Bochi-Mañé dichotomy for 2n-Hamiltonians, Random Perturbation Techniques

UNIVERSIDADE DE LISBOA

Lisbon School of Economics and Management

Bochi-Ma˜

n´

e Dichotomy for 2n-Hamiltonians, Random

Perturbation Techniques

Filipe Andr´

e Paulino Santos

Supervisor

Doctor Jo˜ao Carlos Martinho Lopes Dias, Associate Professor with Aggregation,, Lisbon School of Economics and

Management – Universidade de Lisboa

Thesis submitted for the degree of Doctor in Applied Mathematics for Economics and Management

UNIVERSIDADE DE LISBOA

Lisbon School of Economics and Management

Bochi-Ma˜

n´

e Dichotomy for 2n-Hamiltonians, Random

Perturbation Techniques

Filipe Andr´

e Paulino Santos

Supervisor

Doctor Jo˜ao Carlos Martinho Lopes Dias, Associate Professor with Aggregation,, Lisbon School of Economics and

Management – Universidade de Lisboa

Thesis submitted for the degree of Doctor in Applied Mathematics for Economics and Management

Jury President

Doctor Nuno Jo˜ao de Oliveira Val´erio, Full Professor and President of the Scientific Board, Lisbon School of Economics and Management –

Universidade de Lisboa Vowels

Doctor Jo˜ao Carlos Martinho Lopes Dias, Associate Professor with

Aggregation, Lisbon School of Economics and Management – Universidade de Lisboa

Doctor Jorge Miguel Milhazes de Freitas, Associate Professor with Aggregation, Faculty of Sciences – Universidade do Porto

Doctor M´ario J´ulio Pereira Bessa da Costa, Associate Professor with Aggregation, Faculty of Sciences – Universidade da Beira Interior

Doctor Pedro Miguel Nunes da Rosa Dias Duarte, Associate Professor with Aggregation, Faculty of Sciences – Universidade de Lisboa

Doctor Maria Joana Costa Cruz Oliveira Torres, Assistant Professor, School of Sciences – Universidade do Minho

Acknowledgements

The author is grateful to the (FCT) Funda¸c˜ao para a Ciˆencia e Tecnolo-gia, who financially supported this project with the concession of the grant SFRH/BD/110545/2015 from January 2016 until December 2018 under the doctoral support grants program.

Special thanks are also in order to (ISEG) Instituto Superior de Economia e Gest˜ao, the research center (CEMAPRE) Centro de Matem´atica Aplicada `

a Previs˜ao e Decis˜ao Econ´omica and my Supervisor for his valuable inputs and motivation throughout the course of this work. These Institute and Cen-ter have provided the physical space, numerous conferences and seminars to stimulate this work and pull it forward, and also financial support for some conference trips. In general, all the organization and structure provided by ISEG and CEMAPRE were crucial on the success of the work developed.

Another special thanks to ISEG for being my alma mater during bachelor, masters and now PhD degree and all the Professors that teach and develop their research work there. To my supervisor for all the support and friendship granted during this endeavor.

Finally, I would like to thank my family for the unconditional support along the whole of my academic career and for continuing to believe in me during the toughest of times. It is to them that I would like to dedicate this work and give them the honor of this achievement. To my grandfather and grandaunt, thank you for everything.

Abstract

We prove the high dimensional version of the Bochi-Ma˜n´e dichotomy for Hamiltonian systems, achieving for the first time in a continuous setting such a general result. That is, we find the existence of a C2-residual set of Hamil-tonians for which there is an open mod 0 dense set of regular energy surfaces each being either Anosov or for almost every x, either LE(x) = 0 or there is a partially hyperbolic splitting .

A generalization of Bochi’s random perturbative technique is developed and used in the Hamiltonian framework, to show the extended reach of prob-abilistic methods and their importance on the nesting and iterative process of perturbations. The main technique consists in letting the original dynamics component act on the invariant subspaces while the random component acts on the direction we are iteratively perturbing. We also connect reachability properties of dynamically guided stochastic processes with the existence or lack of domination on orbits.

Contents

1 Introduction and Statement of the Results 5

2 Preliminaries 9

2.1 Symplectic Geometry . . . 9

2.2 Hamiltonian Flows . . . 11

2.3 Oseledets Theorem for Hamiltonians . . . 14

2.4 Dominated Splitting . . . 16

2.5 Poincar´e Linear Transversal Flow . . . 22

3 Exploring and Using Flexibility 27 3.1 Measure on Transversal Sections . . . 28

3.2 Proof of the Main Theorem . . . 29

3.2.1 How to Use the Flexible Sequences . . . 33

4 Non-Dominance: Construction of the Perturbations for each Case 35 4.1 Hamiltonian Setting . . . 35

4.2 Construction Restrictions . . . 36

4.2.1 Rotation Size . . . 36

4.2.2 Iterated Nested Rotations . . . 38

4.2.3 Remaining Case . . . 40

4.3 Cases of Non Dominance . . . 41

4.3.1 Type I . . . 41

4.3.2 Type II . . . 44

4.3.3 Type III . . . 46

4.3.4 Type IV . . . 49

5 Constructing the Random Perturbation 51 5.1 Random Shadowing . . . 51

5.1.1 The Choice of Stochastic Process . . . 52

5.1.2 Comparison Between Perturbation and Random Trajec-tory . . . 53

6 Global Exponent Decay 59 6.1 Kakutani Tower Scheme and Special Flows . . . 59 6.2 Integrated Lyapunov Exponent . . . 60 6.3 Exploiting the Local Exponent Decay . . . 63

7 Extending the Random Perturbation Techniques 71

7.1 Qualitative Approach . . . 71 7.2 Quantitative Approach . . . 73 7.3 Application to the Joint Flexibility Case . . . 75

Chapter 1

Introduction and Statement of

the Results

The dependence of Lyapunov exponents on the dynamics of the flows of Hamil-tonian systems is a topic of high relevance in the field of dynamical systems. An idea that aimed to obtain generic conditions for the value of the Lyapunov Exponent was outlined by Ricardo Ma˜n´e in [1]. The referred result has been proved and actually extended in [2].

Ma˜n´e conjectured that for a compact manifold M , we could find a residual subset of area preserving diffeomorphisms, such that, each non-Anosov dif-feomorphism in the set is either partially hyperbolic or its upper Lyapunov exponent is zero almost everywhere, with respect to the normalized area mea-sure.

The context was later extended from generic area preserving diffeomor-phisms to C1_{-generic symplectic diffeomorphisms and one of the further}

in-terests in studying this idea is to know if the same result can be obtained for Hamiltonian systems. Note that already when working with symplectomor-phisms, Hamiltonian flows are used since they constitute a way of constructing perturbations guaranteed to be symplectic and therefore measure preserving.

The main problem in the latter is to find a way to use the perturbation techniques based on small rotations, while guaranteeing that the perturbed flow is still the flow associated with some Hamiltonian C2-close to the original. In this matter, inspiring results were obtained in [3] and [4].

We complete the Bochi-Ma˜n´e dichotomy in the Hamiltonian framework by applying a version of the dimension reduction techniques more suitable to our cases of non-dominance in the continuous setting. For each connected compo-nent of each regular energy level we create a sequence of cylinders around the flow, allowing us to perturb differently in each cylinder while smooth pasting all of them by assuming some natural regularity conditions and with the aid of bump functions.

with smooth boundary and the Hamiltonians will be real valued functions at least two times differentiable: Cr_{(M, R), (r ≥ 2) but endowed with the} C2_{-topology since the perturbation techniques do not work on more refined}

topologies. A reason for this is that we need to re-scale between the iterations that compose the perturbation in order to assure that its support is contained in a small neighborhood of the point and does not interfere with the pertur-bations elsewhere. This re-scaling despite maintaining the derivative of the flow, leads to the uncontrolled growth of the second derivative, affecting the C2 _estimates.

Moreover, the Hamiltonian H must be constant on each connected com-ponent of the boundary (in the non-empty boundary case). For each energy level

H−1(e) = {x ∈ M : H(x) = e}

we denote its connected components by Se ⊂ H−1(e). The residual character

of Morse functions implies the existence of only finitely many critical points and we will therefore work only on connected components of regular energy surfaces.

The measure µ in M preserved by the Hamiltonian flow will be the one generated by the volume form ωn. One can also obtain a measure µSe

corre-sponding to the restriction of µ to Se defined by the relation between a 2n and

2n − 1 induced volume forms:

ωe(x)(.) = ωn(x)(X, .)

where the . represents the 2n − 1 elements from TxSe and X is a vector field

transverse to Se.

We are in conditions to now state the main result:

Theorem 1.0.1. Let (M, ω) be a 2n-dimensional symplectic and compact smooth manifold. Then there exists a residual set R of Hamiltonians in the C2_{-topology, such that for each H ∈ R, the union of the regular energy surfaces}

Se that either:

• are Anosov

• for almost every x, either LE(x) = 0 or there is a partially hyperbolic splitting

forms an open dense subset of M with full -mod0 measure.

