ads:
Robustly transitive sets
in nearly integrable Hamiltonian systems
MEYSAM NASSIRI
Abstract. We introduce C
r
-open sets (r = 1, 2, . . . , ∞), of symplectic diffeomorphisms
(and Hamiltonian systems), exhibiting large robustly transitive sets. As a consequence
of the constructions we show that, arbitrarily C
∞
-close to certain (nearly) integrable
Hamiltonian systems with more than two degrees of freedom, there exist systems with
unbounded robustly transitive sets.
Contents
1. Introduction and main results 2
2. Iterated function system 7
2.1. Contracting and expanding maps 7
2.2. Recurrent diffeomorphisms 9
3. Symplectic blender 13
3.1. cs-blender with higher dimensional unstable central bundle 15
3.2. Double-blender: affine model 16
3.3. Symplectic blender: affine model 17
3.4. Robustness: relaxing the constructions 17
4. Proof of Theorem A 18
4.1. The perturbations 19
4.2. Constructing symplectic blender 20
4.3. Almost minimality of stable and unstable foliations 21
4.4. Robustness of the almost minimality of foliations 23
4.5. Transitivity and Top ological mixing 24
5. Instabilities in nearly integrable systems 25
5.1. Instability versus recurrency 25
5.2. Proofs of Theorem B and Corollary C 26
6. Some r emarks and open problems 28
References 29
Date: June 12, 2006.
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2 MEYSAM NASSIRI
1. Introduction and main results
The theory of Kolmogorov, Arnold and Moser, (KAM) gives a precise description of the
dynamics of a set of large measure of orbits for any small perturbation of a non-degenerate
integrable Hamiltonian system. These orbits lie on the invariant KAM tori for which the
dynamics are equivalent to irrational (Diophantine) rotations. This theory applies for the
autonomous Hamiltonians, time-periodic Hamiltonians and also for symplectic diffeomor-
phisms. A basic and natural question is what happens for other orbits. What is the possible
behavior of most orbits (in the topological sense) for generic systems?
In the case of autonomous systems in two degrees of freedom or time- periodic systems
in one degree of freedom (i.e., 1.5 degrees of freedom), the KAM theorem proves the stability
of all orbits, in the sense that the actions do not vary much along the orbits. Since each
KAM torus has codimension one in the phase space, its complement is disconnected and
contains two connected components. Thus, any orbit remains between two nearby invariant
tori. This, of course, is not the case if the degree of freedom is larger than two, where
the KAM tori are of codimensions at least two. A natural question arises: Do generic
perturbations of integrable systems in higher dimensions exhibit instabilities?
The problem of instabilities for high dimensional nearly integrable Hamiltonian sys-
tems (i.e. small perturbations of integrable systems) has been considered one of the most
important problems in Hamiltonian dynamics. The first example of instability is due to
Arnold [A2], w ho constructed a family of small perturbations of a non-degenerate inte-
grable Hamiltonian system that exhibits instability in the sense that there are orbits for
which the variation of action is large. This kind of topological instability is sometimes called
the Arnold diffusion. In fact, he had conjectured [A1, pp. 176] that the answer of the above
question should be positive. While there is a large number of works and announcements
towards this conjecture, specially in the recent years (see e.g. [CY], [D], [DLS], [KMV],
[Ma], [X], and references there), few is known about “most of the orbits” in the complement
of invariant or periodic Diophantine tori. Although it is very difficult to prove the existence
of “some” instable orbits in general, it is the simplest expected non-trivial behavior in the
complement of invariant tori. For instance, one may ask about transitivity or topological
mixing.
On the other hand, in the non-conservative dynamics, there are several important
recent contributions about robust transitivity. Recall that a diffeomorphism of a manifold
M is transitive if it has a dense orbit in the whole manifold. Such a diffeomorphism is called
C
r
-robustly transitive if it belongs to the C
r
-interior of the s et of transitive diffeomorphisms.
It has been known since 1960’s, that any hyperbolic basic set is C
1
-robustly transitive. The
first examples of non-hyperbolic C
1
-robustly transitive sets are due to M. Shub [Sh] and
R. Ma˜n´e [M˜n]. For a long time their examples remained the unique ones. Then, L. D´ıaz
(who was mainly interested in the dynamical consequences of hetero-dimensional cycles),
jointly with C. Bonatti, discovered [BD] a semi-local source for transitivity, and C
1
robust
in nature. They called it blender. Using this tool, one may construct examples of robustly
transitive sets and diffeomorphisms. In contrast, Bonatti, D´ıaz, Pujals, Ures, [DPU], [BDP]
ads:
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 3
have surprisingly shown that any C
1
robustly transitive set admits an invariant dominated
splitting on its tangent bundle, and a weak form of hyperbolicity holds. This result has
been extended independently by Horita, Tahzibi [HT] and by Saghin [Sa] to the symplectic
case, where the robust transitivity holds only in the symplectic world. Another important
result in this direction is due to Arnaud, Bonatti and Crovisier [BC], [ABC]. They show
that generically in the C
1
topology any symplectic diffeomorphism on compact manifold
is transitive. They also prove that on non-compact manifolds, generic orbits of generic
diffeomorphisms are not bounded. It is important to note that the C
1
topology is essential
in all these results, because of the use of several basic perturbation lemmas (connecting
lemmas, Franks lemma, etc.) known only in the C
1
topology. For the recent surveys on this
topic and on a related theory about stably ergodic diffeomorphisms on compact manifolds,
developed in the last decade by C. Pugh, M. Shub, and many others, see [BDV, chapters
7,8], [PS], [PSh].
In this paper, the problem of instability is investigated as a consequence of the existence
of large or unb ounded robustly transitive sets. We develop the methods of robust transitivity
into the context of symplectic and Hamiltonian systems. And then we apply them for
the nearly integrable symplectic and Hamiltonian systems with more than two degrees
of freedom. We introduce such Hamiltonians or symplectic diffeomorphisms exhibiting
unbounded or large robustly transitive sets. In particular, a stronger form of instability for
a large set of orbits is obtained.
Let us introduce some definitions before to state the main results. Let f : M −→ M be
a diffeomorphism of a compact manifold M . An f-invariant subset Λ is partially hyperbolic
if its tangent bundle T
Λ
M splits as a Whitney sum of T f-invariant subbundles:
T
Λ
M = E
u
⊕ E
c
⊕ E
s
,
and there exist a Riemannian metric on M and constants 1 > λ > 0 and µ > 1 such that
for every p ∈ Λ ,
m(T
p
f|
E
u
) > µ > T
p
f|
E
c
≥ m(T
p
f|
E
c
) > λ > T
p
f|
E
s
> 0.
The co-norm m(A) of a linear operator A between Banach spaces is defined by m(A) :=
inf{ A(v) : v = 1}. The bundles E
u
, E
c
and E
s
are referred to as the unstable, center
and stable bundles of f, respectively.
An example of a partially hyperbolic set is a hyperbolic set, for which E
c
= 0.
Let f and g be two diffeomorphisms on manifolds M and N, respectively. Supp ose
that Λ ⊂ M is an invariant hyperbolic set for f. We say g is dominated by f|
Λ
if Λ × N is
a partially hyperbolic set for f × g, with E
c
= T N. In this paper we sometimes talk about
“weak” hyperbolic periodic point, meaning that the point is hyperbolic, but contraction
and expansion constants are sufficiently close to one.
In a similar way one may define partially hyperbolic sets in a non-compact manifold.
Let X be a metric space, and F : X → X. A set Y ⊂ X is transitive for F if for any
U
1
, U
2
open in X, such that U
i
∩ Y = ∅, there is some n with F
n
(U
1
) ∩ U
2
= ∅. If in
4 MEYSAM NASSIRI
addition, for any open sets U
1
, U
2
⊂ Y (in the res tricted topology), there is some n with
F
n
(U
1
) ∩ U
2
= ∅, then we say Y is strictly transitive. A stronger property is topological
mixing, where F
n
(U
1
) ∩ U
2
= ∅ holds for any sufficiently large n.
Let D
r
be a subspace of Diff
r
(M) with the C
r
topology. A compact set Y ⊂ M is
D
r
-robustly transitive for f ∈ D
r
, if for any g ∈ D
r
sufficiently close to f , the continuation
(should be well defined) of Y is transitive for g. More generally if M is not compact, a
non-relatively compact set Y ⊂ M is D
r
-robustly transitive if it is a union of compact
D
r
-robustly transitive sets. In the same way one may define robustly (strictly) topological
mixing.
A point x is non-wandering for a diffeomorphism f if for any neighborhood U of x
there is n ∈ N such that f
n
(U) ∩ U = ∅. By Ω(f) we denote the set of all non-wandering
point of f.
Now, let us recall some basic facts and definitions of symplectic topology. A symplectic
manifold is a C
∞
smooth manifold M together with a closed non-degenerate differential
2-form ω. We denote it by (M, ω) but sometimes we just write M. Examples of symplectic
manifolds are orientable surfaces, even dimensional tori and cylinders, and the cotangent
bundle T
∗
N of an arbitrary smooth manifold. A C
1
diffeomorphism f is symplectic if f
preserve ω; i.e. f
∗
ω = ω. We denote by Diff
r
ω
(M) the space of C
r
symplectic diffeomor-
phisms of M with the C
r
topology, 1 ≤ r ≤ ∞. If the symplectic form ω is exact, that is
ω = dα for some 1-form α, and f
∗
α − α = dS for some smooth function S : M → R, then
we say that f is an exact symplectic diffeomorphism.
Our main result concerning symplectic diffeomorphisms is the following.
Theorem A. Let M and N be two symplectic manifolds (not necessarily compact). Let
f
1
∈ Diff
r
ω
(M) such that there exists an open set U ⊂ M whose maximal invariant set Λ is
a hyperbolic transitive compact set. Let f
2
∈ Diff
r
ω
(N) such that:
a) f
2
is dominated by f
1
|
Λ
, and f
2
has a (weak) hyperbolic periodic point.
b) For any
˜
f
2
sufficiently C
r
close to f
2
, Ω(
˜
f
2
) = N .
Then there is a C
r
-arc {F
µ
}
µ∈[0,1]
of C
r
symplectic diffeomorphisms on M × N , such that
F
0
= f
1
× f
2
, and for all µ ∈ (0, 1], Λ × N is robustly strictly topologically mixing in
Diff
r
ω
(M × N). More precisely,
(1) For all F
µ
, µ ∈ (0, 1], the maximal invariant set Γ
F
µ
in U × N is (strictly) topo-
logically mixing. Moreover Γ
F
µ
contains hyperbolic periodic points whose stable and
unstable manifolds are dense in it.
