p)) = Ej(p) if j /∈ {i, i + 1, i∗, (i + 1)∗},
– T̃ i (Ei(p)) = Ei+1(p) and T̃ i (Ei+1(p)) = Ei(p).

A key tool in the proof of Proposition 7.1 is symplectic transitions constructed in Proposition 7.2. These transi-
tions preserve the dominated splitting corresponding to two different periodic points.

Let p,pi ∈ Σ such that

E(p) = F1 ≺ · · · ≺ F2N,

and

E(pi) = E1 ≺ · · · ≺ E(i+1)∗,i∗ ≺ · · · ≺ Ei,i+1 ≺ · · · ≺ E2N,

where Fi and Ei are 1-dimensional eigenspaces and Ei,i+1 and E(i+1)∗,i∗ are 2-dimensional eigenspaces corre-
sponding to the complex eigenvalues. We fix a symplectic basis {f1, . . . , f2N } for E(p). The eigenvalues corre-



656 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
Fig. 1. Constructing a transition whose matrix sends F2N−1 ⊕ F2N to E2N−1,2N .

sponding to E(i+1)∗,i∗ are inverse of the eigenvalues corresponding to Ei,i+1. We only deal with the case i > N ,
that is when Ei,i+1 appears after E(i+1)∗,i∗ in the above dominated splitting. The another case is completely similar
to this case.

Proposition 7.2. There exists a symplectic transition [ti,0] from p to pi such that its matrix Ti,0 satisfies the
following:

• Ti,0(Fj ) = Ej , if j 
= i, i + 1, i∗, (i + 1)∗,
• Ti,0(Fi ⊕ Fi+1) = Ei,i+1, and
• Ti,0(F(i+1)∗ ⊕ Fi∗) = E(i+1)∗,i∗ .

There is also a symplectic transition [t0,i] from pi to p with a similar properties.

To prove the proposition we state two auxiliary lemmas.

Lemma 7.3. Under the hypotheses of Proposition 7.2, if i = 2N − 1 there exists a symplectic transition from p to
pi such that its matrix T satisfies the following properties:

• T (F2N−1 ⊕ F2N) = E2N−1,2N and T (F1 ⊕ F2) = E1,2,

• T (F3 ⊕ · · · ⊕ F2N−2) = E3 ⊕ · · · ⊕ E2N−2.

Proof. Suppose that ti,0 is an arbitrary transition from p to pi . Let Mi and M denote respectively MA(pi) and
MA(p). After a small symplectic perturbation we may suppose

Ti,0(F2N−1 ⊕ F2N) � E1,2 ⊕ E3 ⊕ · · · ⊕ E2N−2.

By domination, for n1 sufficiently large M
n1
i ◦ Ti,0(F2N−1 ⊕ F2N) is close enough to E2N−1,2N . Now by another

small symplectic perturbation S we may send M
n1
i ◦Ti,0(F2N−1 ⊕F2N) inside E2N−1,2N . Let T1 = S ◦M

n1
i ◦Ti,0.

Now, using again the dominated splitting, there is n2 sufficiently large such that T −1
2 (E1,2) := M−n2 ◦ T −1

1 (E1,2)

is close enough to F1 ⊕ F2. It is possible to choose v1, v2 ∈ E1,2 such that

T −1
2 (v1) = f1 +

2N∑
εjfj and T −1

2 (v2) = f2 +
2N∑

ε̃j fj .
j=3 j=3



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 657
Indeed, let Vf2 := F1 ⊕ F3 ⊕ · · · ⊕ F2N . As dim(Vf2) = 2N − 1 and dim(T −1
1 (E1,2)) = 2 there exists w :=

f1 +ε3f3 +· · ·+ε2Nf2N ∈ Vf2 ∩T −1
1 (E1,2). Since w is close enough to F1 ⊕F2 it comes out that εi , i = 3, . . . ,2N ,

are arbitrarily small. So, it is enough to take v1 = T2(w). Similarly we get v2 satisfying the assertion.
Define a symplectic linear map Ĩ as follows.

Ĩ (fi) =

⎧⎪⎪⎨⎪⎪⎩
T −1

2 (vi) if i = 1,2,
fi if i = 2N,2N − 1,
fi + εi∗f2N + ε̃i∗f2N−1 if 3 � i � N ,
fi − εi∗f2N − ε̃i∗f2N−1 if N < i � 2N − 2.