One of the fundamental concepts to prove it is that of a dominated split-ting (sometimes referred in the literature as projective hyperbolicity) which can be roughly thought out as a weaker form of hyperbolicity. Since for sym-plectic maps, dominated splittings are partially hyperbolic [5] (a proof in the Hamiltonian flow setting is presented, based on the cited work), we can ap-proach the problem through the Oseledets decomposition point of view. For

µ-a.e. x ∈ M , and assuming that the Oseledets splitting with respect to the Hamiltonian is not trivial, we have

TxM = E1(x) ⊕ E2(x) ⊕ ....Ek(x)−1⊕ RXH(x)

that can be zipped by grouping together the sub-spaces with positive Lyapunov exponent on E+, with null exponent on E0 and with negative exponent on E−, into

TxM = E−(x) ⊕ E0(x) ⊕ E+(x)

with E0_{(x) including RX}

H(x), the Hamiltonian vector field direction.

The original splitting being dominated will imply the uniform expansive-ness of E+and the uniform contractiveness of E−, that is, the zipped splitting has partially hyperbolic behaviour. This conclusion is much stronger than the one we take from the Oseledets theorem based on non-uniform limits.

This main result focus on the decomposition of the manifold into energy levels, a natural way to select invariant sets in the Hamiltonian framework,

The decomposition into energy levels is very relevant since that allows to obtain a residual subset by the use of lemma I from [1] which allows to achieve residuality in a product space of Hamiltonians × level sets.

In chapter 2 we formally develop the concepts already introduced and many others related to symplectic structures, Hamiltonian dynamics, Oseledets split-ting, dominance through orbits of the system and flexibility of split-sequences. We make the necessary remarks about their role in the understanding of the whole project. After that, chapter 3 treats the exchange of Oseledets directions along an orbit segment by means of Hamiltonian and symplectic perturbations. This mixing of directions, taking use of sequence flexibility, is what causes the decay of the Lyapunov exponent and is only possible in orbits where the split-ting is not dominated. Then we can make the norm decay leadind to the Lyapunov exponent decay, but only in some small neighborhood of the point without dominated splitting (meaning that there is no dominated splitting along its orbit). This establishes the fundamental local result.

Further on, we develop the Hamiltonian setting and explain how to build perturbations for the first three cases of non-dominance, in chapter 4. In chap-ter 5 we explore in depth Bochi’s random approach to the problem of rotating large angles on non-symplectic planes. The ideas there present are worth a further study of the possibility they can be rehashed for other perturbative problems. It is in the sequence of chapter 4, completing the construction of the random perturbation that solves the last case.

The globalization of such result to the whole manifold is obtained through local applications of the previous result and a covering by means of a Kakutani tower scheme using the Vitali covering lemma. Each of the sets of the covering will be a neighborhood of a non-dominated point. Chapter 6 illustrates how to get a global exponent decay from the local decay.

After these main results are obtained we shift focus to the possible ex-tensions of the random perturbation techniques in order to make them more general and develop its foundations. That is the content of chapter 7, with a special study about the ”how to” work towards the application of this improved technique for the Bochi-Mane dichotomy.

Chapter 2

Preliminaries

We start by familiarizing with key concepts on symplectic geometry, Hamil-tonian flows, Oseledets theorems, dominated splittings and linear Poincar´e transversal flows.

2.1

Symplectic Geometry

Definition 2.1.1. We say that some property holds as generic depending on the following contexts:

• In a measure theoretical sense, if it holds for full measure sets. • In a topological sense, if it holds for a dense open set.

• In a topological sense on the specific case of a Banach space, if it holds on a countable intersection of open dense sets (residual set).

The central object of symplectic geometry is the concept of symplectic form, which is a special type of differential form.

Definition 2.1.2. The kth exterior powerVk

S of a vector space S is a vector space equipped with a map denominated exterior multiplication

ψ : S × S... × S −→

k

^

S (2.1.1)

such that ψ is alternating multilinear and a basis for Vk

S is given by

{si1 ∧ ... ∧ sik : 1 ≤ i1 < ... < ik≤ n} (2.1.2)

Definition 2.1.3. An exterior k-form in Rn _{is a map ω that associates to each}

p ∈ Rnan element ω(p) ∈Vk

(Rnp∗) (the set of k-multiniear maps on Rn ) and

can be generically written as

ω(p) = X

i1<...<ik

ai1...ik(p)(dxi1 ∧ ... ∧ dxik)p, ij ∈ {1, ..., n}

ai1,...,ik are real functions in R

n_{. When they are differentiable, we call ω a}

differential k-form.

We work with a symplectic manifold (M, ω) with M being of dimension 2n and ω being a symplectic form.

Definition 2.1.4. Let X be a 2n-dimensional vector space over the real num-bers. The bilinear map ω : X × X → R is a symplectic form if:

ω(x1, x2) = −ω(x2, x1) for all x1, x2 ∈ X

X_ω⊥= {x1 ∈ X : ω(x1, x2) = 0, for all x2 ∈ X} = {0}

Altough that is the definition of symplectic form on a vector space, we refer to symplectic form or symplectic structure as a mapping p → ωp that for each

p ∈ M gives a form on TxM (as in the definition of differential form) in such a

way that dω = 0 along M . This last condition is called closedness of the form. From such symplectic form we derive a volume form

ωn= ω ∧ .... ∧ ω

which in its turn associates with a Lebesgue measure µ by Liouville theorem. Then it is possible to define diffeomorphisms that preserve this structure and consequently the measure.

Definition 2.1.5. Given symplectic manifolds M1 and M2 with symplectic

structures given by the forms ω1 and ω2, we say that a diffeomorphism between

them, f : (M1, ω1) → (M1, ω2) is a symplectomorphism if f∗ω2 = ω1, that is,

ω1(X, Y ) = ω2(DfX, DfY )

with X and Y vector fields in M1.

Just like for inner products, we can define an isomorphism J : V → V , where V is a linear space such that ω(v1, v2) = J v1.v2 and by compactness of

the space and continuity of the form we have the existence of C > 0 such that the following inequalities hold for any v1, v2

|ω(v1, v2)| ≤ C||v1||.||v2||

C−1||v1|| ≤ ||Jv1|| ≤ C||v1||

To better structure the action of the perturbations, namely, the rotation possibilities at our disposal, we make the distinction between types of sub-spaces.

Definition 2.1.6. Consider a linear space V and its linear subspace S. Then, we say that:

S is symplectic if S ∩ S⊥= {0} S is isotropic if S ⊂ S⊥, S is lagrangian if S = S⊥

Here, S⊥ = {v ∈ V : ω(v, s) = 0 for all s ∈ S} is the symmplectic comple-ment.

If a plane is symplectic it is easy to build a rotation in the plane that doesn’t change the space complementary to said plane, making the problem effectively 2-dimensional. However, if S is an isotropic plane, a symplectic map that ”freely” rotates it while leaving the orthogonal complement invariant doesn’t exist. We actually show in a chapter 4 example that it is possible to build rotations on isotropic planes but they are very restricted.

2.2

Hamiltonian Flows

A central object in our project is the Hamiltonian flow, a concept with countless applications in mathematics and physics. It is defined by the contraction of the symplectic form ω with a vector field as follows: Let H be a real function in the manifold M and consider the vector field XH : T M → T M for which

the contraction (also known as interior product) with ω verifies (XH a ω).v = ω(XH, v) = dH(v)

that is, XH a ω becomes an exact 1-form equal to the differential of the

0-form H. Then we denominate Hamiltonian flow as the group of symplector-morphisms ϕt H : M → M generated by XH as  _d dtϕ t H = XH ◦ ϕtH ϕ0 H = Id

To work in a more manageable space, namely, symplectic flowbox coordi-nates, we use the following theorem proved in [3] since it works for an arbitrary number of even dimensions.

Theorem 2.2.1. Fix p ∈ [2, +∞]. Let (M, ω) be a Cp _{and 2N dimensional}

symplectic manifold. Consider an Hamiltonian H ∈ Cp_{(M, R) and a regular}

point for H, x ∈ M . Then, there exists an open set U ⊂ M containing x and a local Cp−1 _{symplectomorphism ψ that changes the original coordinates and}

symplectic form to the canonical coordinates and canonical symplectic form, that is: ψ : (U, ω) −→ (R2N, n X i=1 dxi∧ dxn+i)

such that in the new coordinates, the flow is rectified, H(x) = [ψ(x)]n+1, x ∈ U

Some estimates are needed to build the Hamiltonian perturbations. The following pair of lemmas will be fundamental to achieving that.

Lemma 2.2.2. Let H be a Hamiltonian on R2n_{, a > 0 and S : R}2n _{→ R}2n

be a symplectic linear map. Define Hamiltonians H1(x) = H(ax)_a2 and H2(x) =

H(S(x)). Then for the second diferentials we have D2H1(x)(v, w) = D2H(ax)(v, w)

D2H2(x)(v, w) = D2H(S(x))(Sv, Sw)

and for the flows

ϕt_H1(x) = ϕ t H(ax) a ϕt_H2(x) = S−1◦ ϕt H ◦ S(x)

Proof. For the second order derivatives its only needed to check the relation between Hessian matrices. Since

∂2[H1(x)] ∂xi∂xj = ∂ 2_a−2_[H(ax)] ∂xi∂xj = ∂ 2_H ∂xi∂xj (ax) and ∂2_[H 2(x)] ∂xk∂xj = 2n X i,s=1 ∂2_H(S(x)) ∂xs∂xi .msj.mik = [STHH(S(x))S]kj

where mij is the (i, j) element of matrix S, the first two equalities are obtained.