(2) For any bounded domain N
c
in N , and any ν > 0, there exists a neighborhood U of
{F
µ
: µ ∈ (0, 1]} in Diff
r
ω
(M × N) such that for any G ∈ U the set Γ
G,N
c,ν
, the
continuation of Λ × N
c,ν
, is transitive and topological mixing, where N
c,ν
⊂ N
c
and
N
c
\ N
c,ν
is an open set in N of Lebesgue measure smaller than ν.
The Theorem A, roughly speaking, says that if the product of a hyperbolic basic set Λ
by any non-wandering dynamics on N is partially hyperbolic then we can perturb it such
that (the continuation of) Λ × N become a robustly topological mixing set.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 5
Remark that the non-wandering hypothesis (b) is obviously satisfied if the manifold
N is compact or has a finite volume.
As a matter of fact, all known results about instability concern with nearly integrable
systems. There are two reasons for it. First, for nearly integrable systems the stability seems
a priori highly probable, and the invariant KAM tori are the “obstructions for instability”.
So, to study the instabilities in general, one considers perturbations of integrable systems as
the most crucial examples. Second, KAM theory gives useful dynamical informations of the
system, and these informations are crucial in the classical methods for proving instabilities.
One of the advantages of Theorem A is that the initial system F
0
is not necessarily close to
the integrable systems, e.g. f
1
could be an Anosov diffeomorphism.
Theorem A is also related to an interesting example of Shub and Wilkinson [SW]. They
proved that the product of “Anosov × Standard map” on T
4
is C
∞
approximated by (sym-
plectic) stably ergodic systems. The ergodicity implies transitivity, but not topologically
mixing. In the proof they use the central tool in the theory of stable ergodicity, namely,
the accessibility. Two things seem essential in their proof. The first one is global (partial)
hyperb olicity and the second one is compactness. See also Rem ark 6.1. On the other hand,
there is no ergodic nearly integrable system, because the union of invariant KAM tori has
positive Lebesgue measure. Theorem A can be seen as a local and topological version of
this example.
Let (M, ω) be a symplectic manifold and H : R × M → R a C
r
function, called the
(time dependent) Hamiltonian. For any t ∈ R, the vector field X
H
t
determined by the
condition
ω(X
H
t
, Y ) = dH
t
(Y ) or equivalently i
X
H
t
ω = dH
t
is called the Hamiltonian vector field associated with H
t
:= H(t, ·) or the symplectic gradient
of H
t
. A diffeomorphism is called Hamiltonian diffeomorphism if it is the time-one map of
some time periodic Hamiltonian flow.
Remark 1.1. The Theorem A can be stated in the context of exact and Hamiltonian diffeo-
morphisms, and also time-dependent Hamiltonians. The statements are analogues and it is
left to the reader.
Theorem B. Let M and N be two symplectic manifolds (not necessarily compact), and h
1
and h
2
be two C
r
Hamiltonians on M and N, respectively. Let f
1
and f
2
be the time one
map of the hamiltonian flow generated by h
1
and h
2
, respectively. Suppose that
(i) h
1
is time periodic and f
1
has a transversal homoclinic point,
(ii) f
2
is dominated by a hyperbolic invariant set of f
1
,
(iii) the whole manifold N is the non-wandering set for h
2
.
Then the Hamiltonian h
1
+ h
2
is approximated in C
∞
topology by time-periodic Hamilto-
nians on M × N exhibiting a topologically mixing partially hyperbolic invariant set Ξ × N.
Moreover, the continuation of this set is well defined, and either it remains topologically
mixing or it contains wandering points converging to the infinity.
6 MEYSAM NASSIRI
When the manifold N is of dimension two, then as we mentioned before, existence of
invariant KAM tori provides the stability of all points. In particular, there is no wandering
point. The following corollary concerns with the class of integrable systems that contains
the so-called a priori unstable integrable Hamiltonian systems H (cf. [CY], [DLS], [X]).
Corollary C. Let H
0
(p, q, x, y, t) = h
2
(p) + h
1
(x, y, t) be a time-periodic Hamiltonian,
where t ∈ T := R/Z is the time, (p, q) ∈ R × T, and (x, y) ∈ R
n
× T
n
. Suppose that
h
2
(p) = p
2
, and let h
1
be an arbitrary Hamiltonian with some non-hyperbolic periodic
orbit. Then, C
∞
-arbitrarily close to H
0
, there are C
r
(r ≥ 5) open sets of time periodic
Hamiltonians exhibiting instability, namely, there exist topologically mixing invariant sets
containing arbitrary large regions of the action variable p.
Let us say a few words on the proofs. First ingredient is a new tool in symplec tic
dynamics called symplectic blender, a semi-local source of robust transitivity. It is based on
the seminal work of Bonatti and D´ıaz [BD]. T he symplectic blender provides robustness of
the density of stable and unstable manifolds of a hyperbolic periodic point, in any compact
region, which implies robustness of transitivity or even topological mixing.
Another main ingredient is that we reduce the problem to the one of iterated function
system. Indeed, in comparison with the classical methods for instability, here we follow the
dynamical consequence s of the whole structure of homoclinic intersections of a normally hy-
perbolic submanifold, instead of only one of such intersections. Any homoclinic intersection
gives a holonomy map (or the outer maps of [DLS]). Hence considering the whole struc-
ture of homoclinic intersections we will have infinitely many different outer maps, and this
allows us to obtain instability and transitivity. We found the iterated function system as a
natural and nice context to set down this idea. As a model one may consider perturbations
of the product of a horseshoe and an integrable twist map and then results on the iterated
function system yield minimality of (strong) stable and unstable foliations. Then using the
symplectic blender one can show that transitivity (or even topological mixing) appears in
an action variable and in a robust fashion.
Note specially that we do not use any KAM-type invariant sets in the proof. For
instance, recurrency has an important role. And therefore, the classical problem, the large
gap problem doe s not make sense here, although the large gaps b etween Diophantine tori
may appear in a normally hyperbolic manifold N .
This paper is organized as the following. In section 2 we study transitivity of two
deferent kinds of the iterated function systems (IFS). Namely, the IFSs of expanding maps,
and the IFSs of recurrent diffeomorphisms. We use the former ones in sec tion 3, where
we introduce the symplectic blenders. In section 4 we prove Theorem A. In section 5 we
prove Theorem B and Corollary C. Finally, in section 6 several remarks and open problems
related to the main results are discussed.
Acknowledgment. I would like to thank my thesis advisor Enrique Pujals for many
insightful discuss ions. I am indebted to him and Jacob Palis for their advices, supports and
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 7
enormous encouragements. I am also grateful to Marcelo Viana for useful conversations.
Finally, I wish to acknowledge financial supports of IMPA-CNPq.
2. Iterated function system
In this se ction we study transitivity of some iterated function system (IFS). In the IFSs,
instead of taking iteration by only one map, one considers all the possible compositions and
iterations of several maps. As a consequence, a point x may have an infinite number of orbits.
The transitivity of the iterated function system of expanding maps has a fundamental role
in the construction and properties of blenders (see section 3) and of the symplectic maps
will be used in the proof of density of (strong) stable and unstable manifolds (see section
4).
Let g
1
, g
2
, . . . , g
n
be functions defined on the metric space X. The iterated function
system G(g
1
, g
2
, . . . , g
n
) is the action of the group generated by {g
1
, g
2
, . . . , g
n
} on X. We use
the notion of multi-index σ = (σ
1
, . . . , σ
k
) ∈ {1, 2, . . . , n}
k
for g
σ
= g
σ
k
◦· · ·◦g
σ
1
. We also de-
note |σ| := k. An orbit of x ∈ X under the iterated function system G = G(g
1
, g
2
, . . . , g
n
) is a
sequence {g
Σ
k
(x)}
∞
k=1
where Σ
k
= (σ
1
, . . . , σ
k
) and {σ
i
}
∞
i=1
∈ {1, 2, . . . , n}
N
. By Orbit
+
G
(x)
we denote the orbit of x ∈ X under the IFS G. Similarly, we denote Orbit
−
G
(x) as the back-
ward orbit of x, i.e. the orbit of x under the IFS G
−1
= G(g
−1
1
, g
−1
2
, . . . , g
−1
n
). The orbit of
a subset U ⊂ X is defined as the union of all its orbits, i.e. Orbit
+
G
(U) = ∪
x∈U
Orbit
+
G
(x).
Definition 2.1. The IFS G(g
1
, g
2
, . . . , g
n
) is said transitive if the G-orbit of any open set is
dense. A set U is transitive for G if the G-orbit any open subset of U is dense in U. This is
equivalent to the existence of some point with dense G-orbit in U.
2.1. Contracting and expanding maps. In this subsection we study the transitivity for
the iterated function system of contracting and expanding maps. The result presented here
we be used in the construction of blender in section 3.
The simplest examples of contracting (expanding) maps are linear maps of R
n
, for
which the absolute value of all eigenvalues are smaller than one (resp., bigger than one). In
other words, a matrix A is contracting iff, A := sup{|Av|/|v| : |v| = 0} < 1 for some
norm | · | in R
n
.
In general, a map φ on a metric space (X, d) is contracting iff there is a constant
0 < K < 1 such that d(φ(x), φ(y)) < Kd(x, y), for all x, y ∈ X. The contraction bound (if
exists), is a number λ ∈ (0, 1) for which, φ in addition satisfies λd(x, y) < d(φ(x), φ(y)),
for all x, y ∈ X. This constant does not exist for any contracting map. For example, if
some points converges super-exponentially fast to the unique fixed point of φ, and it can
easily be constructed. For generic smooth contracting map φ on R
n
, the contraction bound
does exist if we consider only a compact set U . In this case, the constant is equal to
min{m(Df
z
) : z ∈ U }.
Proposition 2.2. Let U ⊂ R
n
be an open disk containing 0 and φ : U → U be a contracting
map with the contraction bound λ and φ(0) = 0. Then there exists k ∈ N such that for any
8 MEYSAM NASSIRI
Figure 1. The covering and well-distributed properties. The disk D is the
largest one and the other disks are its images under φ
i
’s.
ε > 0 small there exist vectors c
1
, . . . , c
k
∈ B
ε
(0) and a number δ > 0 such that
B
δ
(0) ⊂ Orbit
+
G
(0),
where G = G(φ, φ + c
1
, . . . , φ + c
k
). Moreover,
δ ≥
ε
1 − λ
, and k < C(n) · λ
−n
.