Let us check that Ĩ is a symplectic transformation. It is enough to verify that ω(Ĩ (fj ), Ĩ (fk)) = ω(fj , fk) for all
1 � j , k � 2N , j 
= k. We have the following cases:

(i) Let j = 1,2 and 3 � k � N . One just prove for j = 1 (the case j = 2 is analogous). We have

ω
(
Ĩ (f1), Ĩ (fk)

) = ω

(
f1 +

2N∑
j=3

εjfj , fk + εk∗f2N + ε̃k∗f2N−1

)
= ω(f1, εk∗f2N) + ω(εk∗fk∗ , fk) = εk∗ − εk∗ = 0 = ω(f1, fk).

(ii) Let j = 1,2 and N < k � 2N − 2. Again we just give the proof for j = 1. We have

ω
(
Ĩ (f1), Ĩ (fk)

) = ω

(
f1 +

2N∑
i=3

εjfj , fk − εk∗f2N − ε̃k∗f2N−1

)
= ω(f1,−εk∗f2N) + ω(εk∗fk∗ , fk) = −εk∗ + εk∗ = 0 = ω(f1, fk).

(iii) Let 3 � j, k � 2N − 2. In this case we have

ω
(
Ĩ (fj ), Ĩ (fk)

) = ω(fj ± εj∗f2N ± ε̃j∗f2N−1, fk ± εk∗f2N ± ε̃k∗f2N−1) = ω(fj , fk).

(iv) Let j = 1,2 and k = 2N − 1,2N . Let us just show this case for the when j = 1 and k = 2N :

ω
(
Ĩ (f1), Ĩ (f2N)

) = ω

(
f1 +

2N∑
i=3

εifi, f2N

)
= ω(f1, f2N).

This complete the proof that Ĩ is symplectic.
Hence, T := T2 ◦ M

n2
p ◦ Ĩ maps F1 ⊕ F2 and F2N−1 ⊕ F2N respectively to E1,2 and E2N−1,2N .

It remains to prove that

T (F3 ⊕ · · · ⊕ F2N−2) = E3 ⊕ · · · ⊕ E2N−2.

Take 2 < j < 2N − 1 and assume that T (fj ) /∈ E3 ⊕ · · · ⊕ E2N−2. So, there is v ∈ E1,2 ∪ E2N−1,2N such that
ω(T (fj ), v) 
= 0. Let w := T −1(v) ∈ (F1 ⊕ F2) ∪ (F2N−1 ⊕ F2N). Then,

0 = ω(fj ,w) = ω
(
T (fj ), T (w)

) = ω
(
T (fj ), v

) 
= 0,

which is a contradiction. This completes the proof of the lemma. �
Lemma 7.4. Under the hypotheses of Proposition 7.2, if i < 2N − 1 there exists a symplectic transition T from p

to pi with the following properties:

• T (F2N) = E2N and T (F1) = E1,

• T (F2 ⊕ · · · ⊕ F2N−1) = E, where E = E2 ⊕ · · · ⊕ E2N−1 or E2,3 ⊕ E4 ⊕ · · · ⊕ E2N−3 ⊕ E2N−2,2N−1.



658 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
Proof. Let ti,0 be an arbitrary symplectic ε-transition from p to pi . After a small symplectic perturbation we
may suppose Ti,0(F2N) /∈ E1 ⊕ E2 ⊕ · · · ⊕ E2N−1. So, by the dominated splitting, for n1 sufficiently large M

n1
i ◦

Ti,0(F2N) is close enough to E2N . Now, by another small symplectic perturbation S we may send M
ni

i ◦ Ti,0(F2N)

inside E2N . Let T1 = S ◦ M
n1
i ◦ Ti,0. Then T1(F2N) = E2N . All perturbations are symplectic, but we do not know

whether T1(F1) = E1 or not.
Let {f1, . . . , f2N } be a symplectic basis for E(P ) and e2N = T1(f2N). For any e1 ∈ E1 we have

ω
(
T −1

1 (e2N),T −1
1 (e1)

) = ω(e2N, e1) 
= 0.

So, we conclude that T −1
1 (E1) has a non-null component in the F1 direction. Now, using the dominated splitting,

for n2 sufficiently large we have T −1
2 (e1) := M−n2 ◦ T −1

1 (e1) is close enough to f1 for some e1 ∈ E1. More
precisely for some e1 ∈ E1 we can write:

T −1
2 (e1) = f1 + ε2f2 + ε3f3 + · · · + ε2Nf2N

where fj ∈ Fj and εi are small enough whenever n2 is sufficiently large.
Now we define a symplectic perturbation of the identity map defined on the basis {f1, f2, . . . , f2N } as follows:

Ĩ (fi) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

f1 +
2N∑
i=2

εifi if i = 1,

f2N if i = 2N ,

fi + εi∗f2N if 1 < i � N ,

fi − εi∗f2N if N < i < 2N .