In order to get the equations for the Hamiltonian flows, establish relations between vector fields

XH1(x) = J ∇

T

H1(x) = a−1J ∇H(ax) = a−1XH(ax)

Replacing x with ϕt

H1(x) and using the flow properties it can be seen as a

differential equation solved by ϕt

H1(x) since ∂ϕt H1(x) ∂s = a −1 XH(aϕtH1(x))

Due to local unicity of solution and since the flows are defined for every t, if f0(t)=a−1XH(afx(t))

x has another solution with fx(0) = x they must coincide.

Plugging in fx(t) = a−1ϕtH(x) we see it is indeed a solution.

For the H2(x) Hamiltonian, the vector field XH2(x) is equal to

J ∇TH2(x) = J ST∇TH(S(x)) = J STJ−1XH(S(x)) = S−1XH(S(x))

Similarly to H1, a differential equation solved by ϕtH2(x) is obtained, f

0_(t)=S−1_X

H(Sfx(t))

x

with fx(0) = x. Plugging in fx(t) = S−1◦ ϕtH(S(x)) we again reach the

Lemma 2.2.3. Let H : R2n _{→ R be a smooth function that is constant outside}

of a compact set. Then the associated Hamiltonian flow ϕt_{H : R}2n _{→ R}2n is well defined for t ∈ R, and we have the following bounds for the flow and its derivative, for all x ∈ R2n _{and all t ∈ R:}

||ϕt H(x) − x|| ≤ |t| sup x∈R2n ||DH|| ||Dϕt H(x) − Id|| ≤ exp  |t| sup x∈R2n ||D2_H||  − 1

Proof. The flow is well defined for all time because in the regularity conditions stated for the Hamiltonian function H the associated vector field XH will be

complete. Now write f (t, x) = ϕt

H and apply the multivariate Lagrange mean

value theorem to f as a function of t. ||f (t, x) − f (0, x)|| =     t∂f ∂s(u, x)     , for some 0 < u < t This yields ||ϕt H(x) − x|| ≤ |t|     ∂ϕu H ∂s (x)     = |t|.||XH(ϕuH(x))||

Plugging in the relation between the Hamiltonian vector field and the Hamil-tonian gradient we get

||XH(ϕuH(x))|| = ||J.∇ T H(ϕ u H(x))|| ≤ ||∇ T H(ϕ u H(x))||

and we easily get that ||∇T

H(ϕuH(x))|| is bounded by sup ||v||=1 ||∇T_H(ϕu_H(x))v|| = sup ||v||=1 ||DH(ϕu_H(x))(v)|| ≤ ||DH(x)||

Consider the function g(t, x) = 1+||Dϕt_H(x)−Id||. Computing its time partial derivative yields

∂g

∂s(u, x) = limt→u

g(t, x) − g(u, x) t − u = limt→u ||Dϕt H(x) − I|| − ||DϕuH(x) − I|| t − u ≤ lim t→u ||Dϕt H(x) − DϕuH(x)|| t − u ≤ limt→u     ∂Dϕv H ∂s (x)     , u < v < t     ∂Dϕv H ∂s (x)     =     D∂ϕ v H ∂s (x)     ≤ sup x ||D2_H||

where the last inequality follows from the first bound. Since g(t, x) ≥ 1 then g as a function of t verifies

∂g

∂s(t, x) ≤ supx

for all x. We are therefore under the conditions of the Grownwall’s lemma which guarantees that

g(t, x) ≤ exp Z t 0 sup x ||D2_H||ds  = exp (|t| sup x ||D2_H||)

resulting in the referred bound for g(t, x) − 1.

2.3

Oseledets Theorem for Hamiltonians

Oseledets theorem has many versions, applicable to different settings. Here we present, although without a complete proof, the Oseledets theorem for Hamiltonians.

Theorem 2.3.1. Given µ-a.e. point x ∈ M we either have: A trivial splitting TxM = Ex with dim(Ex) = 2n and

lim t→±∞ 1 t log ||Dϕ t H(x)v|| = 0, v ∈ Ex or a splitting TxM = Ex1⊕ ... ⊕ E d−s x ⊕ E 0 x⊕ E d+s+1 x ⊕ ... ⊕ E 2d x

with d ≤ n dependent on x, s = dim(E0x)

2 , RXH(x) ⊂ E 0_{(x) and} lim t→±∞ 1 t log ||Dϕ t H(x)vj|| = λj(H, x), vj ∈ Exj

Due to preservation of measure it is also true that

lim t→±∞ 1 t log Det(Dϕ t H(x)) = d−s X j=1 λj(H, x)dim(Exj) = 2d X j=d+s+1 λj(H, x)dim(Exj)

Finally, the angle between the subspaces of the splitting evolves subexponentially lim

t→±∞

1

t log (sin(θt)) = 0

Proof. Applying the usual Oseledet’s theorem for cocycle diffeomorphisms we can get all the assymptotic results from the theorem for Dϕk

H for k ∈ N. Using

that for any t ∈ R we can write it has t = k + p for k as previously and p < k and also that Dϕp_H for fixed p is uniformly bounded, then the limits do not change if we use Dϕk

H(x) or DϕtH(x) = Dϕ p

H(ϕkH(x)).DϕkH(x). The invariance

property is also mantained when using t instead of k.

Finally we need to check that RXH(x) ⊂ E0(x). By the construction of

the Hamiltonian vector field, Dϕt

direction in the Oseledet’s sense. The Lyapunov exponent in this direction will be lim t→±∞ 1 t log ||Dϕ t H(x)XH(x)|| = lim t→±∞ 1 t log ||XH(ϕ t H(x))|| = 0

due to the vector field XH being C1 and M being compact.

More information is extracted from this result, namely:

The symmetry between the exponents due to the measure preserving prop-erty of the flow

λj(H, x) = −λj−d−s(H, x) for d + s < j ≤ 2d

The gradient ∇H direction also has null exponential growth, despite not being an Oseledets direction due to not verifying the invariance property.

Lemma 2.3.2. The Lyapunov exponent of a symmplectic diffeomorphism f : M → M is invariant through charts.

Proof. Let f∗ = φi+1◦f ◦φ−1i be the representation of f : M → M using charts

φi chosen from an atlas such that all their norms and norms of their inverses

are bounded by a uniform constant. Each chart φi : Ui → R2n contains fi(x).

We can then obtain a new splitting with subspaces Eyi = Dφi(f

i_(x))E fi_(x)

and similarly for Fyi for yi = φi(f

i_(x)).

First we check that the new subspaces indeed constitute an invariant split-ting for f∗ _{in R}2n_{. Since the Dφ}

i are isomorphisms between linear spaces, both

the properties of trivial intersection between fibers of E and F and their direct sum being the total space, extend to E and F . Furthermore

Df∗Eyi = D(φi+1◦ f ◦ φ

−1

i )(yi)Dφi(fi(x))Efi_(x) =

= Dφi+1(f ◦ φ−1i (yi))Df (φ−1i (yi))Efi_(x)

where we canceled the inverse terms. Now using the coordinates in M we get Dφi+1(fi+1(x))Df (fi(x))Efi_(x) = Dφ_i+1(fi+1(x)))E_fi+1_(x) = E_y

i+1

where we used the original invariance of the splitting for f . yi+1= f∗(yi) and

that concludes the proof of invariance.

It remains to compute the Lyapunov exponents for the new splitting. An elementary computation shows that to get the nth-composition of the ex-pression of f in charts we just need to get the chart representation of the nth-composition of f , that is, f∗n = φi+n◦ fn◦ φ−1i . Then we compute the

exponent lim n→±∞ 1 nlog ||Df ∗n (y)|Ey|| = lim n→±∞ 1 nlog ||Dφi+n(f i+n_(x))Dfn_(fi_(x))|E fi_(x)||

Now using the bound given by the atlas and the lower bound given by com-pacity and exclusion of critical points (the set of regular points will also have full measure) we can crunch the limit of

Xn,i= 1 n log ||Dφi+n(f i+n (x))Dfn(fi(x))|Efi_(x)|| as follows lim n→±∞ 1 n log 1 Ki ||Dfn_(fi_(x))|E fi_(x)|| ≤ lim n→±∞Xn,i≤ ≤ lim n→±∞ 1 nlog Ki||Df n (fi(x))|Efi_(x)||

which ends up leading to limn→±∞Xn,i = λ+(f, x) since there is a universal

bound for all Ki, it concludes the proof.

With this last result we can just make a perturbation g such that it will have the chart g∗ = φi+1◦ g ◦ φ−1i representation in R2n. Note that the charts

are still chosen so that each Ui contains a point of the orbit of f but that is

no problem due to the construction choice for the maps we will select later.

2.4

Dominated Splitting

Let Λ ⊂ R ⊂ M be an invariant set under ϕt_H, where R is the full-measure set of Oseledets regular points and consider the splitting TΛM = EΛ⊕ FΛ.