These properties are robust in the following sense:
Let φ
0
= φ and φ
i
= φ + c
i
. The same is true for the IFS of any family of contracting maps
˜
φ
i
close to φ
i
if their contraction bounds are also close to those of φ
i
’s.
In order to prove this proposition we start with a non-perturbative version of it, which
also clarifies the robustness of transitivity.
Remark 2.3. For smooth maps λ = m(Dφ(0)). No matter if the eigenvalues of Dφ(0) are
complex or real.
Definition 2.4. We say that an iterated function system G(φ
1
, . . . , φ
k
) of contracting maps
has the covering property if there is a open set D such that
D ⊂
k
i=1
φ
i
(D).
The set of (unique) fixed points z
i
’s of φ
i
’s is well-distributed if any open ball of diameter
d and centered in D contains some z
i
, where
d ≥ λ
−1
· max{r | ∀x ∈ D, ∃i, B
r
(x) ⊂ φ
i
(D)}
and λ is the minimum of the contraction bounds of φ
i
’s.
Proposition 2.5. Let φ
i
: R
n
−→ R
n
, i = 1, 2, . . . , k, be contracting maps, and φ
i
(z
i
) = z
i
be their unique fixed points. Suppose that the iterated function system G = G(φ
1
, . . . , φ
k
)
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 9
has covering property on D. Then for any x ∈ D there exists a sequence {σ
j
}
∞
j=1
such that
for all j ∈ N, σ
j
∈ {1, 2, . . . , k}, and
φ
−1
σ
j
◦ φ
−1
σ
j−1
◦ · · · ◦ φ
−1
σ
1
(x) ∈ D.
In addition, if the set {z
i
}
k
i=1
is well-distributed in D then
D ⊂ Orbit
+
G
(0).
Proof. To prove the first part notice that given a point x ∈ D, the covering property
says that there is σ
1
∈ {1, 2, . . . , k} such that φ
−1
σ
1
(x) ∈ D. Then, inductively, one constructs
a sequence {σ
j
}
∞
j=1
such that φ
−1
σ
j
◦ φ
−1
σ
j−1
◦ · · · ◦ φ
−1
σ
1
(x) ∈ D.
Now we prove the se cond part. The well-distributed property yields that for any small
ball B
r
(x
0
) in D, either it belongs to some φ
i
(D) or it contains the fixed point of some φ
i
.
The latter case could be weakened to “or it contains some few G-iterations of the fixed point
of some φ
i
”. Now, If the ball B
r
(x
0
) is very small then it belongs to the domain of some
φ
i
, i.e. B
r
(x
0
) ⊂ φ
i
(D), and so there is x
1
∈ D such that B
λ
−1
·r
(x
1
) ⊂ φ
−1
i
(B
r
(x
0
)) ⊂
D. We may continue this process inductively. Since, the ratio of the balls is increasing
exponentially, after some iteration, it would be large enough to contain the fixed p oint of
some φ
i
. This completes the proof.
Proof of Proposition 2.2. It is enough to show that there exist a number k, and
certain (small) translations of the map φ, the c overing property and the well-distributed
hypothesis holds in some open ball B
ε
(0). Then using Proposition 2.5 we obtain the density
of G-orbit of 0. It is not difficult to see that k < C(n) ·λ
−n
. The persistency follows the fact
that the covering property and the well-distributed hypothesis are C
0
robust properties if
the contraction bounds of the nearby maps are close to the initial ones.
2.2. Recurrent diffeomorphisms. In this subsection we study the transitivity for the
iterated function system of recurrent diffeomorphisms. T he results of this subsection s hall
be used in the proof of the main theorem.
let us first to recall some definitions.
An orbit is quasi periodic if its closure T is diffeomorphic to the torus and the dynamics
on T is conjugate to an irrational rotation on the torus.
A Hamiltonian on a 2n-dimensional manifold is called completely integrable if it has n
integrals in involution. Recall that an integral is a smooth real function on N or N ×R which
is constant along the orbits of the Hamiltonian flow. A Hamiltonian is called integrable if it
is locally completely integrable. A diffeomorphism is called integrable if it is the time-one
map of some integrable Hamiltonian flow.
The Liouville-Arnold theorem says that if f ∈ Diff
r
ω
(N) is integrable then N = ∪N
i
,
where
• N
i
’s are mutually disjoit open sets,
• for any i, N
i
is invariant and diffeomorphic to D
n
× T
n
by a diffeomorphism h
i
,
• any torus h
−1
i
({x} × T
n
) is f -invariant and its dynamics is conjugate to a rotation.
10 MEYSAM NASSIRI
We may also suppose that
• the family {N
i
} is locally finite in N.
Lemma 2.6. Let f
1
be an integrable symplectic diffeomorphism on the symplectic manifold
N. Then arbitrarily close to f
1
there is another integrable symplectic diffeomorphism f
2
which is conjugated to f
1
by a smooth change of coordinates on N such that
(1) any f
1
-invariant torus intersects transversally some f
2
-invariant torus, and vice
versa,
(2) given t wo open sets U, V ⊂ N, there is a chain of tori T
j
, j = 1, 2, . . . , s, invariants
for f
σ
j
, σ
j
= 1 or 2, such that, each T
j
, (j < s), intersects transversally T
j+1
, T
1
intersects U and T
s
intersects V .
Proof. We construct a symplectic diffeomorphism φ ∈ Diff
r
ω
(N) close to the identity
such that f
2
= φ ◦ f
1
◦ φ
−1
has the desired properties. As mentioned before, N = ∪N
i
,
where N
i
is diffeomorphic to D
n
× T
n
by a diffeomorphism h
i
. It is convenient to consider
the polar coordinate system on D
n
× T
n
, that is, any point is represented by
(r
1
, . . . , r
n
, θ
1
, . . . , θ
n
),
where 0 ≤ r
i
< 1 and θ
i
∈ T.
The construction of φ has two steps.
Step 1. Let ψ
1
∈ Diff
r
ω
(R
2
) be the time one map an integrable Hamiltonian flow such that
in the polar coordinate we have
• ψ
1
(r, θ) = (r, θ), if r ≥ 1,
• ψ
1
({r = c}) = {r = c}, if 1 > c ≥ 0,
• any two open set in the unit disk {r < 1} are connected by a chain of circles {r = c
j
}
and ψ
1
({r = c
i
}).
Note that it is not difficult to define ψ
1
explicitly. Now let
ψ =
n times
ψ
1
× · · · × ψ
1
.
Define ϕ ∈ Diff
r
ω
(N) by
ϕ =
h
−1
i
◦ ψ ◦ h
i
on N
i
id on N \ ∪N
i
The smoothness of ϕ on each N
i
is trivial, and on the boundary of N
i
’s follows from the
fact that N
i
’s are a locally finite family in N, they are mutually disjoint and ψ is equal to
the identity on the b oundary of D
n
× T
n
.
Step 2. Let i > j such that ∂N
i
∩ ∂N
j
contains a regular hypersurface (codimension one)
S
ij
. Then for any such i, j we consider a small open neighborhood U
ij
of some point of the
hypersurface S
ij
. The sets U
ij
are pairwise disjoint. Let U
+
ij
= U
ij
∩ N
i
and U
−
ij
= U
ij
∩ N
j
.
Then consider a symplectic diffeomorphism ϕ
ij
supported in U
ij
such that
ϕ
ij
(U
−
ij
) ∩ U
+
ij
= ∅ and ϕ
ij
(U
+
ij
) ∩ U
−
ij
= ∅.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 11
Now we take the composition of the all the above diffeomorphisms to define φ ∈
Diff
r
ω
(N), that is
φ := (◦
ij
ϕ
ij
) ◦ ϕ.
It is not difficult to see that f
2
= φ ◦ f
1
◦ φ
−1
has the desired properties.
Proposition 2. 7. Let T
1
be an integrable symplectic diffeomorphism on the symplectic
manifold N such that almost all points are quasi periodic. Then arbitrarily close to T
1
there
is an integrable symplectic diffeomorphism T
2
on N such that the iterated function system
G(T
d
1
, T
d
2
) has a dense orbit, for an y d, d
∈ Z. Moreover, almost all points have dense
G-orbits.
Proof. Suppose that T
2
is an integrable diffeomorphism so that almost all points are
quasi periodic. Let S
0
be the set of all quasi periodic points for T
1
, which is T
1
-invariant.
Similarly, let S
0
the set of all quasi periodic points for T
2
, which is T
2
-invariant. It follows
that the complements of S
0
and S
0
have zero Lebesgue measure, and Lebesgue measure
is invariant under T
1
and T
2
. Let S the set of all points whose orbits under the iterated
function system G(T
1
, T
2
) belong to S
0
∩ S
0
.
Claim. The set S has total Lebesgue measure.
Proof of Claim. We use an inductive process. Let the sequence of sets S
k
and S
k
,
k ∈ N defined as the following:
S
k+1
:=
n∈Z
T
n
1
(S
k
),
S
k+1
:=
n∈Z
T
n
2
(S
k
).
By the definitions, S
k
is T
1
-invariant and S
k
is T
2
-invariant. The complements of these sets
have zero Lebesgue measure. Furthermore, if x ∈ S
k
then for all n, m ∈ Z, T
m
2
◦ T
n
1
(x) ∈
S
k−1
, since S
k
⊂ S
k−1
. So S
k
contains the set of all points in N whose first k-th iterations
under G(T
d
1
, T
d
2
) belong to S
0
, for any d, d
∈ Z. More precisely,
S
k
= {x ∈ N | ∀n
1
, m
1
, . . . , n
k
, m
k
∈ Z, T
n
k
2
◦ T
m
k
1
◦ · · · ◦ T
n
1
2
◦ T
m
1
1
(x) ∈ S
0
}.
This shows that S = ∩
∞
k=0
S
k
. The complement of this set has zero Lebesgue measure. This
completes the proof of the claim.
Now we apply Lemma 2.6 for T
1
. Then we obtain φ ∈ Diff
r
ω
(N) close to the identity
and T
2
= φ ◦ T
1
◦ φ
−1
, such that given two open sets U, V , there is a chain of tori T
j
,
j = 1, 2, . . . , s, invariants for T
σ
j
, σ
j
= 1 or 2, such that, each T
j
intersects (transversally)
T
j+1
, T
1
intersects U and T
s
intersects V . It is not difficult to find an orbit of G which
shadows this chain. For any z ∈ S, there is n
z
such that T
n
z
σ
j
(z) is close to T
j+1
if z is
sufficiently close to T
j
. The set S is G(T
1
, T
2
)-invariant. So if z ∈ S is sufficiently close
to T
1
, then it has a G-orbit shadowing all T
j
, and therefore there is an orbit from U to V .