Let us verify that Ĩ is a symplectic transformation. It is enough to verify that ω(Ĩ (fj ), Ĩ (fk)) = ω(fj , fk) for
all 1 � j, k � 2N,j 
= k. We have the following cases:

(i) Let j = 1 and 1 < k < N . Observe that for k � N we have ω(fk∗ , fk) = −1. So, we have

ω
(
Ĩ (f1), Ĩ (fk)

) = ω

(
f1 +

2N∑
i=2

εifi, fk + εk∗f2N

)
= ω(f1, εk∗f2N) + ω(εk∗fk∗ , fk)

= −εk∗ + εk∗ = 0 = ω(f1, fk).

(ii) Let j = 1, N < k < 2N . Note that for k > N , ω(fk∗ , fk) = 1. Then,

ω
(
Ĩ (f1), Ĩ (fk)

) = ω

(
f1 +

2N∑
i=2

εifi, fk − εk∗f2N

)
= ω(f1,−εk∗f2N) + ω(εk∗fk∗ , fk)

= εk∗ − εk∗ = 0 = ω(f1, fk).

(iii) Let 1 < j,k < 2N . In this case we have

ω
(
Ĩ (fj ), Ĩ (fk)

) = ω(fj ± εj∗f2N,fk ± εk∗f2N) = ω(fj , fk).

(iv) Let j = 1, k = 2N . Then

ω
(
Ĩ (f1), Ĩ (f2N)

) = ω

(
f1 +

2N∑
i=2

εifi, f2N

)
= ω(f1, f2N).

Hence, T := T2 ◦ Ĩ is a symplectic transition from p to pi with T (F2N) = E2N and T (F1) = E1. Moreover, the
fact that T is symplectic implies that for all 1 < j < 2N

ω
(
T (fj ), T (f2N)

) = ω
(
T (fj ), T (f1)

) = 0.



V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661 659
This implies that T (fj ) cannot have coordinates in the e1 and e2N directions. Therefore, T sends the eigenspace
F2 ⊕ · · · ⊕ F2N−1 into E. �
Proof of Proposition 7.2. If i = 2N − 1 we apply Lemma 7.3 and then use the method in the proof of Lemma 7.4
inductively to get the transition Ti,0.

Otherwise, if i < 2N −1 we apply Lemma 7.4 successively 2N − i −1 times to reduce the problem to the above
case. �
Proof of Proposition 7.1. The proof of this proposition follows from the same arguments of [5, Lemma 5.4]. Let
us outline the main steps of the proof and point out how to complete the proof in the symplectic case.

After a small symplectic perturbation we may suppose that Mp is diagonalizable for some p ∈ Σ and there is
pi ∈ Σ such that pi has complex eigenvalue of rank (i, i + 1). Let

(1) E(p) = F1 ≺ · · · ≺ F2N

and

(2) E(pi) = E1 ≺ · · · ≺ E(i+1)∗,i∗ ≺ · · · ≺ Ei,i+1 ≺ · · · ≺ E2N.

By Proposition 7.2 we can construct a symplectic transition [t i] := [t0,i][ti,0] from p to itself whose matrix T i

preserves all subspaces Fj , j 
= i, i + 1, i∗, (i + 1)∗, Fi ⊕ Fi+1, and Fi∗ ⊕ F(i+1)∗ .
Using again Proposition 7.2 and the complex eigenvalues inside Ei,i+1 we can obtain a new transition [t̃ i]

such that T̃ i interchanges Fi and Fi+1. This transition can be constructed exactly as in [5, Lemma 5.7] using
Lemmas 4.3 and 4.2. The unique difference is that to complete the proof in the symplectic case we should prove
that T̃ i interchanges Fi∗ and F(i+1)∗ too.

Given a non-zero vector f(i+1) ∈ F(i+1), let fi = T̃ i (f(i+1)). We show that T̃ i (fi∗) ∈ F(i+1)∗ for any fi∗ ∈ Fi∗ .
Since Fi∗ ⊕F(i+1)∗ is preserved by T̃ i , there exists constants r, s ∈ R such that T̃ i (fi∗) = rfi∗ + sf(i+1)∗ . Then,

0 = ω(fi∗, f(i+1)) = ω
(
T̃ i (fi∗), T̃

i(fi+1)
) = ω(rfi∗ + sf(i+1)∗ , fi) = rω(fi∗ , fi).