When there is no space for confusion we shall omit the index set Λ or suitable interchange the notation while referring to the subspaces of the splitting. The common definition for dominated splitting in the literature is that of an m-dominated splitting,

Definition 2.4.1. Let Λ ⊂ R be an Dϕt

H-invariant set. A splitting TΛ= E⊕F

is m-dominated for natural m if it is Dϕt_H-invariant, the subbundles that compose it have constant dimension and

||Dϕm H|E(x)|| m(Dϕm H|F (x)) ≤ 1 2 x ∈ Λ

however that is not the best suited for the Hamiltonian flow framework so we also introduce a continuous version, the (β, t0)-domination.

Definition 2.4.2. Let β > 0 and t0 > 0. The splitting TΛM = EΛ ⊕ FΛ

is (β, t0)-dominated if it is DϕtH-invariant, the composing subbundles have

constant dimension and for all x ∈ Λ, any v ∈ Ex\ {0}, u ∈ Fx \ {0} and

t ≥ t0, we have kDϕt H(x) vk kDϕt H(x) uk ≤ e−β(t−t0)kvk kuk

It is dominated if it is (β, t0)-dominated for some (β, t0). We say that E −→ F

(E dominates F ).

One of the main results is the relation between domination and partial hyperbolicity so it is the next definition to be introduced

Definition 2.4.3. Λ is partially hyperbolic if dim(E_x0) ≥ 2 and there exists t0 > 0 and β > 0 such that for all vs ∈ Ex−, vu ∈ Ex+ and x ∈ Λ we have for

t ≥ 0,

||Dϕt_H(x)vs|| ≤ e−β(t−t0)||vs|| (2.4.1)

and

||Dϕ−t_H(x)vu|| ≤ e−β(t−t0)||vu|| (2.4.2)

Remark 2.4.1. m(A) is the co-norm and corresponds to the lowest growth vector when A is applied to it. If A is invertible then it is the same as ||A−1||−1. We could have used this notation for the (β, t0)-domination and will

inter-change between them as it is more suitable.

For ease of exposition we will refer to the domination ratios as the following φE,F,H(x, t) =

||Dϕt

H|E(x)||

m(Dϕt

H|F (x))

Also note that the general definition of domination, between more than two invariant subspaces can be decomposed in domination between two subspaces. Definition 2.4.4. Let Λ be an invariant set of ϕt

H : M → M . Assume that

we have a decomposition TΛM = E1⊕ E2⊕ ... ⊕ Ek invariant under DϕtH. We

say that it is a dominated splitting provided

TΛ= (E1⊕ ... ⊕ Ej) ⊕ (Ej+1⊕ ... ⊕ Ek)

is a dominated splitting for any j = 1, ...k − 1.

The partition of TΛ into two subsets: E+ = (E1 ⊕ ... ⊕ Ej) and E−0 =

(Ej+1⊕ ... ⊕ Ek) such that the first j subsets are the ones with stricly positive

Lyapunov exponent is called the zipped Oseledets splitting.

Since the m-domination and (β, t0)-domination are the main blocks of the

domination concept it is important to see that they are interchangeable Theorem 2.4.2. m-domination is equivalent to (β, t0)-domination.

Proof. Let TΛM = E ⊕ F be (β, t0)-dominated for DϕtH. It is easy to see that

φE,F,H(x, t0) ≤ e−β(t−t0)for t0 > t. Making t = ln(2)

β + t0 and m = t

0 _{the smallest}

To prove the reverse we first note that if a splitting is m-dominated, the same behaviour is evidenced on the multiples of m.

||Dϕ2m H (x)|E(x)|| m(Dϕ2m H (x)|F (x)) ≤ ||Dϕ m H(ϕ m H(x))|E(ϕ m H(x))||.||Dϕ m H(x)|E(x)|| m(Dϕm H(ϕmH(x))|F (ϕmH(x))).m(DϕmH(x)|F (x))

Doing the same k times,

||Dϕmk H (x)|E(x)|| m(Dϕmk H (x)|F (x)) ≤ 1 2 k

Between mk and m(k + 1) we have the estimate ||Dϕmk+p_H (x)|E(x)|| m(Dϕmk+p_H (x)|F (x)) ≤  1 2 k Cp with p < m, Cp = supx∈Λ ||Dϕp_H(x)|E(x)||

m(Dϕp_H(x)|F (x)) Now consider k and p such that t =

mk + p and C = sup0≤p≤m,x∈Λ

||Dϕp_H(x)|E(x)||

m(Dϕp_H(x)|F (x)) > Cp for all p. Since k > t m − 1 we get ||Dϕt H|E(x)|| m(Dϕt H|F (x)) ≤ 1 2 k C ≤ 1 2 _mt−1 C

After some simple computations to solve (1₂)mt−1_{.C = e}−β(t−t0) _{for β and}

t0 we conclude that the splitting is (β, t0)-dominated with β = ln(2)_m and t0 =

(1 + ln(C)_ln(2)).m

How does domination compare to hyperbolicity? They both have a unifor-mity clause since the same m must work for every x ∈ Λ. In the hyperbolic case we are assured that the map is expanding uniformly in the directions on one of the subspace of the splitting and is contracting on the other subspace. This by itself implies that the splitting is dominated because we can let enough time go by in order to any definition of dominated splitting be verified.

In a dominated splitting we just need that one of the subspaces expands much more than the other, not necessarily one expanding and the other con-tracting.

The picture changes in the symplectic case due to the growth symmetry between conjugate directions and that is why we have the following theorem. This proof follows closely [8].

Theorem 2.4.3. An invariant and compact set Λ ⊂ S is partially hyperbolic iff dim(E0_{) > 0 and E}+_{⊕ E}0_{⊕ E}− _{is a dominated splitting. Here, E}0_{⊕ E}− _is

Proof. First we check on [8] that E0_⊕E−_{is invariantly split from E}−0_{. We need}

to prove that E+⊕ E0 _{is dominated by E}−_{, to have T}

xM = E+⊕ E0⊕ E− be

a dominated splitting. To see this, let us pick an arbitrary unit vector vc∈ E0

and write the vector J vc as wsu+ wc. Then,

ω(vc, J vc) ≥ C0−1||vc||.||wsu+ wc|| ≥

C₀−1

b ||vc||.||wc||

Note that ω(vc, wc) = ω(vc, J vc− wsu) = ω(vc, J vc) due to ω(vc, wsu) being

zero and so the above bound works for ω(vc, wc) also. This means that for

each vc∈ E0 we can find wc∈ E0 with the above properties. In an analogous

way we can find a ws ∈ E− associated with vu ∈ E+ and wu ∈ E+ associated

with us∈ E−, that is, for any of these pairs

ω(v1, v2) ≥ γ||v1||.||v2||

for some uniform constant γ > 0. So, for fixed x ∈ Λ and unit vectors vc ∈

E0(x) and vs ∈ E−(x) we can find vectors wc ∈ E0(x) and wu ∈ E+(x) such

that ω(wc, vc) ≥ γ and ω(DϕtH(x)wu, DϕtH(x)vs) ≥ γ||DϕtH(x)wu||.||DϕtH(x)vs|| then, ||Dϕt H(x)vc|| ≥ C₀−1 ||Dϕt H(x)wc|| |ω(Dϕt H(x)wc, DϕtH(x)vc)|

which is greater than C

−1 0 γ

||Dϕt H(x)wc||

. The domination of E−0 by E+ _gives

||Dϕt_H(x)vc|| ≥ C₀−1γ ||Dϕt H(x)wu||e−β(t−t0) ≥ C −2 0 γ2 eβt0 e βt_||Dϕt H(x)vs||

which concludes that E− −→ E0_{. Using the domination existent between the}

sub-bundles we get ||Dϕt H(x)(vu+ vc)|| ||vu+ vc|| ≥ b||Dϕ t H(x)vu|| + ||DϕtH(x)vc|| ||vu|| + ||vc|| ≥ ≥ b min ||Dϕ t H(x)vu|| ||vu|| ,||Dϕ t H(x)vc|| ||vc||  ≥ b min(eβ00(t−t000)_{, e}β0(t−t00)₎||Dϕ t H(x)vs|| ||vs|| which leads to ||Dϕt H(x)(vu+ vc)|| ||vu+ vc|| ≥ be−βt0_.eβt||Dϕ t H(x)vs|| ||vs|| with t0 = max(β 00_,β0₎ min(β00_,β0₎ max(t 00 0, t 0 0) and β = min(β 00_{, β}0_{) proving that E}− _−→ E+_{⊕ E}0_.

Let us fix x ∈ Λ and vu ∈ E+ unitary vector. We also consider the

space A = Rvu ⊕ E−0(x). Since dim(E−0) ≥ dim(E+) we have dim(A) > n

and therefore A ∩ Jx(A) 6= ∅ where Jx is the isomorphism introduced in the

symplectic geometry section, at the point x. That guarantees the existence of a nonzero vector v ∈ A such that w = J_x−1(v) ∈ A. Choose v to be unitary and C0 such that ||w|| < C0. We can write v = k1vu+ vsc and w = k2vu+ wsc

with k1, k2 ∈ R. Before we proceed, an important result is needed, related to

the angles between the subspaces of the splitting.