Moreover, given any point x ∈ S and any open set U, there is a finite se quence of tori T
i
,
i = 1, . . . , n, invariant for T
1
or T
2
(alternatively), such that x ∈ T
1
, T
n
∩ U = ∅, and for
12 MEYSAM NASSIRI
any i, T
i
intersects transversally T
i+1
. Then it follows that there exists Σ = (σ
1
, . . . , σ
m
)
such that T
Σ
(x) ∈ U . This c ompletes the proof.
Remark 2.8. If the set of quasi periodic points is residual then following the same argument
in the proof, we conclude that the set of all points with dense orbit for G(T
1
, T
2
) is also
residual.
Example 2.9. Let N = R × (R/2πZ) and
T
1
: (I, θ) −→ (I, θ + h(I)).
In this case, we choose the change of coordinates
φ : (I, θ) −→ (I + cos θ, θ).
And then we define T
2
= φ ◦ T
1
◦ φ
−1
. Now the above argument works well.
Definition 2.10. A point x is recurrent for a homeomorphism f if
lim inf
n→∞
dist(x, f
n
(x)) = 0.
A homeomorphism or diffeomorphism is said recurrent if almost all points are recurrent.
Theorem 2.11. Let T ∈ Diff
r
ω
(N) be a recurrent diffeomorphism. Then for every > 0,
(1) there exist T
1
, T
2
∈ B
(T ) ⊂ Diff
r
ω
(N) such that G(T, T
1
, T
2
) is transitive,
(2) for any open ball V ⊂ N and any bounded domain N
c
⊂ N, there exist k ∈ N
and T
1
, T
2
, . . . , T
k
∈ B
(T ) ⊂ Diff
r
ω
(N) such that N
c
⊂ Orbit
−
G
(V ), where G =
G(T, T
1
, T
2
, . . . , T
k
).
Proof. If T = id, then we choose φ
1
an integrable symplectic diffeomorphism on the
manifold N such that almost all points are quasi periodic, and d
C
r
(φ
1
, id) <
1
2
. Proposition
2.7 implies that for any open set V there exists, φ
2
in Diff
r
ω
(N) and -close to the identity
in the C
r
topology, such that, Orbit
−
G
φ
(V ) ∩ Orbit
+
G
φ
(V ) is (open and) dense in N, where
G
φ
= G(φ
1
, φ
2
). In other words, G
φ
is transitive. This completes the proof of (1) in the case
that T = id.
For an arbitrary recurrent T , let R be the set of recurrent points of T , which is also
invariant for φ
1
and φ
2
. This set is dense. In fact, following an argument similar to the
Claim in the proof of Proposition 2.7 this s et has total Lebesgue measure (and also is
residual).
Let V is an open set in N, and z ∈ R ∩ Orbit
−
G
φ
(V ). This intersection is obviously
non-empty. Then, there are d ∈ N and Σ = (σ
1
, . . . , σ
d
), σ
i
= 1, 2, such that
z ∈ (φ
Σ
)
−1
(V ).
Moreover, for any i = 1, 2, . . . , d, and any l
j
∈ Z, j = 1, 2, . . . , i,
˜z
i
:= (T
l
i
◦ φ
σ
i
) ◦ · · · ◦ (T
l
1
◦ φ
σ
1
)(z) ∈ R.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 13
So, using recurrency, for some (large) l
j
∈ N, the orbit ( ˜z
i
)
i
shadows (z
i
)
i
, where z
i
=
φ
σ
i
◦ · · · ◦ φ
σ
1
(z). This shows that for some l
j
∈ N, the point ˜z
d
belongs to V . But ( ˜z
i
)
i
is
an orbit of z under the iterated function system of
G
2
= G(T, T ◦ φ
1
, T ◦ φ
2
).
In other words, ˜z
d
∈ V ∩ Orbit
+
G
2
(z). Recall that, R ∩ Orbit
−
G
φ
(V ) is dense in N . So, the
G
2
-orbit of any point in a dense set, intersects V . The same is true for backward G
2
-orbits.
Thus, Orbit
±
G
2
(V ) is (open and) dense in N , and G
2
is transitive. This completes the proof
of (1).
Given N
c
⊂⊂ N b ounded, and V ⊂ N open, we let X = B
1
(N
c
) \ Orbit
−
G
2
(V ). X
is a compact set with empty interior. So for any x ∈ X there exists, h
x
in Diff
r
ω
(N)
and -close to the identity in the C
r
topology, such that h
−1
x
(x) ∈ V
−
:= Orbit
−
G
2
(V ).
Since V
−
is open, there is a neighborhood U
x
of x such that h
−1
x
(U
x
) ⊂ V
−
. The family
{U
x
} is open cover of the compact set X. So there exist k ∈ N, x
1
, x
2
, . . . , x
l
∈ X and
h
x
1
, h
x
2
, . . . , h
x
k
∈ B
(id) ⊂ Diff
r
ω
(N) s uch that
X ∩ h
−1
x
1
(X) ∩ · · · ∩ h
−1
x
k
(X) = ∅.
Thus
T
−1
(X) ∩ (h
x
1
◦ T )
−1
(X) ∩ · · · ∩ (h
x
k
◦ T )
−1
(X) = ∅.
Therefore,
N
c
⊂⊂ T
−1
(V
−
) ∩ (h
x
1
◦ T )
−1
(V
−
) ∩ · · · ∩ (h
x
k
◦ T )
−1
(V
−
).
If we define G := G(T, T ◦ φ
1
, T ◦ φ
2
, h
x
1
◦ T, . . . , h
x
k
◦ T ), then we have
N
c
⊂⊂ Orbit
−
G
(V ).
Remark 2.12. If N is compact, by the Poincar´e recurrence theorem, T is recurrent. For
non-compact manifold N we know that almost all points are either recurrent or converge
to infinity. Moreover, in the interior of the non-wandering set of T , generic points (in a
residual set) are recurrent. So, when the non-wandering set has (large) non-empty interior,
as the same as above, there is an iterated function system of its nearby systems exhibiting
transitivity in the interior of the non-wandering set. See also section 5.1.
3. Symplectic blender
Definition, existence and properties of symplectic double blender are discussed in this
section.
Bonatti and D´ıaz in [BD] introduce blenders, geometric models for certain hyperbolic
sets originating in the unfolding of heterodimensional cycles, that play an imp ortant role
as a mechanism for creation of cycles, and semi-local source of transitivity. Although their
methods may be modified for conservative case, the symplectic case is more delicate.
14 MEYSAM NASSIRI
In [BD] a cs-blender, roughly speaking, is a hyperbolic (locally maximal) invariant set
with a splitting of the form E
ss
⊕ E
u
⊕ E
uu
, dimE
u
= 1, such that a convenient projection
of its stable set has larger topological dimension than the stable set itself. This phenomenon
is robust in the C
1
topology. Similarly, one may define cu-blender.
Their constructions are essentially 3-dimensional, i.e., the central bundle is one-dimen-
sional. Here we call the bundle E
u
the central bundle. On the other hand, to apply this local
tool for systems with higher dimensional ce ntral bundles they use a chain of blenders with
one-dimensional central bundles and different indices (i.e. dimension of the stable bundle)
connected to each other. This allows them to use such blenders in more situations. This
is of course impossible in the symplectic case, since all eigenvalues are pairwise conjugate
and so all hyperbolic periodic points have the same index. So in the symplectic case we
would involve the higher central dimensions in the creation of blender. We construct a new
class of such blenders in the symplectic (or Hamiltonian) systems that work like a chain of
cs-blenders and a chain of cu-blenders simultaneously.
In section 3.1, regardless of the symplectic case, we study blenders with higher central
dimensions when the central bundle is uniformly unstable (stable, respectively) and we
construct a cs-blender (cu-blender, respectively). In section 3.2, we consider the case that
the central bundle splits into two stable and unstable subbundles, that is, the maximal
invariant set is hyperbolic of the form E
ss
⊕ E
s
⊕ E
u
⊕ E
uu
, and we create a blender which
exhibits the features of both cu- and cs- blenders. We call it double-blender. Note that this
case is very compatible with the symplectic case where the eigenvalues of periodic points
are pairwise conjugate. In section 3.3, we study the symplectic case, and we introduce the
symplectic version of the above phenomenon, which we call symplectic blender.
In order to give a more clear picture of the dynamics of the above phenomena we start
by a simple affine model for each one and then we relax the construction to the more flexible
versions, robust under C
1
small perturbations, which is the subject of section 3.4. In fact,
we may also define blenders in another way which takes in to account their properties,
rather than their construction (see also [BDV, chapter 6]). We continue using the above
informal definitions until subsection 3.4, where the formal definition of blender is presented.
Throughout this section we consider the diffeomorphism f of R
2
which is the Smale
horseshoe on U := [0, 1]
2
and is of the form that follows.
The vertical sub-rectangles X
1
= I
1
× [0, 1] and X
2
= I
2
× [0, 1] are connected compo-
nents of f(U ) ∩ U and also the horizontal sub-rectangles Y
1
= f
−1
(X
1
) and Y
2
= f
−1
(X
2
)
are connected components of f
−1
(U) ∩ U. The restrictions of f to Y
1
and to Y
2
are affine
maps with linear part
±
1
4
0
0 ±4
.
From now on we suppose x ∈ E
ss
, y ∈ E
uu
, associated to f and we denote by (x
0
, y
0
) the
unique fixed point of f in X
1
.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 15
Figure 2. IFS of expanding maps
3.1. cs-blender with higher dimensional unstable central bundle. The following
proposition about the iterated function system of expanding maps is a sp e cial case of Propo-
sition 2.2.
Proposition 3.1. For i = 0, 1, 2, 3, let g
i
(x, y) := (a
i
x + b
i
, c
i
y + d
i
) where 1 < a
i
=
c
i
=
16
15
< 2 and b
i
, d
i
’s are such that the fixed points of g
0
, .., g
3
are respectively, P
0
=
(0, 0), P
1
= (1, 1), P
2
= (−0.1, 1.1), P
3
= (1.1, 0.1) . See Figure 1. Given any open rectangle
Σ ∈ [0, 1]
2
there is g
σ
∈ G(g
1
, g
2
, g
3
, g
4
) such that (0, 0) ∈ g
σ
(Σ). This property persists for
all (uniformly) expanding maps ˜g
i
close to g
i
if their expansion bounds (i.e. the contarction
bounds of ˜g
−1
i
) are also close to those of g
i
’s.