The fact that ω(fi∗ , fi) 
= 0 implies r = 0. Therefore T̃ i (fi∗) belongs to F(i+1)∗ . Similarly we can prove that
T̃ i (F(i+1)∗) = Fi∗ . �
End of proof of Proposition 5.6. In Proposition 7.1 we construct transitions [t̃ i] whose action on the finite set
{Fi(p)}1�j�2N of eigenspaces of M

Ã
(p) is the transposition (i, i + 1) which interchanges Ei(p) and Ei+1(p).

In what follows a combinatorial argument exactly as done in [5] shows that after a small perturbation we ob-
tain a totally elliptic periodic point. We verify that the arguments in their paper works also in the symplectic
case.

Given 0 � k < 2N denote by σk the cyclic permutation defined by σk(Ej (p)) = Ej+k(p), where the sum i + j

is considered in the cyclic group Z/(2N)Z.
As any permutation is a composition of transpositions, for every 0 � k < 2N there exists an element [S̃k] in the

semi-group generated by transitions [t̃ i] such that if S̃k is the matrix corresponding to the word [S̃k] then one has
S̃k(Ej (p)) = Ej+k(p).

Let [Sk] be the word of matrices corresponding to the perturbation [S̃k] in the semi-group generated by the
initials [tj ]. As the [tj ] are ε1-transitions from p to itself, any word in the semi-group generated by the [tj ],
in particular the [Sk] is also an ε1-transition from p to itself. Let us write [S0] = [S2N ] the empty word whose
corresponding matrix is the identity.

By definition of transitions, for any n ∈ N there is a point xn ∈ Σ such that the word [M]A(xn) is ε1-close to the
word [Wn] corresponding to the matrix Wn defined by

Wn := W2N−1,n ◦ · · · ◦ W1,n ◦ W0,n,

where Wi,n := S2N−i ◦ (MA(p))n ◦ Si ◦ S2N−iSi .



660 V. Horita, A. Tahzibi / Ann. I. H. Poincaré – AN 23 (2006) 641–661
We know that for any i, the matrix S̃2N−i ◦ S̃i acts trivially on the set of spaces {Ej(p)}. Let us denote by

• λ̃k the eigenvalues of M
Ã
(p) corresponding to Ek ,

• μi,j the eigenvalue of S̃2N−i ◦ S̃i corresponding to the Ej(p).

Consequently, for every j and any n ∈ N the space Ej(p) is an eigenspace of the matrix

W̃i,n := S̃2N−i ◦ (
M

Ã
(p)

)n ◦ S̃i ◦ S̃2N−i S̃i ,

whose corresponding eigenvalue is μ2
i,j λ̃

n
i+j .

As the transitions are symplectic we have λ̃i λ̃i∗ = 1 and μi,jμi,j∗ = 1. It comes out that

2N∏
i=1

λ̃i = 1

and

Cj :=
2N−1∏
i=0

μ2
i,j =

(
2N−1∏
i=0

μ2
i,j∗

)−1

= C−1
j∗ .

The word [W̃n] corresponding to the matrix defined as

W̃n := W̃2N−1,n ◦ · · · ◦ W̃1,n ◦ W̃0,n

is (ε0 + ε1)-close to [Wn] and so it is an ε0 + 2ε1-perturbation of the word [M]A(xn). So we conclude that Ej(p)

is an eigenspace of W̃n with eigenvalue Cj .
Observe that Cj are not necessarily close to 1. Consider Bn matrices having Ej(p) as eigenspaces and (Cj )

−1/n

as their eigenvalues. Observe that Bn is symplectic too and (Bn)
n = W̃−1

n . Denote by [M]
Â
(p) the word obtained

from [M]
Ã
(p) by replacing its first letter Ã(p) by Ã(p)◦Bn. For n large enough this new word is an ε1-perturbation

of [M]
Ã
(p), so by item (i) of Proposition 7.1 it is also 2ε1-perturbation of [M]A(p). As Bn commutes with M

Ã
(p)

we get(
M

Ã
(p) ◦ Bn

)n = Mn

Ã
(p) ◦ W̃−1

n .

So, the word [Ŵn] obtained by changing the initial subword [M]n
Ã
(p) of [W̃n] by [M]

Â
(p) is (ε0 + 2ε1) < ε

close to the word [M]A(xn) and its corresponding matrix Ŵn = W̃n ◦ W̃−1
n = Id. This completes the proof.

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