If a splitting TxM = E(x) ⊕ F (x) is (β, t0)-dominated then the angle between

the subspaces E(x) and F (x) is bounded away from 0. To see that we pick any unitary vectors u ∈ E(x) and z ∈ F (x). Then by direct application of the previous definition we have:

||Dϕt H(x)u||e β(t−t0)_{≤ ||Dϕ}t H(x)(z − u)|| + ||Dϕ t H(x)u|| and therefore, ||Dϕt H(x)||.||z − u|| ≥ m(Dϕ t H(x)) e β(t−t0)_{− 1}

By the compacity of M and ϕt

H ∈ C1, we get

||z − u|| ≥ m(Dϕt

H(x))(eβ(t−t0)− 1)||DϕtH(x)|| −1

> 0 This implies that ||u ⊕ z|| > b(||u|| + ||z||) for some scalar b > 0. Thus we get the following:

1 = ||v|| > b(||k1vu|| + ||vsc||) ⇐⇒ |k1| + ||vsc|| <

1 b and similarly to w. Therefore we get the that |k1|,

|k2|

C0 , ||vsc|| and

||wsc||

C0 are all

bounded from above by 1_b. Now we want to obtain bounds for:

ω(w, v) ≤ |ω(vsc, k2vu)| + |ω(wsc, k1vu)| + |ω(vsc, wsc)|

From the Dϕt_H-invariance and symplectic form properties we get:

|ω(vsc, k2vu)| ≤ C2 0 b ||Dϕ t H(x)vsc||.||DϕtH(x)vu||

With the hypothesis of (β, t0)-domination we get:

|ω(vsc, k2vu)| ≤ C0 b e −β(t−t0)_||Dϕt H(x)vu|| 2 .||vsc||

We obtain also something similar for the second term on the right and side that |ω(wsc, k1vu)| ≤ C_b0e−β(t−t0)||DϕtH(x)vu||

2

.||wsc||, and for the last term,

|ω(vsc, wsc)| ≤ e−β(2t−t0)||DϕtH(x)vu|| 2

Using the bounds for ||vsc|| and ||wsc|| and summing up we get:

ω(w, v) ≤ C0 b

2

(e−2βt+ 2e−βt).||Dϕt_H(x)vu||2

With ω(w, v) = v.v = 1 we obtain uniform expansion in E+_:

||Dϕt_H(x)vu|| ≥ eβ ∗_(t−t 0) where β∗ = β₂ and t0 = _ln(√√b 3C0) −β∗ + .

Since E+ _{dominates E}0 _{⊕ E}−_{, it dominates E}− _{and the angle between}

E+ and E0 is bounded away from zero. Applying the same reasoning to ||Dϕ−t_H (x)v−0|| we use the result about splitting the central space. Now fix

x ∈ Λ and for any unit vector vs ∈ E−(x) we can find unit vector wu ∈ E+(x)

such that C0 ≥ ω(DϕtH(x)vs, DϕtH(x)wu) ≥ γ||DϕtH(x)vs||.||DϕtH(x)wu|| and

||Dϕt H(x)vs|| ≤ C0 γ||Dϕt H(x)wu|| ≤ C0 γ e −β(t−t0)

In order to exclude periodic points we use the lemma below.

Lemma 2.4.4. Consider an invariant splitting TxM = E ⊕ F for H such that

the smallest Lyapunov exponent from E (λe(x, H)) is strictly larger than the

largest one of F (λf(x, H)). If x is periodic for ϕtH, the splitting is dominated

at x with β = λe(x, H) − λf(x, H) and t0 = ln(C)_β + T where T is the period of

x and C is a constant depending on H.

Proof. Suppose x is Oseledets regular and periodic with period T such that λe(x, H) − λf(x, H) > 0. Where λe = lim t→∞ 1 t log ||Dϕ t H(x)|Ex|| and λf = lim t→∞ 1 t log ||Dϕ t H(x)|Fx|| respectively. But since ϕT

H(x) = x we get an upper bound for φH(x, t)

φH(x, t) = ||Dϕt H(x)|Fx|| m(Dϕt H(x)|Ex) = ||Dϕ t−nT H (ϕ nT H (x)). Qn i=1Dϕ T H(x)|Fx|| m(Dϕt−nT_H (ϕnT H (x)). Qn i=1DϕTH(x)|Ex)

where n is determined as the smallest natural number such that (n + 1)T > t > nT . Then using elementary norm properties and defining the constant C = sup_{(T >s>0),x∈M} ||Dϕs(x)|Fx||

m(Dϕs_(x)|E

φH(x, t) ≤ C. ||DϕT H(x)|Fx|| n m(|DϕT H(x)|Ex) n = C enT λf(x,H) enT λe(x,H)

The last equality comes from noting that the subsequence going just through the multiples of T is constant

1 nT log ||Dϕ nT H |Fx|| = 1 T log ||Dϕ T H|Fx||

and by Oseledets theorem the limit converges on x giving that log ||DϕT

H|Fx|| =

eT λf(x,H)_{. The same argument works for the denominator.}

Finally we check that when considering non-dominated segments we can exclude periodic points.

Remark 2.4.5. We were able to obtain equality when separating the norms on the last computation because the invariant subspace for x is repeating itself after T iterations and for a linear invertible map A : X → X on a finite dimensional linear space X, ||An|| = ||A||n.

Using the last bound and choosing t0 such that C = e(λe(x,H)−λf(x,H))(t0−T )

one finally obtains

φH(x, t) ≤ Ce−(λe(x,H)−λf(x,H))nT ≤ e−(λe(x,H)−λf(x,H))(t−t0)

Therefore we get a (β, t0) dominated splitting with

 β = λe(x, H) − λf(x, H)

t0 = C_β + T

2.5

Poincar´

e Linear Transversal Flow

A way to go around the appearance of the Hamiltonian vector field on the splitting is by considering the linear transversal Poincar´e flow. For that we first split the tangent bundle as TxM = RXH(x) ⊕ Nx where Nx is the orthogonal

complement of RXH(x) in TxM . Then we can see the map acting on this

splitting as follows,

Dϕt_H : TRM −→ TRM that makes (x, v) −→ (ϕtH(x), DϕtH(x)v)

In this work we will also consider only the regular non critical points of the Hamiltonian. By the density of Morse functions on the set of possible Hamiltonians and the finiteness of critical points on those functions we will also be sure to be excluding a zero measure set for our purposes. The map Dϕt_H : TRM −→ TRM is not guaranteed to be NR invariant because there is

not established preservation of orthonormality. So we use the quotient space NR= TRM/RXH(R), that is, we ignore the vector field direction component

and define the Poincar´e linear flow as the unique map Pt

H in NR verifying

ψ ◦ P_Ht = Dϕt_H ◦ ψ where ψ is an isometry between NR and NR.

Finally, we restrict this map to a energy level set to obtain the Poincar´e transversal linear flow, which will be a symplectomorphism for the form in-duced on the subbundle whose fibers are the intersections between normal fibers and the fibers of the referred energy level. We will use Φt

H for the

Poincar´e transversal linear flow associated with the Hamiltonian H. This is a technique particular to the Hamiltonian setting since for other, more general symplectic diffeomorphisms, we do not have additional information on any center manifold directions.

Remark 2.5.1. For Πx the orthogonal projection on the tangent bundle (x

fiber) then Ψt_H(x)v = Πϕt

H(x)◦ Dϕ

t H(x)v.

The use of the transversal linear Poincar´e flow is only relevant if we can tie its behaviour to that of the original dynamics. First we see that their Lyapunov exponents are actually the same.

Theorem 2.5.2. Given an Oseledets regular point x, its Lyapunov exponents related to the Φt

H-invariant decomposition are equal to the ones related to the

Dϕt

H-invariant decomposition.

Proof. Consider the Oseledets splitting for Dϕt

H with exponents λj(H, x)

asso-ciated to the subspaces E_xj. Now consider the projections of each of the spaces on the normal part to RXH(x), that is, Nxj = Πx(Exj). Let vj ∈ Exj, so the

vectors of Nj

x are of the form nj = aXH(x) + vj for some a ∈ R.

lim t→±∞ 1 t log ||Φ t H(x)nj|| = lim t→±∞ 1 t log ||ΠϕtH(x)Dϕ t H(x)vj|| = lim t→±∞ 1 t log |sin(θ j t)|.||DϕtH(x)vj|| = = lim t→±∞ 1 t log |sin(θ j t)| + lim_t→±∞ 1 t log ||Dϕ t H(x)vj||

where we used sin(θ_tj) =

||Π ϕt_H(x)Dϕ t H(x)vj|| ||Dϕt H(x)vj|| with θ j

t being the angle between the

subspaces XH(ϕtH(x)) and E j ϕt

H(x)

. By the subexponential evolution property of the angles θ_tj we finally obtain that

lim t→±∞ 1 t log ||Φ t H(x)nj|| = lim t→±∞ 1 t log ||Dϕ t H(x)vj|| = λj(H, x)

With this result we can even obtain equivalence of hyperbolicity, Theorem 2.5.3. Let Λ be a ϕt

H-invariant and compact set. Then Λ is partially

hyperbolic for ϕt

Proof. As stated in the 2.5.2 theorem we have for each pair of subspaces Nj and Ej ||Φ±t_H(x)nj|| = | sin(θj±t)|.||Dϕ ±t H(x)vj|| ||Φ±t_H (x)nj|| ||nj|| = ||Dϕ ±t H(x)vj|| ||vj|| .hj(±t, x) with hj_{(±t, x) =} ||vj|| ||nj||.| sin(θ j

±t)|. Using the fact that ||vj||

||nj|| =

1

cos(π₂−θj₀) and

theorem 2.3.1 we establish that hj_{(0, x) = 1 and}

lim t→±∞ 1 t log(h j_{(t, x)) = lim} t→±∞ 1 t log( 1 hj_{(t, x)}) = 0

This implies that if one of the splittings has uniform exponential expan-sion or contraction in some direction then so has the other in the associated direction.