Now, consider the diffeomorphism F of R
4
such that in B := [−1, 1]
2
× [0, 1]
2
is of the
form:
F (p, q; x, y) := (g
i
(p, q); f(x, y)), if (x, y) ∈ B
i
and (p, q) ∈ [−1, 1]
2
,
where B
1
= X
1
∩ Y
1
, B
2
= X
2
∩ Y
2
, B
3
= X
1
∩ Y
2
, B
4
= X
2
∩ Y
1
, and g
i
’s are the expanding
maps taken in Proposition 3.1 Observe that F (B) ∩ B contains the four boxes [−1, 1] ×
[−1, 1]× X
1
, [−1,
3
4
]× [−1, 1] × X
1
, [−1, 1] × [−1,
3
4
]× X
2
and [−1,
3
4
]× [−1,
3
4
]× X
2
and that
Q = (0, 0, x
0
, y
0
) is the (unique) hyperbolic fixed saddle of F of index 3. Let W
s
loc
(Q) =
{0} × {0} × [0, 1] × {y
0
} be the connected component of W
s
(Q) ∩ B that contains Q.
Definition 3.2. A vertical strip with respect to Q, or simply vertical strip, is a rectangle
∆ = Σ × {x} × [0, 1], where x ∈ [0, 1] and Σ is a closed rectangle (with non-empty interior)
in [0, 1]
2
.
The next proposition gives the main geometric property of cs-blender.
Proposition 3.3. Every vertical strip ∆ with respect to Q intersects W
s
(Q).
16 MEYSAM NASSIRI
Proof. This proposition may be reduced to the transitivity of the iterated function
system G(g
0
, g
1
, g
2
, g
3
). Any vertical strip intersects all B
i
’s, and the map F restricted to
each of B
i
is equal to g
i
× f. The image of any vertical strip ∆ contains a union of four
vertical strips ∆
j
each of which intersects all B
i
s. So the G-orbit of (0, 0) corresponds to
some points in W
s
(Q), and is the same as the projection of W
s
(Q) to the central direction
along the E
s
⊕ E
uu
. The Proposition 3.1 shows that the orbit of (0, 0) is dense in [0, 1]
2
.
This means that the projection of W
s
(Q) to the central direction along the E
s
⊕ E
uu
is
dense in [0, 1]
2
. So W
s
(Q) intersects every vertical strip ∆.
The following is a direct consequence of the above proposition:
Proposition 3.4. Suppose that there is a hyperbolic periodic point P of F of index 1 whose
one-dimensional unstable manifold crosses B along a vertical segment {p}×{q}×{x}×[0, 1]
on the north-east of W
s
loc
(Q) (i.e., p, q ∈ (0, 1)). Then W
s
(P ) ⊂ W
s
(Q).
Thus the one-dimensional stable manifold of Q looks like a 3-dimensional manifold, as
its closure contains the 3-dimensional manifold W
s
(P ).
3.2. Double-blender: affine model. In the 3-dimensional cs-blenders, if one projects
the cube and its pre-image along of stable direction a figure like Smale horseshoe appears
but two right and left rectangles overlap, while in the projections along the weak unstable
direction do not overlap. Having this in mind, consider a 4-dimensional horseshoe with the
splitting of the form E
ss
⊕ E
s
⊕ E
u
⊕ E
uu
such that the projection along E
ss
give a figure
like 3-dimensional horseshoe but its two wings overlapping and the same feature for the
inverse map and E
uu
. This led us to the following affine model.
Consider the following maps on R, which are maps in central bundle
ψ
1
(p) :=
4
5
p ϕ
1
(q) :=
5
4
q
ψ
2
(p) :=
4
5
p +
2
5
ϕ
2
(q) :=
5
4
q −
1
2
Note that ϕ
i
:= ψ
−1
i
, and (p, q) ∈ E
c
.
Let F be a diffeomorphism on R
4
such that in B := [−1, 1]
2
× [0, 1]
2
is of the following
form:
F (p, q; x, y) := (ψ
i
(p), ϕ
j
(q); f (x, y)), if (x, y) ∈ X
i
∩ Y
j
and (p, q) ∈ [−1, 1]
2
.
The dynamics of F inside the box B is hyperbolic, E
ss
⊕ E
s
⊕ E
u
⊕ E
uu
. The maximal
invariant set in B, i.e., Λ =
F
n
(B) is a cs-blender, if we consider E
ss
⊕ E
s
as stable
direction, E
u
as central and E
uu
as strong unstable directions. Similarly Λ is a cu-blender
if we consider E
ss
as strong stable direction, E
s
as central and E
u
⊕ E
uu
as unstable
directions. Therefore Λ is a double-blender. Note that using the results of the previous
subsection, we may consider multi-dimensional central bundle, i.e., both of weak stable and
unstable bundles of arbitrary dimension ≥ 1.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 17
3.3. Symplectic blender: affine model. We consider the following maps on R, which
are maps in central bundle
ψ(p) := λp , ϕ(q) :=
1
λ
q,
where 1 − λ > 0 is small enough.
The symplectic diffeom orphism F on R
4
is defined as the product of the above maps:
F (p, q; x, y) := (ψ(p), ϕ(q); f(x, y)).
We shall perturb F by the time-one map of the flow of a Hamiltonian vector field such
that the resulting map is a diffeomorphism with the properties of the model in the previous
subsection.
Let α and β be smooth bump functions on R such that for all t ∈ R, 0 ≤ α(t) ≤ 1, and
α(t) = 1 if t ∈ I
1
∪ I
2
and α(t) = 0 if t /∈ J
1
∪ J
2
,
where J
1
and J
2
are disjoint neighborhoods of I
1
and I
2
, respectively.
Similarly, for all t ∈ R , 0 ≤ β(t) ≤ 1, and β(t) = 1 if t ∈ [−1, 1] and β(t) = 0 if
t /∈ [−
3
2
,
3
2
].
We define F
ε
:= Φ
ε
◦ F , where Φ
ε
is the time-one map of the flow ass ociated to the
Hamiltonian
H
ε
:= εα(x)α(y)β(p)β(q)((i − 1)p − (j − 1)q) , if x ∈ J
i
and y ∈ J
j
, i, j ∈ {1, 2}.
The support of H
ε
is the disjoint union of four boxes of dimension 4. Then we have the
following
Theorem 3.5. Let F
ε
be the Hamiltonian diffeomorphism as defined above. If ε > 1−λ > 0
are small enough, then F
ε
has t he form of the affine model of double blender and so the
maximal invariant set for F
ε
inside the B := [−1, 1]
2
×[0, 1]
2
is a symplectic double-blender.
3.4. Robustness: relaxing the constructions. We use cone field structures to make
sure that the feature that we explained in the affine cases remains for nearby systems.
In the above affine models we have four cone fields C
ss
, C
s
, C
u
, C
uu
, invariant under the
derivative DF . These cone fields will define invariant foliations in the b ox B. Of course,
these foliations in the affine models coincide with the vertical and horizontal segments and
strips. We may repeat all the above process by replacing these vertical and horizontal
segments with the almost vertical/horizontal strips/segments, and reducing the iterated
function system in central bundles.
Now we prove the robustness of the main features of blenders. We know that these
cone fields remain invariant for any C
1
nearby system G. So for nearby systems we will
have almost vertical/horizontal strips/segments. These almost vertical/horizontal segments
allow us to introduce the corresp onding iterated function systems of expanding/contracting
maps in central bundles. The new iterated function systems are close to the initial ones,
and so thanks to the results of section 2.1, we have robustness of transitivity property
of iterated function system of such expanding maps. Therefore the dynamical feature of
blender appears also for any G in a C
1
neighborho od of F.
Let us state here the formal definition of symplectic and double blenders.
18 MEYSAM NASSIRI
Definition 3.6. Let B is an open embedded ball on which there are four cone fields C
ss
, C
s
,
C
u
, C
uu
, invariant under the derivative DF. A vertical strip (or u-s trip) is an embedded
(u + uu)-dimensional disk in B, which contains the uu-leaves of each its points. Similarly
we define horizontal strip (or s-strip).
Definition 3.7. The pair (P, B) is a double blender for the diffeomorphism F if satisfies the
following features:
B-1 P is a hyperbolic saddle periodic p oint of F .
B-2 B is an open embedded ball on which there are four cone fields C
ss
, C
s
, C
u
,
C
uu
, invariant under the derivative DF .
B-3 For any G sufficiently close to F in the C
1
topology, the stable manifold of P
G
intersects any u-strip in B, and the unstable manifold of P
G
intersects any s-strip
in B. Here P
G
is the continuation of P .
Definition 3.8. A symplectic blender is a double blender for a symplectic (or Hamiltonian)
diffeomorphism.
We summarize this section in the following theorem (see also subsection 4.2).
Theorem 3.9. Let M and N be two symplectic manifolds (not necessarily compact). Let
r = 1, 2, . . . , ∞. Suppose that f
1
∈ Diff
r
ω
(M) has a hyperbolic periodic point p
1
with
transversal homoclinic intersections and f
2
∈ Diff
r
ω
(N) has a hyperbolic periodic point
p
2
such that its hyperbolicity is weak enough. Then, there is a C
r
-arc {F
µ
}
µ∈[0,1]
of C
r
symplectic diffeomorphism on M × N such that,
(1) F
0
= f
1
× f
2
.
(2) There is a neighborhood V of {F
µ
}
µ∈(0,1]
in Diff
1
(M × N ) such for any G ∈ V,
the pair (P
G
, B) is a double blender, where P
G
is the continuation of hyperbolic
P
0
= (p
1
, p
2
) and B is an embedded open disk in M × N .
Note that, this is not the only way to create blenders. In fact, one may create them
by a perturbation of a system exhibiting a quasi transversal homoclinic or heteroclinic
intersection, by the similar ways, but with more technical details (see [BD] and [N]). Here
we only considered the case where the unperturbed system is a product of two systems,
one of them with a transversal homoclinic intersection and the other one with a hyperbolic
saddle with weak hyperbolicity. Because it is s ufficient for the proof of our main theorems.