Finally, the following result gives a complete picture of the relations be-tween dominated splittings, partial hyperbolicity and extensions of dominated splittings both on the original dynamics and on the transversal ones.

Theorem 2.5.4. Let H ∈ C2_{(M, R) and a regular energy surface S. If Λ ⊂ S}

has a dominated splitting for Φt

H, then Λ is partially hyperbolic.

Proof. First we check that if Λ has a dominated splitting for Φt

H then the same

happens for Λ. Consider a sequence xn → x ∈ S. Due to local compactness

we have subsequences xnj such that each subspace of the splitting E

i

x_nj → E i x

with the same dimension. Then TxS = E 1 x ⊕ E

2

x because the subpaces that

compose a dominated splitting cannot merge as their angle is always bounded away from zero.

We also have sequences Ei ϕn H(xnj)→ E i ϕn H(x) = Dϕ n H(E i

x) and due to uniform

continuity of the normal projection: Ni ϕn H(xnj) = Πϕ n H(xnj)(E i ϕn H(xnj)) → Πϕ n H(x)(E i ϕn H(x)) = Πϕ n H(x)Dϕ n H(E i x) which is equal to Ni_ϕn

H(x) and so we get for y = ϕ

n H(x), n ∈ Z ||Φm y |N 2 y|| m(Φm y |N 1 y) ≤ 1 2

which proves that N2_y ⊕ N1_y is a dominated splitting and by uniqueness is equal to N2 ⊕ N1_{. In this derivation we used that the splitting for Dϕ}t

H is

also dominated. That comes from the 2.4.3 theorem. Then we have for the set Λ that

Where we also used the 2.5.3 theorem. It only remains to justify that Φt H

is indeed a symplectomorphism. When constructing the Poincar´e transversal linear map we are considering only the space tangent to some energy level (and therefore excluding the gradient direction) and ignoring the vector component associated to the direction of the vector field XH. Considering unitary vectors

of these directions we get

1 = ||∇H||2 = DH(∇H) = ω(XH, ∇H)

meaning that the directions XH and ∇H span a symplectic 2-dimensional

plane. So Φt_H is acting in a 2N − 2 dimensional space preserving its symplectic structure because it is a restriction of the symplectomorphism Dϕt

H excluding

a symplectic subspace.

Theorem 2.5.1. Let H ∈ C3_{(M, R) and a regular energy surface S. If Λ ⊂ S}

has a dominated splitting for Φt_H, then µS(Λ) = 0 or S is Anosov or it has a

partially hyperbolic splitting with center subspace of dimension at least 2. Proof. By the extension of a dominated splitting theorem if Λ ⊂ S has a dominated splitting for Φt

H then Λ is partially hyperbolic, which means it

either has center subspace of dimension at least 2 or it is hyperbolic. If it is hyperbolic then by the theorem about the properties of hyperbolic sets we either have that µS(Λ) = 0 or Λ = S (i.e. Anosov case).

Theorem 2.5.2. The following is also true for the splitting E−(x) ⊕ E+_(x)

and φE−_,E+_,H(x, t).

• If the splitting E−0_{(x) ⊕ E}+_{(x) is continuous then φ}

E−0_,E+_,H(x, t) is C0

in the space variables for all t.

• If the same splitting is (β, t0)-dominated then

φE−0_,E+_,H(x, t + τ ) < φ_E−0_,E+_,H(ϕτ_H(x), t)

for τ > t0.

Proof. First we prove the continuity of φ. Making use of the triangle inequality for norms we get

|(||Dϕt

H(x)|E(x)|| − ||DϕtH(y)|E(y)||)| ≤ ||DϕtH(x)|E(x) − DϕtH(y)|E(y)||

This norm is a supremum over v ∈ E. Let Px : E → Ex be the projection

on the x fiber. With this notation we have that the right hand side of the previous equation equals ||Dϕt

H(x)P(x) − DϕtH(y)P(y)||. It is now easy to see

that h(x) = Dϕt_{H(x)P(x) is continuous since ϕ}t_H(x) is C1 and the projection is continuous by the hypothesis on the splitting. Since φ is simply the product of two norms (the other being the inverse of the co-norm), the argument extends

to the whole function. The compacity of M guarantees the norms do not ”explode” with x and so the first claim is proved. Using the same strategy as in the proof of the equivalence of dominantion concepts, we get the inequality φ(x, t + τ ) ≤ φ(ϕτ

H(x), t)φ(x, τ ). Now, if the splitting is (β, t0)-dominated and

we choose τ > t0 we get φ(x, τ ) ≤ e−β(τ −t0) < 1 and therefore φ(x, t + τ ) <

φ(ϕτ

Chapter 3

Exploring and Using Flexibility

To be able to assert what we understand as symplectic perturbation, we es-tablish the definition of symplectic neighborhoods.

Definition 3.0.1. For  > 0, the -basic neighborhood V(id, ) of the identity in Sympl1_ω(M ), is the set of all h ∈ Sympl_ω1(M ) such that for each x ∈ M

(1) h(x) ∈ B(x)

(2) ||Dhx− I|| ≤ 

For a general symplectomorfism, its -basic neighborhood is defined as V(f, ) = {g ∈ Sympl1

ω(M ) : f

−1_{◦ g ∈ V(id, ) ∨ g ◦ f}−1 _{∈ V(id, )}}

Then we can also define flexibility using the concept of split-sequence as in [2]. Note that here we are extending to a continuous setting since our split-sequence is indexed to real t. However since we only perturb at discrete iterates, the original concept is sufficient.

Definition 3.0.2. Let  > 0 and κ > 0. We say that the split-sequence {At_{(x), E}

x, Fx} of length T is (, κ)-flexible if for all η > 0 there is a bounded

open set in R2n _{containing the origin such that we can find H ∈ C}s_{(M, R)}

(with s > 2)  − C2-close to H verifying:

(1) H = H outside the T − length flowbox at x associated with H (2) DXH(y) = DXH(y) in the hypersurface edges of the flowbox

(3)||D(A−ti _{◦ ϕ}ti

H) − Id|| <  uniformly for some 0 ≤ ti ≤ T and i =

1, ..., k(T )

(4) There exists G ⊂ U such that µ(G) > (1 − κ)µ(U ) and ∠(DϕTH(x).Ex, Dϕ

T

H(x)FϕT

H(x)) < η, x ∈ G

Remark 3.0.1. The use of k(T ) is a generalization since we could simply chose ti = i and k(T ) = T .

Definition 3.0.3. Given a Hamiltonian H and natural numbers 1 ≤ p ≤ d − 1 and real numbers β, t0 we define the following invariant sets under the flow of

H:

• O(H) as the set of Oseledets regular points, that is, the full measure set of points for which the propositions of the Oseledets theorem are valid • Dp(H, β, t0) as the set of points x for which there is a (β, t0)-dominated

splitting of index p along the orbit of x.

• Γp(H, β, t0) = M |Dp(H, β, t0) as the set with lack of dominance until

some time tt0. • Γ0_p(H, β, t0) = {x ∈ Γp(H, β, t0) ∩ O(H) : λp(H, x) > λp+1(H, x)} • Γ00_p(H, β, t0) = {x ∈ Γ 0 p(H, β, t0) : x not periodic } • Γp(H, ∞) = ∩t0,β∈R+Γp(H, β, t0) • Γ0_p(H, ∞) = ∩t0,β∈R+Γ 0 p(H, β, t0)

In order for the measure from transversal sections to make sense on the flowbox framework we show how to relate such measures. The impact of these definitions will become more apparent on the chapter 6 about globalization results.

3.1

Measure on Transversal Sections

Theorem 3.1.1. For all , t > 0 there exists r > 0 such that for any measur-able set A ⊂ B(x, r) = {(u1, ..., ud−1, v1, ..., vd−1, w) ∈ R2d−1 :

q Pd−1 i=1 u2i + v2i < r, |w| < r} we have |µ(A) − α(t)µ(Pt H(x)A)| <  where α(t) = ||XH(ϕtH(x))|| ||XH(x)|| and Pt H(x) = ϕ τ (x,t)

H with τ (x, t) being the first time the flow hits the normal section

at ϕt_H(x).