4. Proof of Theorem A
In this section we give the proof of Theorem A. The proof is constructive. It is divided
in five parts. First we introduce the perturbations in subsection 4.1. Then in the four
sequel subsections we prove that the perturbed systems satisfy the desired properties. In
subsection 4.2 we prove the existence of a symplectic blender. Then in subsection 4.3 we use
the results of iterated function systems of recurrent diffeomorphisms (section 2.2) to prove
that the strong stable and unstable manifolds of almost all points in the central manifold
intersects the constructed blender. In subsection 4.4 we show that this property is robust
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 19
under small perturbation, and here we use the dynamical properties of the blender. Then
we complete the proof in subsection 4.5 by proving that there is a hyperbolic periodic point
such that its stable and unstable manifolds are both dense in the set Λ × N in a robust way,
concluding the robustly topological mixing.
4.1. The perturbations. Let r = 1, 2, . . . , ∞, f
1
∈ Diff
r
ω
(M) and f
2
∈ Diff
r
ω
(N) as in
the Theorem A. Let U ⊂ M be a small simply connected open set, such that for some
k ∈ N, Λ :=
n∈Z
f
kn
1
(U) is an invariant hyperbolic compact set for f
k
1
. By choosing U
suitable and k large enough, we may suppose that f
k
1
|
Λ
is conjugate to a shift of d symbols
{1, . . . , d}. The required number d of symbols in the proof depends to dimN and f
1
× f
2
.
By taking f
k
1
, and f
k
2
instead of f
1
and f
2
, we may assume that Λ is f
1
- invariant and
Λ :=
n∈Z
f
n
1
(U) is conjugate to a shift of symb ols {1, . . . , d}, where d is sufficiently large.
We identify the set identify Λ with that set {1, . . . , d}
Z
.
In order to define our local perturbation we first consider open sets A
ij
and pairwise
disjoint open sets
A
ij
in the way that
A
ij
∩ Λ = {(x
i
)
i∈Z
| x
0
= i, x
1
= j} and A
ij
⊂
A
ij
.
In a similar way we define A
I
and
A
I
as neighborhoods of I
Z
, where I ⊂ {1, 2, . . . , d}. In
addition we set A
i,∗
= ∪
j
A
ij
.
By the assumptions, f
2
has a hyperbolic periodic point p
2
, with sufficiently weak
hyperb olicity. Suppose that T
p
2
N = E
s
p
2
⊕ E
u
p
2
. We consider p
1
∈ Λ a hyperb olic fixed
point for f
1
. Let P
0
= (p
1
, p
2
).
Let φ
s
be the linear contracting map given by Df
2
|
E
s
p
2
. Proposition 2.2 gives a number
l as a required number of elements of IFS to obtain transitivity in some small disk. This
number only depe nds on the dimension of N and the contraction bound of φ
s
.
We fix the number d = 2l + 4, and its related k and U as above. And let I =
{1, 2, . . . , d − 4}, J
1
= {1, d − 3, d − 2} and J
2
= {1, d − 1, d}.
Let δ > 0 is small enough and ε : [0, 1] −→ [0, δ]
2
is an smooth simple curve such that
ε(0) = (0, 0).
Let F
0
= f
1
× f
2
.
For µ ∈ (0, 1], F
µ
is defined as the following. Let (ε
1
, ε
2
) := ε(µ), and consider Hamiltonians
ε
1
˜
h
1
and ε
2
˜
h
2
supported on pairwise disjoint sets as follows. Let ψ
ε
1
and ψ
ε
2
, respectively
their associated diffeomorphism. Now let Ψ
µ
= ψ
ε
2
◦ ψ
ε
1
. Since the support of ψ
ε
i
’s are
pairwise disjoint, they may commute with each others.
We define
F
µ
:= Ψ
µ
◦ F
0
.
The aim of this section is to show that F
µ
has the properties claimed in Theorem A. One
may describe briefly the perturbation Hamiltonians as the following:
(1) Let Hamiltonian
˜
h
1
: M × N −→ R supported on (
A
I
\
A
1∗
)× N, such that ψ
ε
1
◦F
0
has a symplectic blender. The detailed definition of
˜
h
1
is presented in subsection
4.2 and there we show the existence of a blender (P
0
, B).
20 MEYSAM NASSIRI
Figure 3. Support of local perturbations projected to Λ. The blocks
with the same color are in the support of the same Hamiltonians. No
perturbation is made in the black or white parts
(2) The Hamiltonian
˜
h
2
: M × N −→ R is supported on (
A
J
1
\
A
1∗
) × N and its
restriction to Λ × N is locally constant with respect to variables in M . More
precisely,
ψ
ε
2
◦ F
0
(x, q) = (f
1
(x), φ
1
◦ f
2
(q)), if x ∈ A
d−3,∗
,
ψ
ε
2
◦ F
0
(x, q) = (f
1
(x), φ
2
◦ f
2
(q)), if x ∈ A
d−2,∗
,
where φ
1
and φ
2
are obtained in the proof of Theorem 2.11 (1). In subsection 4.3
using the symbolic dynamics and the result of section 2.2 we show that for almost
every z in the fibers {x} × N,
W
ss(uu)
(z; F
µ
) ∩ B = ∅.
4.2. Constructing symplectic blender. Here we s how that how to define the perturba-
tion
˜
h
1
: M × N −→ R in order to create a symplectic blender B = A
I
× B. In fact, based
on the affine models of section 3, we also sketch the proof of Theorem 3.9. Notice that
Theorem A satisfies the hypotheses of Theorem 3.9.
Let ∗ = s, u and φ
∗
be the linear contracting map given by Df
2
|
E
∗
p
2
. Proposition 2.2
gives the vectors c
∗
1
, c
∗
2
, . . . , c
∗
l
∈ E
∗
p
2
, |c
∗
i
| <
1
, such that the corresponding IFS is transitive
in some small disk D
∗
. We let B ⊂ D
s
× D
u
, where the product is taken in a local chart
on N.
Then we define Hamiltonian
˜
h
1
in order to realize above IFS’s. Recall that ψ
ε
1
is the
time one map of ε
1
˜
h
1
.
.
For i = 1, 3, . . . , l,
F
µ
(x, q) = ψ
ε
1
◦ F
0
(x, q) = (f
1
(x), φ
s
i
◦ f
2
(q)), if x ∈ A
i+1,I
, q ∈ N.
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 21
And for j = l + 1, . . . , 2l,
ψ
ε
1
◦ F
0
(x, q) = (f
1
(x), f
2
(q)), if x ∈ A
I,j+1
, q ∈ N.
Then, similar to the affine models, given a u-strip ∆ = γ
uu
× D × {(q, p)}
s
, then
F
µ
(∆) ⊃ ∪
j
∆
j
, where ∆
j
= γ
uu
i
× φ
u
j
(D) × {(q
i
, p
i
)}
s
. By induction we have,
F
k
µ
(∆) ⊃
Σ∈I
k
γ
uu
i
× φ
u
Σ
(D) × {(q
i
, p
i
)}
s
.
We project this set along the strong unstable foliation and also along the stable fo-
liation. Then the fixed point of φ
u
1
corresponds to the local stable manifold of P
0
. The
iterations of the fixed point of φ
u
1
under the IFS of φ
u
j
’s, also correspond to some parts of
the global stable manifold of P
0
. The results of section 2.1 shows that the IFS of φ
u
j
’s is
transitive. Therefore, the projection of W
s
(P
0
) along the strong unstable foliation in B is
dense on D
u
. This implies that W
s
(P
0
) intersects any u-strip in B.
Similarly we can show that W
u
(P
0
) intersects any s-strip in B. In other words, the
pair (P
0
, B) is a symplectic blender for F
µ
.
In addition, we have the following proposition which is a consequence of the first part
of Proposition 2.5.
Proposition 4.1. Under the hypotheses of Theorem 3.9 it is possible to create symplectic
blender with the following additional property:
B-4 Any forward and backward iteration of a uu-segment (ss-segment) in B, in-
tersects B in a uu-segment (ss-segment, respectively).
Consequently, the set of all points whose strong (un)stable manifolds intersect B, is an
invariant set.
4.3. Almost minimality of stable and unstable foliations. In this subsection that
the strong stable and unstable manifolds of almost all points in the central manifold N
0
:=
{p
1
} × N intersects the c onstructed blender. We refer to this property by the almost
minimality of the strong stable and unstable foliations.
Proposition 4.2. Let p ∈ Λ be a fixed point of f
1
which is not in the support of our
perturbations. Then there is an open and dense set R ⊂ N with total Lebesgue measure
such that for every q ∈ R and for any n ∈ Z, W
uu
(F
n
µ
(p, q)) ∩ B = ∅.
Proof. The key elements in the proof are the symbolic dynamics, the results of section
2.2 and the Proposition 4.1.
We consider restriction of f
1
to Λ. For any p = (p
i
)
i∈Z
∈ Λ = {1, 2, . . . , d}
Z
, the local
and global unstable manifolds of p for f are
W
u
loc
(p ; f|Λ) = {(x
i
) | ∀n ≤ 0, x
i
= p
i
}
W
u
(p ; f|Λ) = {(x
i
) | ∃n
0
∈ Z, ∀n ≤ n
0
, x
i
= p
i
}
22 MEYSAM NASSIRI
The above remark implies that the local and global strong unstable manifolds of (p, q) for
F
µ
are
W
uu
loc
(p, q; F
µ
|Γ) = W
u
loc
(p; f|Λ) × {q} = {(x
i
) | ∀n ≤ 0, x
i
= p
i
} × {q},
W
uu
(p, q; F
µ
|Γ) =
n≥0
F
n
µ
(W
uu
loc
(F
−n
µ
(p, q); F
µ
|Γ)).
Let T
1
= f
2
, T
2
= φ
1
◦ f
2
and T
3
= φ
2
◦ f
2
, where φ
1
and φ
2
are given as in the proof
of Theorem 2.11 (1).
Let q ∈ Rec(f
2
) ⊂ N such that there is a finite sequence (σ
i
)
n
i=1
such that σ
i
∈ {1, 2, 3}
and
T
σ
n
◦ T
σ
n−1
◦ · · · ◦ T
σ
1
(q) ∈ T
−2
1
(B).
We denote the set of all such points by R
1
.
Now, we consider
x = (x
i
) = (
W
u
loc
(p)
. . . , p
−2
, p
−1
, p
0
;
IF S
a
1
, a
2
, . . . , a
n
0
, 1, 1,
arbitrary
x
n
0
+3
, . . .),
where for i = 1, 2, . . . , n
0
,
a
i
= 1 if σ
i
= 1,
a
i
= d − 3 if σ
i
= 2,
a
i
= d − 2 if σ
i
= 3.