Proof. Define µ as the measure associated to the volume form ωn(v1, ..., v2n−1)(x) = ωn(

XH(x)

||XH(x)||

, v1, ..., v2n−1)(x)

where ωn _{is the invariant volume form associated to the original measure µ.}

We can find a relation between the standard Poincar´e map and the linear transversal Poincar´e map and we see how the pullback of this map acts on the volume form ωn_:

Pt H(x)

∗

= ωn( XH(ϕ t H(x)) ||XH(ϕtH(x))|| , P_Ht(x)v1, ..., PHt(x)v2n−1)(ϕtH(x)) = = ωn(Dϕt_H(x) XH(x) ||XH(ϕtH(x))|| , P_Ht(x)v1, ..., PHt(x)v2n−1)(ϕtH(x)) = = ω(Dϕt_H(x) XH(x) ||XH(ϕtH(x))|| , Dϕt_H(x)∇H(x))∧...∧ω(P_Ht(x)vn−1, PHt(x)v2n−1) = = ω( XH(x) ||XH(ϕtH(x))|| , ∇H(x)) ∧ ... ∧ ω(vn−1, v2n−1) = = ωn( XH(x) ||XH(x)|| . ||XH(x)|| ||XH(ϕtH(x))|| , v1, ...v2n−1) = = 1 α(t).ω n₍ XH(x) ||XH(x)|| , v1, ...v2n−1) = 1 α(t).ω n_.

Note that sometimes we omit the point (ϕt_H(x)) when it is clear from context.

As we can see the Poincar´e standard map does not preserve measure on normal sections but we have a estimation for how it changes it. It is immediate that α(t)det(P_Ht(x)) = 1 . Using Pt H(y) = ϕ t H(x) + P t H(x).y + R 2_{(y) (3.1.1)}

where R2 _{is the rest with terms of order larger then 2 on y, it is possible to}

choose r > 0 so small such that for y ∈ B(x, r) we have det(DPt

H(y)) -close

to det(P_Ht(x)). With that, a bound as the following is obtained

|µ(A) − α(t)µ(Pt H(x)A)| ≤ |α(t)µ(P t H(x)A) − α(t)µ(P t H(x)A)|+ +|α(t)µ(P_Ht(x)A) − µ(A)|

For the first element of the right hand side |α(t)µ(P_Ht (x)A)−α(t)µ(P_Ht(x)A)| = |

Z

A

(det(DP_Ht (y))−det(P_Ht(x))dµ(y))| < 

The second element vanishes as a direct consequence of det(Pt

H(x)) = α(t),

resulting in our final claim.

3.2

Proof of the Main Theorem

A sequence of steps relating dominance with flexibility and residuality are nec-essary to conclude theorem 1.1. and can be resumed as proving the following:

1. that lack of a dominated splitting along an orbit segment can only present itself in the form of 4 different types.

2. that for each of the 4 types, the split sequence DϕH(ϕi(x)) acting on the

zipped Oseledets splitting E+_{⊕ E}−0 _{along the associated non dominated}

orbit is (, κ)-flexible. The global constants  and κ are set from the α set on the separation of the four non-domination cases. In sum with the previous item, non-dominance implies flexibility. This follows from the constructions of perturbations that support flexible sequences on section 4.

3. that if an orbit supports a flexible sequence we can find an Hamiltonian H, C2-close to H such that LE(H, M ) is strictly smaller than the original exponent. The first result present is the local decay of the exponent. To prove this item we must follow section 6 where it is globalized.

4. the set of Hamiltonians H for which we cannot cause such Lyapunov ex-ponent decay (e.g. due to existence of a dominated splitting) is residual. where LE(H, Λ) = lim t→∞ Z Λ log(||Φt_H(x)||)dµ(x)

We start by proving item (4) using the previous items and see how such result allows to conclude theorem 1.1.

Lemma 3.2.1. The functions LE(H, Λ) for an invariant set Λ are upper semi-continuous.

Proof. The LE(H, Λ) were determined to be an infimum over of continuous functions and therefore are upper semi-continuous.

Then by the previous lemma, since the integrated Lyapunov exponent, as a function of the Hamiltonian, is upper semicontinuous then the set of its continuity points is a residual set. So we select said residual set

R1 = {H ∈ C2(M, R) : lim inf H0→H

LE(H0, Λ) = LE(H, Λ)}

Using item 2 in conjunction with item 3 we see that the cases where we can cause the decay of the integrated Lyapunov exponent happen when there is no dominance since when proving those items we get

non-dominance =⇒ flexibility =⇒ Oseledets directions exchange . Then, excluding the trivial splitting case, continuity implies the existence of a dominated splitting and so there is a residual set of Hamiltonians such that their zipped oseledets splittings are either dominated or trivial. Actually

this set is constituted by partially hyperbolic splittings with central mani-fold of dimension at least 2 since in the symplectic and Hamiltonian settings, dominated and partially hyperbolic splittings are equivalent.

Consider the product set M = M × C2_{(M, R) endowed with the standard}

product topology. Define the following set where Sp(H) represents an energy

surface that passes through p:

A = {(p, H) ∈ M : Sp(H) is a regular Anosov surface } (3.2.1)

Then A is open by structural stability of Anosov systems.

We can also pick a residual set R2 formed by surfaces without hyperbolic

sets of positive measure due to the general result for hyperbolic sets on chapter 2.

Now while R1 is a set of Hamiltonians, residual in the C2(M, R), R2 is a

set of surfaces. So for R1 we use the decomposition:

A = ∪H∈R1nH × {H}

where nH is the residual subset of M with the property that for H ∈ R1∪ A

and p ∈ nH, the energy surface of H that goes through p either is Anosov or for

each x on that surface either all Lyapunov exponents are null (LE(H, x) = 0) or H has a partially hyperbolic splitting.

Note that since A is open by structural stability of Anosov diffeomorphisms than R2∪ A is again residual and the main theorem is proved.

Finally we just need to prove items 1 through 3. Item 1 is extensively proved in [2] and is a result from symplectic algebra, independent from the particular setting. We now prove item 3 and leave item 2 for the next chapters where we effectively build the flexible sequences.

The theorem we want to prove for item 3 is:

Theorem 3.2.1. Let H ∈ C2_{(M, R) with aperiodic flow and V a C}2_{-neighborhood}

of H and δ > 0, 0 < κ < 1. For t0 sufficiently large and β > 0 there exists a

measurable function T : Γp(H, β, t0) → R+ with the following properties:

For almost every x ∈ Γp(H, β, t0) and every t ≥ T (x) there exists r =

r(x, t) > 0 such that the following holds

• The flowbox with base Br(x) is non self-intersecting until time t0;

• for any 0 < r0 < r there exists an Hamiltonian H ∈ V such that H = H outside of the T (x)-length flowbox with DX_H(y) = DXH(y) in the

hypersurface edges of the flowbox. • There is a set G ⊂ B_r0

(x) such that µ(G) > (1 − κ)µ(Br0(x)) and

1

t log(|| ∧

p

Φt_H(y)||) ≤ λp−1(H, x) + λp+1(H, x) + J (H, x) + δ

for all y ∈ G and J (H, x) = λp(H,x)−λp+1(H,x)

where λp(H, x) is the Lyapunov exponent on the subset of the splitting of

index p and λp(H, x) =Pp

i=0λi(H, x)

Proof. Since the sequence ΦH(ϕtH(x)) : E+⊕ E

−0 _{is (, κ)-flexible we pick} κ 2

instead of κ and  > 0 sufficiently small depending on the choice of charts (note that we can bound all possible charts). Now we separate the case where λp(H, x) = λp+1(H, x) and the case where λp(H, x) > λp+1(H, x).

For the regular non-periodic points in the first case the conclusion is trivial since we take T (x) large so that if t > T (x) then 1_tlog(|| ∧p ΦH(x)||) is δ₂

close to λp_{(H, x) since J (H, x) = 0 and λ}p−1_{(H, x) + λ}

p+1(H, x) = λp(H, x).

Then for each t ≥ T (x), taking r = r(x, n) small so that the flowbox is not self-intersecting and Φt_H(y) is close to Φt_H(x) for all y ∈ Br(x). Then it would

suffice to choose H = H

Now consider the set Γ of points from Γp(H, β, t0) that are also

non-periodic, Oseledets regular and such that λp(H, x) > λp+1(H, x). We assume

that µ(Γ) > 0 otherwise there is nothing to prove. If A are the points for which the lack of domination holds then Γ = ∪tϕtH(A). Fix C > supH∈V,x∈M||ΦH±1(x)||

and recall sublemma 6.2 from [2] switching Df for ΦH. Let x ∈ Γ be fixed,

t ≥ T (x) and some order s = s(x, t) as in the referred sublemma. Then we define the point y = ϕs

H(x).

We now discretize this process so that sublemma 6.2 is directly applicable to our setting. We also recall from the preliminaires that the assymptotic re-sults for Φt

H on the normal sub-bundles are interchangeable with the original

dynamics Dϕt_H. The sublemma states that if we can find a sequence of sym-plectic linear maps Li : Tϕi

H(z) → Tϕ i

H(z) such that for sufficiently small γ > 0

we have

∠(L[t0]....L0.E

+_{(z), E}−0

) < γ then we will get

1 t log(|| ∧ p_[Φt−s−t0 H (ϕ s+t0 H (x))L[t0]....L0Φ s H(x)]||) < < λp−1(H, x) + λp+1(H, x) + J (H, x) + δ 2

Since this decay must hold not only for x but for the majority (depending on κ) of points nearby we use the same argument from the paper this sublemma originates:

We select points z0, ..., z[t] that are on the ϕH orbit from 0 to t excluding

the segment [s, s + t0− 1] and such that d(zi, ϕiH(x)) < γ for all i.