It is clear that x ∈ W
u
(p, f
1
|Λ) and so (x, q) ∈ W
uu
(p, q; F
µ
|Γ). We now take the
iterations of the point (x, q) under F
µ
. Since F
µ
restricted to A
a
i
,J
1
× N is equal to
f
1
× T
σ
i
, inductively we have:
(f
i
1
(x), T
σ
i
◦ T
σ
i−1
◦ · · · ◦ T
σ
1
(q)) = F
i
µ
(x, q) ∈ W
uu
(F
i
µ
(p, q); F
µ
).
In particular, for i = n
0
+ 1, F
n
0
+1
µ
(x, q) ∈ B. So,
W
uu
(F
n
0
+1
µ
(p, q); F
µ
) ∩ B = ∅.
Now we apply the Propos ition 4.1. It implies that for all n ∈ Z,
W
uu
(F
n
µ
(p, q); F
µ
) ∩ B = ∅.
Let R the set all points q ∈ N such that the above intersection holds. We proved that
R
1
⊂ R. The set R is open, because B is open and (the compact parts of) the strong stable
and unstable manifolds depends continuously to the points. On the other hand, in section
2.2 it was shown that the set R
1
has total Lebesgue measure. This completes the proof.
Remark 4.3. As a matter of fact, any skew product symplectic diffeomorphisms on a con-
nected manifold is in fact a direct product of two symplectic diffeomorphism. Let us explain
it for the Hamiltonians. Let U ⊂ R
2n
× R
2m
, and h : U × R → R be a Hamiltonian function
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 23
and f be the time-one map of its corresponding flow. If f(x, y; p, q) = (x, y, g(x, y; p, q)),
where x
i
and y
i
are symplectic conjugate variables and the same for p
i
and q
i
, then
˙x
i
= −
∂h
∂y
i
= 0, ˙y
i
=
∂h
∂x
i
= 0, ˙p
i
= −
∂h
∂q
i
, ˙q
i
=
∂h
∂p
i
.
The first two equalities implies that h does not depend to x and y. So f is the product
id × g.
This is no longer true for disconnected invariant sets. So, we see that F
µ
is a skew-
product on the disconnected invariant set Λ × N, while it could not be a skew product on
M × N.
4.4. Robustness of the almost minimality of foliations. The hypothesis (b) in The-
orem A implies that F
0
is partially hyperbolic on Γ
F
0
:= Λ × N which is locally maximal.
More precisely by the results of [HPS] we have:
H-1 Γ
F
0
is normally hyperbolic and F is plaque expansive (see [HPS, p.116 and
Theorem 7.2]).
H-2 There is a neighborhood U ⊂ Diff
1
(M × N) of F such that every G ∈ U has
a (locally maximal) invariant Γ
G
homeomorphic to Λ × N and is a continuation of
Γ
F
0
.
H-3 There is a G-invariant foliation on Γ
G
by manifolds diffeomorphic to N that
is, the continuation of fibration defined on Λ × N. So G induces a homeomorphism
˜
G on the quotient of Γ
G
by the foliation. It then follows that
˜
G is conjugate to f
1
|
Λ
(see [HPS, Theorem 7.1]).
H-4 G restricted to Γ
G
is conjugate to a skew product G
∗
: (x, w) −→ (f
1
(x), g
x
(w))
on Λ × N, which depends continuously on G.
Given N
c
⊂⊂ N, let Γ
G,N
c
⊂ Γ
G
the continuation of Λ × N
c
, that is, the image of
Λ × N
c
by the homeomorphism given in H-2 above. We let N
0
:= {p
1
} × N, and
˜
N
0
is the
continuation of N
0
for G.
In order to have all the above properties it is enough to consider the family {F
µ
} in
the set U, by taking δ > 0 s mall enough.
If W
uu
(p, q; F
µ
)∩B = ∅, then there is L > 0 large enough, such that W
uu
(p, q; F
µ
)∩B
contains a uu-segment of B. B is open, so there is a neighborhood V
(p,q )
of (p, q) such that
for any point z ∈ V
(p,q )
, W
uu
(z; F
µ
) ∩ B contains a uu-segment of B.
In section 4.3 we proved that the set R of points whose strong stable and unstable
manifolds intersects B is dense (and of total measure) in N
0
. Now we see that R contains
an open dense subset of N
0
.
We call X = N \ R the exceptional set, which is a closed set with empty interior and
of zero measure.
Then, given any compact set R
c
⊂ R, there is some large L such that for any z ∈ R
c
,
W
uu
L
c
(z; F
µ
) ∩ B contains a uu-segment of B.
24 MEYSAM NASSIRI
Since the compact parts of (strong) stable and unstable manifolds depends continuously
to the diffeomorphism, there exists W
µ,R
c
, a neighborhood of F
µ
, such that for any G ∈
W
µ,R
c
, and any z ∈ Γ
G,R
c
, W
uu
L
c
(z; G) ∩ B contains a uu-segment of B (w.r.t. G).
In other words, we have robustness of the almost minimality of strong stable and
unstable foliations.
4.5. Transitivity and Topological mixing. Recall the following two general fact on
symplectic diffeomorphisms.
S-1 A normally hyperbolic invariant submanifold of symplectic diffeomorphism is
a symplectic submanifold (with a canonical 2-form which is the restriction of the
given symplectic form) and
S-2 The restriction of a symplectic diffeomorphism to its normally hyperbolic invari-
ant submanifolds is preserving the restricted symplectic form.
Therefore, using H-1 - H-4, S-1 and S-2, the hypothesis (b) in Theorem A, yield that
if the neighborhood U of F
0
is sm all enough, then for any G ∈ U, G|
˜
N
0
is (sm oothly)
conjugate to a diffeomorphism g which is C
r
close to f
2
in Diff
r
ω
(N) and so all points in
˜
N
0
are non-wandering for G. As mentioned before, the family F
µ
is constructed in U.
Let N
c
be any open and bounded domain in N . Given ν > 0, let X
c,ν
= B
ν
(N
c
∩ X).
And let R
c,ν
= N
c
\ X
c,ν
⊂ R.
Now for any G ∈ W
µ,R
c,ν
, we first show that,
˜
R
c,ν
⊂ W
s
(P
G
) ∩ W
u
(P
G
),
recall that, P
G
is the continuation of the hyperbolic point (p
1
, p
2
).
Let ∆ be an open set in Γ
G
such that ∆ ∩
˜
R
c,ν
= ∅. Then, for any large number n
0
,
there is a point z
∗
∈ ∆ such that G
n
(z
∗
) ∈ ∆ ∩
˜
R
c,ν
for some n ≥ n
0
.
Let γ = W
uu
(z
∗
; G)∩∆, then for some large n > 0, G
n
(γ) has diameter larger than L
c
and so contains W
uu
L
c
(G
n
(z
∗
); G). Since G
n
(z
∗
) ∈ R
c,ν
, we conclude that G
n
(γ) contains a
uu-segment in B. Thus G
n
(∆) contains a u-strip in B. The property B-4 of double-blender
implies that W
s
(P
G
; G) intersects G
n
(∆) and so
W
s
(P
G
; G) ∩ ∆ ∩ Γ
G
= ∅.
For any op e n set ∆
in Γ
G
such that ∆
∩
˜
R
c,ν
= ∅, similarly we can show that some
iteration of ∆
contains a s-strip in the blender B, and so W
u
(z
G
; G) intersects ∆
in Γ
G
.
In other words, the closure of stable and unstable manifolds of P
G
for G are both
contain
˜
R
c,ν
, for any
˜
N
c
and ν.
Now, H-4 and the density of f
1
- stable and unstable manifold of p
1
in Λ implies that,
Γ
G,R
c,ν
⊂
W
s
(P
G
; G) ∩ W
u
(P
G
; G).
In particular for any F
µ
,
Γ
F
0
⊂ W
s
(P
0
; F
µ
) ∩ W
u
(P
0
; F
µ
).
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 25
Whenever the stable and unstable manifolds of a periodic hyperbolic point are both
dense on some set, the λ-lem ma provides transitivity and topological mixing.
Thus, for any N
c
and ν > 0,
i) R
c,ν
is topological mixing for G.
ii) Γ
G,R
c,ν
is strictly topological mixing for G.
And in particular,
i) N
0
is topological mixing for any F
µ
ii) Γ
F
0
= Λ × N is strictly topological mixing for any F
µ
.
The proof of Theorem A is completed.
Remark 4.4. In the perturbations introduced in the proofs we could use the generating
functions instead the Hamiltonians. It lets us to unify the proof of Theorem A and its
variation for the Hamiltonians. The Hamiltonian version of Theorem A shall be used in the
proof of Theorem B.
5. Instabilities in nearly integrable systems
5.1. Instability versus recurrency. The following basic lemma shall be used in the proof
of Theorem B.
Lemma 5. 1. There is a residual subset R of int(Ω(f)) such that any point in R is a
(positively and negatively) recurrent point.
Proof. Let B = {U
i
: i ∈ N} be a countable topological base in int(Ω(f)). For every
i ∈ N, there is n
i
∈ N such that f
n
i
(U
i
) ∩ U
i
= ∅. Let x
i
∈ V
i
:= f
−n
i
(U
i
) ∩ U
i
. Since
B
k
= {U
i
: i ≥ k} is also a topological base, the set {x
i
}
∞
r=k
is dense in int(Ω(f)). So
∞
i=k
V
i
is open and dense subset of int(Ω(f)). Then, R
+
:=
∞
k=1
∞
i=k
V
i
is residual. We
claim that R
+
⊂ Rec
+
(f). Since B is a top ological base, for any > 0 there is a k
such
that, if i > k then diam(U
i
) < . Now, for any x ∈ R
+
and for i > k
, x ∈ V
i
. So there is
n
i
∈ N such that, d(f
n
i
(x), x) < diam(U
i
) < . Since > 0 was arbitrary, this implies that
x is a positively recurrent point. We could do it for f
−1
to obtain a residual subset R
−
of
negatively recurrent points. Any point in the residual set R = R
−
∩ R
+
is positively and
negatively recurrent.
We say that a point x converges to infinity if for any bounded se t U ther is a number
n
0
such that for any n > n
0
, f
n
(x) /∈ U.
The following lemma is a corollary of a variation of Poincar´e recurrence theorem for
unbounded m easures (due to Hopf) which yields that for conservative homeomorphisms on
the manifolds with unbounded measure, almost all points e ither are recurrent or converge
to infinity.
Lemma 5.2. Let f be a conservative homeomorphism on a non-compact manifold with
unbounded Lebesgue measure. Then Lebesgue almost all points in Ω(f)
converge to infinity,
in the future and also past iterations.