Then with aid of the chosen charts φ we define iy_x : TyM → TxM with iyx = [Dφy(x)]

−1

we are able to relate the mappings between different points and for a new collection of maps Li and ||L

±1 i || ≤ C such that 1 t log(|| ∧ p [Φt−s−t0 H (zs+t0)L[t0]....L0Φ s H(z0)]||) < < λp−1(H, x) + λp+1(H, x) + J (H, x) + δ

Since y ∈ A because we return to a point where the lack of dominance condition holds, the sequence is (, κ)−flexible.

By the definition of flexibility there exists an Hamiltonian H and a subset of U , G such that

• H = H outside of the H−flowbox

• X_H = XH on the hypersurface edges of the flowbox

• µ(G) > (1 − κ)µ(U ) • ∠(Φt_(g).iy g.E+(y), i ϕt H(y) ϕ Ht(g).E −0_(ϕt0 H(y))) ≤ γ for all g ∈ G. Now let G = ϕ−s_H (G) ⊂ B_r0

(x). For any h ∈ G, if we define hi = ϕi_H(h) for

0 ≤ i ≤ [t] and L = Φ_H(hi+[s]) for 0 ≤ i < [t0] and we get the desired decay.

3.2.1

How to Use the Flexible Sequences

The angle control and concatenation of iterative steps constitutes the exten-sion of realizability to flexibility. We can think of a flexible sequence as the concatenation of realizable maps such that their combined action exchanges the Oseledets directions.

If a split-sequence is (, κ)-flexible we can find -close perturbations that send tangent vectors between invariant directions for (1 − κ) percent of points of an open set. The proof of that in the symplectomorphic setting is construc-tive but requires the division of the non-dominance property in four cases, qualitatively dependent on the Hamiltonian dynamics DϕH.

The notion of flexibility or its closely related realization is what determines if a perturbation is good. The decay on the Lyapunov exponent is possible when we can build a perturbation that verifies the flexibility properties. In that case, a few other conclusions are semi-direct:

• A flexible sequence conjugated by linear symplectic maps is again a flex-ible sequence (although with different parameters).

Proposition 3.2.2. Consider the sequence of symplectic linear maps h0, ...hn on the Euclidean space R2d and let fi and gi be same length split

sequences acting on splittings E1

i⊕Ei2 and Fi1⊕Fi2 respectively. If h0, ...hn

is such that hi(E

0

i) = F

0

i and hi+1(fi) = gi(hi) then the split sequence fi

being (, κ)-flexible implies that gi is (((maxi(||hi||)) 2

), κ)-flexible. This result allows to prove flexibility in an Euclidean space using canoni-cal symplectic coordinates and extrapolate it to the tangent space where the original dynamics act, provided the symplectic maps described above exist.

• If a subsequence is flexible for some values of κ and  then so are the sequences that contain it.

Proposition 3.2.3. Let fi be a split sequence of length n acting on

E1

i ⊕ Ei2. If the shorter split sequence of fi with i from 0 < n0 < n to

n0 < n1 < n is (, κ)-flexible, then so is the full n length fi split sequence,

for the same (, κ).

Due to this results and while building the flexible sequences for each case, that conclude item 2, we only need to focus on certain subsequences and we can construct those sequences on rectified cylinders by making use of the symplectic flowbox theorem.

Chapter 4

Non-Dominance: Construction

of the Perturbations for each

Case

4.1

Hamiltonian Setting

In this chapter we go through the construction of Hamiltonian perturbations and how to approach the problem for the different manifestations of non-domination. This subdivision in types of non-domination is key to find a solution.

Recall that our perturbation must not only be a symplectomorphism but also correspond to the time-1 map of an Hamiltonian flow for some Hamiltonian close to the original. Therefore the perturbation space is restricted to the group Hamiltonian symplectomorphisms Ham(M, ω), a normal subgroup of Symp(M, ω).

The goal remains similar but with a slightly adjusted definition of flexibility where the Hamiltonian function is the central object to be perturbed instead of directly on the mapping. The flexibility concept will be introduced in the next chapter, when we prove that these chapter’s constructions follow such property.

We make some simplifications in the multidimensional case, the major one being the use of the linear transverse Poincar´e flow so that we can straighten out the orbits of the flow along the time direction using the symplectic flowbox theorem from [3] that we already stated. The complete proof can be found on [3] since it is applicable to any even dimensional symplectic manifold but we present here a sketch of that proof for easier reference and ilustration of the method:

1. Fix an energy level for H and an energy level for a function G such that their vector fields are transversal and ω(XH, XG) 6= 0. This gives the

possibility of a tangent space splitting like TxM = TxSe,G⊕ RXH(x) ⊕

RXG(x). Se,G is the set H−1(e) ∪ G−1(0) ∪ U and it is easy to check that

the form ω restricted to Se,G is symplectic.

2. By the Darboux theorem we change the previous restricted form to the canonical one in R2n−2 _{and extend it to R}2n _{using the flows generated}

by XH and XG. This results in the following symplectomorphism

g(x) = (−τ (x), h1◦ ϕ e−H(x) G ◦ ϕ τ (x) H (x), H(x), h2◦ ϕ e−H(x) G ◦ ϕ τ (x) H (x))

where τ (x) is the time it takes x to reach G−1(0) through the flow of H and (h1, h2) = h the symplectomorphism give by the Darboux theorem.

In order for the concept of flexibility to be preserved when switching be-tween the original coordinates and the ones straightened with the symplectic flowbox theorem we must assure that a flexible split sequence built on the new coordinates will have an associated flexible split sequence with the origi-nal ones. For that we refer to the same article [3] with a brief resume of the process utilized:

Since concatenation maintains the flexibility of the sequence, we only need to care about two types of flexible sequences: the trivial ones consisting of only terms such as ΦH; the rotational terms like φH◦ R where R is some symplectic

rotation.

In either case, using the transformation g such that H = g ◦ H0 and the

Hamiltonian H0 obtained from the definition of flexibility applied to H0

ac-companied with the respective G0 ⊂ U0 we define the origin centered map

gx = g − g(x) and get that by H0, (0, κ0)-flexibility that

• µ(G0) ≥ (1 − κ)µ(U0) =⇒ µ(G) ≥ (1 − κ)µ(U ) with G = gx(G0)

• For H = H0◦ gx and the boundedness of the derivatives of g we get

||H − H||_C2 = ||(H₀ − H₀) ◦ g_x|| ≤ ₀× const ≤ 

• The last part, proving that the rotation between spaces in the original coordinates will be the correct one requires a different approach from [3] and consists in selecting specific choices for the mapping g as outlined for each non-dominance type later in this chapter.

4.2

Construction Restrictions

4.2.1

Rotation Size

Consider a map f : M → M which we want to perturb by composing with some map h so that we get the perturbation g = f ◦ h. Now imagine that

the goal of the perturbation is to change some assymptotic result from an orbit of f . We choose that assymptotic value to be the Lyapunov exponent for the point x on that orbit, where x is assumed to be Oseledets regular. Note that by Oseledets theorem, the set of Oseledets regular points has full measure for any f -invariant measure. Furthermore consider the family of R2 maps F

U = {hα, α ∈ [0,π₂]} such that ||hα(x) − x|| <  if x ∈ U and Dhα = Rα

where

Rα =

cos α − sin α sin α cos α



Notice that by defining h in R2 we are making a simplification to the euclidian manifold. However we have already shown that it suffices to consider such maps and we will actually continue this construction by assuming f : R2 → R2. For a nontrivial splitting with an upper Lyapunov exponent positive we have TxM = Ex⊕ Fx such that:

lim n→±∞ 1 nlog(||Df n_(x)|E x||) = λ+(f, x) > λ−(f, x) = lim n→±∞ 1 n log(||Df n_(x)|F x||)

Let m be large enough to be determined later and β = ∠(Efm_(x), F_fm_(x)). Then

we define the perturbation map g in the following way g = f ◦ hβ in U

f in M ∩ Vc

where U is a small enough neighborhood of fm(x) and V such that U ⊂ V . This example goes to show how we build perturbations that decay the Lyapunov exponent and simultaneously one of its limitations. When we chose hβ the value of β is determined by the angle between the subspaces of the

splitting at some point of the orbit. If that angle does no get arbitrarily close to zero then we can’t choose a β that guarantees the C1_{-closeness we seek. For}

instance

||g − f ||C1 ≥ ||Dg − Df || = ||Df (Dh_β − I)|| ≥ m(Df )||Dh_β− I||

We can find a lower bound for this by choosing a canonical direction say v = [1, 0] and get ||g −f ||C1 ≥ C(1−cosβ) which can not be made infinitesimal

unless the subspaces of the splitting that determines β get infinitesimally close in the projective space, which does not generally happens on all the cases where we need to build these kind of perturbations.

The idea of close rotational proximity can still be used if there are natural conditions (related to our dynamics) that can be used to perturbatively create that proximity. This idea consists in three steps:

• Change an invariant direction d1by a small angle using some Hamiltonian

perturbation around a single point, obtaining a new direction d2 not ϕH