26 MEYSAM NASSIRI
As a matter of fact, similar results may be stated on each fiber of the invariant sets such
as Γ
G
in Theorem A. That is, “almost all points” means “almost all points with respect to
the Lebesgue measure on each fiber”, also residual and open sets in the restricted topology
in fibers.
Now, suppose that the ass umption (b) in Theorem A fails. For instance, suppose
that Ω(f
2
) = N but for some
˜
f close to f
2
, Ω(
˜
f
2
) N. In this case, the same results on
transitivity and topologically mixing hold on the interior of non-wandering set. Indeed, we
used the hypothesis Ω(
˜
f
2
) = N , only in the last step of the proof to show that some of the
arbitrary large iterations of generic points in
˜
N
c
remain in some desired compact set
˜
N
c
.
This follow from the Lemma 5.1.
In contrast, let U
c
⊂⊂ M×N be an open set and Γ
G,c
= U
c
∩Γ
G
such that Γ
G,c
Ω(G).
Then, almost all points in some open subset of U
c
converge to infinity in the past and in
the future. Moreover, there is an open set V
c
⊂ U
c
such that
(1) V
c
∩ Γ
G,c
= ∅.
(2) Almost all (w.r.t the restricted Lebesgue measure) points in V
c
∩N
x
c
goes to infinity
both in the past and the future, where N
x
c
is the intersection of some fiber N
x
with
U
c
. In this case, we have a sense of instability, that is, orbits which come from
infinity and stay for some iterations near a transitive invariant set and then go back
to infinity.
These facts together with Theorem A leads to a dichotomy in this context:
existence of large robustly transitive sets or
existence of wandering orbits converging to infinity.
5.2. Proofs of Theorem B and Corollary C. In this section we complete the proofs
of Theorem B and Corollary C. First we recall the following result of Zenhder [Z] and
Newhouse [Ne].
Theorem 5.3 (Zenhder-Newhouse). There is a residual set R ⊂ Diff
r
ω
(M), 1 ≥ r ≥ ∞,
such that if f ∈ R, then any quasi-elliptic periodic point of f is a limit of transversal
homoclinic points of f.
A periodic point p of f of period n is called quasi-elliptic if T
p
f
n
has a non-real
eigenvalue of norm one, and all eigenvalues of norm one are non-real. Notice that if f is
Anosov, then robustly there is no quasi-elliptic periodic point. Indeed, C
r
generically every
periodic point is either hyperbolic or quasi-elliptic (cf. [Ne]).
Proof of Theorem B. Let f
1
and f
2
be the time one map of the flow generated by
the Hamiltonians h
1
and h
2
respectively. Since f
2
is integrable, it is dominated by f
1
|
Λ
,
and moreover a generic small perturbation of f
2
has some hyperbolic pe riodic point with
arbitrary weak hyperbolicity. Let
ˆ
f
2
be a small perturbation of f
2
such that its non-
wandering set is the whole manifold N and has a hyperbolic periodic point (with weak
hyperb olicity). If
ˆ
f
2
is enough close to f
2
then it is also dominated by f
1
|
Λ
. Now we
may re peat the prove of Theorem A for F
0
= f
1
×
ˆ
f
2
. Note that all the perturbations
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 27
had been done by some Hamiltonians. Then we obtain a family of Hamiltonians H
µ
for
each of which the time one map F
µ
of the corresponding flow satisfies the properties (1)
and (2) in Theorem A. Fix N
c
⊂⊂ N and ν > 0. As in the theorem A, there exists a
neighborho od W
c,ν
of the constructed family {H
µ
: µ > 0} such that if H ∈ W
c,ν
and
G is its corresponding time one map, then one of the following possibilities hold, either
˜
R
c
⊂ Ω(G) or not. Here R
c,ν
is a compact set no exceptional point (see the definition in
subsection 4.5) and
˜
R
c,ν
is its continuation w.r.t. G. If
˜
R
c
⊂ Ω(G) then we may follow the
final part of the proof of Theorem A to show that
˜
R
c,ν
is topologically mixing. Otherwise,
if
˜
R
c
Ω(G) then we use the results of subsection 5.1. In this case, for a residual subset of
˜
R
c
∩ Ω(f)
all points converge to infinity, both in past and in the future. This completes
the proof.
Proof of Corollary C. Let M = R
n
× T
n
and N = R × T
1
. First we perturb the
hamiltonian h
1
on M to obtain a transversal homoclinic intersection. Since h
1
has a non
hyperb olic periodic point, by a small perturbation we make it quasi-elliptic. Theorem 5.3
yields that for any C
r
generic perturbation
˜
h
1
of h
1
, this orbit is accumulated by hyperbolic
periodic points with homoclinic transversal intersections. Note that h
2
is dominated by
the restriction of
˜
h
1
on the hyperbolic basic set obtained from the homoclinic transversal
intersection.
Now, we take another small (generic) perturbation
˜
h
2
of the integrable Hamiltonian
h
2
on N to create a weak hyperbolic periodic point.
Since r ≥ 5, N is of dimension two and the integrable hamiltonian h
2
is non-degenerate,
then the KAM theorem implies that the non-wandering set robustly contains the manifold
N. In other word, the time one map f
2
of the flow generated by h
2
satisfies the hypothesis
(b) of Theorem A. In particular all the hypotheses of Theorems A and B hold for
˜
h
1
and
˜
h
2
(and their associated time on maps). Now we use Theorems A and B, and it completes
the proof.
Remark 5.4. If the dimension of N is two, then either any point in N
0
belongs to some
compact invariant region limited by two invariant curves or there is an unbounded Birkhoff
region of instability. In the former case we obtain transitivity since the hypothesis (b) of
Theorem A holds. In the latter case the instability region contains orbits starting near to
one boundary and converge to infinity (this is a classical result of Birkhoff). As in the
Corollary C, if the integrable system on N is non-degenerate and r ≥ 5, then using the
KAM theorem the hypothesis (b) holds and the second case does not occur. In the lower
regularity or in the degenerate case the hypothesis (b) does not hold in general. In this case
the union of the images of the non-wandering se t in N
0
under all the su-holonomy maps,
contains the boundary of the Birkhoff instability region. It implies that the orbit of any
open set intersecting the non-wandering set in N
0
, is unbounded and its closure contains
the non-wandering set in N
0
.
28 MEYSAM NASSIRI
6. Some remarks and open problems
The main results of this paper arise several natural questions. Here we mention some
of them. The first remark is concerned with a possible alternative approach to prove tran-
sitivity.
Remark 6.1. In the context of Theorem A, the accessibility with the density of recurrent
point implies transitivity (but not mixing). Without the global hyperbolicity it is difficult
to obtain “stable” accessibility. First, it seems essential to suppose that the Hausdorff
dimension of the hyperbolic set Λ to be large enough. Second, for stability of accessibility
one needs the continuity of Hausdorff dimension of the projections of the (hyperbolic) Cantor
set along the invariant foliations. Unfortunately, the stable and unstable foliations are not
smooth, and so the Hausdorff dimension of the projections do not vary continuously. A
similar difficulty occurs in the persistence of homoclinic tangencies in higher dimensions.
1. Transitivity and partial hyperbolicity. The first question concerns the genericity of the
robustly mixing partially hyperbolic sets. Theorem A suggests that the answer of the
following problem would be positive. See also [N].
Problem 6.2. Does there exist a residual set R ⊂ Diff
r
ω
(M), 1 ≤ r ≤ ∞, such that
if f ∈ R, then any normally hyperbolic invariant submanifold N for f with transversal
intersection between its stable and unstable manifolds is topologically mixing, provided that
N ⊂ int(Ω(f))?
In contrast, as in the case of C
1
topology (see [DPU], [BDP] and [HT]), we believe that
the partial hyperbolicity condition is necessary for robustness of mixing in any C
r
topology.
This problem is also related to the C
r
stability conjecture which is still open.
Problem 6.3. Let (M, ω) be a symplectic manifold. Suppose that Γ is robustly topological
mixing invariant set for f in Diff
r
ω
(M). Is it a partially hyperbolic set?
2. Ergodicity and stable ergodicity. Let f
1
and f
2
as in Theorem A. Suppose that N is
compact. Then the topologically mixing invariant set obtained in the Theorem A is lami-
nated by central manifolds diffeomirphic to N. This lamination is normally hyperbolic. See
H-1–H-4, S-1 and S-2, in section 4. As a matter of fact, this implies that for all symplectic
diffeomorphism G near to f
1
× f
2
, there is an invariant measure ρ
G
supported the continu-
ation of Λ × N. Moreover, the measure ρ
G
is a skew product of the Lebesgue measure on
the fibers (i.e. the volume form obtained by the restriction the symplectic 2-form on the
fibers) over the Bernoulli measure of the shift on Λ = {1, . . . , d}
Z
.
As was mentioned in the introduction, Theorem A can be seen as a local and topological
version of the example Shub and Wilkinson [SW], where they proved that the product of
“Anosov × Standard map” on T
4
is C
∞
approximation by (symplectic) stably ergodic
systems. A natural problem arises:
Problem 6.4. Is it possible to C
∞
approximate the product f
1
× f
2
of Theorem A by
symplectic diffeomorphisms G for which t he invariant ρ
G
supported the continuation of
Λ × N is ergodic or stably ergodic? Is the compactness assumption on N necessary?
ROBUST TRANSITIVITY IN HAMILTONIAN SYSTEMS 29
3. Other contexts. Other problems concern natural extensions and applications of our
results and method in similar contexts. For instance,
(1) analytic symplectic and Hamiltonian systems,
(2) geodesic flows on manifolds of dimensions larger that two,
(3) perturbations of geodesic flows on surfaces by perio dic potentials,
(4) the dynamics near the (quasi) elliptic periodic points in dimensions ≥ 4,
(5) generic energy levels of time independent Hamiltonian systems,
(6) specific mechanical problems such as restricted 3-body problem.
4. On t he abundance of instability. Let the Hamiltonian H
0
is written as the sum of two
functions which depend to different variables. In this paper we have proved that, if H
0
is integrable or has a partially hyperb olic invariant set, then H
0
+ h exhibits instability
(Arnold diffusion) and large topological mixing set, where h =
˜
h
0
+
1
˜
h
1
+
2
˜
h
2
, the C
r
-
norm of
˜
h
i
’s are one, and h
0
is generic (open dense), h
1
is not generic, but h
2
isarbitrary.
Moreover, 0 <
i
< ε
i
(h
1
, h
0
).
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IMPA, Rio de Janeiro, Brazil.
E-mail address: [email protected]
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