The dynamics of conformal Hamiltonian flows: dissipativity and conservativity

We study in detail the dynamics of conformal Hamiltonian flows that are defined on a conformal symplectic manifold (this notion was popularized by Vaisman in 1976). We show that they exhibit some conservative and dissipative behaviours. We also build many examples of various dynamics that show simultaneously their difference and resemblance with the contact and symplectic case.


Introduction
Symplectic dynamics models many conservative movements.Yet, other phenomena are dissipative and require another setting.This is the case of the damped mechanical systems: they are modelled by conformal Hamiltonian dynamics, which alter the symplectic form up to a scaling factor.
This notion of conformal symplectic dynamics can be placed in a broader context.To define such a dynamics, we only need to know in charts an equivalence class of 2-forms for the relation where ω 1 " ω 2 if ω 1 " f ω 2 for some non-vanishing function f .A manifold endowed with such an equivalence class of local 2-forms, one of them being closed, is called a conformal symplectic manifold, a notion popularized by Vaisman in [12].An equivalent notion is the notion of conformal structure pM, η, ωq, a manifold M endowed with a 1-form called the Lee form and a 2-form called the conformal form, whose precise definition is given in section 2.1.A proof of the equivalence of the two notions is given in [2].
We will study autonoumous conformal Hamiltonian flows (CHF in short) pϕ s q sPR of compact manifolds, see definition in Section 2.2.They alter the conformal form up to a non-constant scaling factor.As the volume ω n can increase or decrease at different points of the manifold under the action of the dynamics, we can expect different behaviours, some of them being conservative e.g.completely elliptic periodic orbits, invariant foliations with compact leaves and some other being dissipative, e.g.attractors or repulsors.
A precise definition of what we call conservative or dissipative requires the introduction of a notion related to the shape of the orbits.The winding of a point x P M through time is defined as the map t Þ Ñ r t pxq (r stands for "rotation"), r t pxq :" ż t 0 ηpB s ϕ s pxqqds, @t P R.
Then ϕ t ω " e rt ω, see Lemma 1, and a point x P M is ‚ either (positively) dissipative when lim tÑ`8 |r t pxq| " `8; ‚ or (positively) conservative.Our main result, Proposition 7, asserts that for every CHF pϕ H t q, if D `is the set of positively dissipative points and if C `the set of positively conservative points, then up to a set of zero volume, C `coincides with the set of positively recurrent points, and then D `with the set of positively non-recurrent points.Also, the ω-limit set ωpxq of every x P D `is contained in tH " 0u.Some examples of conservative and dissipative points are ‚ every attractor intersects tH " 0u, has non-trivial homology and almost every point in its basin of attraction that doesn't belong to the attractor is in D `, Corollary 9; ‚ if x is a periodic points that is not a critical point of H, then when x P C `, the first return map to a Poincaré section preserves a closed 2-form and a foliation into (local) hypersurfaces; when x P D `, then Hpxq " 0 and the first return map to a Poincaré section alters a certain closed 2-form up to a constant factor that is different from 1; ‚ every fixed point of the flow is conservative.We will provide also an example of wild conservative points: points that are recurrent, in tH ‰ 0u but whose ω-limit set intersects tH " 0u, see Section 3.5.Hence these points satisfy lim inf tÑ`8 |r t pxq| " `8.The origin of most of our examples is contact geometry.In particular, in Section 2.4, we introduce a notion of twisted conformal symplectization that is crucial to the elaboration of examples and counter-examples.In a similar way, switching H to ´H, the set C ´of negatively conservative points is the set of x P M such that lim tÑ´8 |r t pxq| " `8 and D ´" M zC ´is the set of negatively dissipative points.We prove in Proposition 10 that C ´and C `are always equal up to a set of zero volume.It is not true that for a general flow, the set of positively recurrent points is equal to the set of negatively recurrent points up to a set of volume zero.
A CHF pϕ t q is (positively) conservative when C `" M and dissipative when C has zero volume.We highlight a strong relation between the topology of tH " 0u and the property of being convervative: when tH " 0u has a neighbourhood V such that for every loop γ : T Ñ V , ş γ η " 0, then pϕ H t q is conservative, Section 4.2.This contains the case when H doesn't vanish, Section 4.1.But there exist some examples of conservative CHF that are not in this case, Section 4.5.As the nonvanishing property is open in C 0 -topology, we obtain C 0 -open sets Hamiltonians H such that the associated CHF flows are conservative.
Among the conservative CHF, the Lee flows are those that correspond to the Hamiltonian H " 1 for some choice of representative pη, ωq of the conformally symplectic structure.They are an extension of the Reeb flows in the contact setting.We will provide in every dimension examples of Lee flows ‚ that are transitive, Section 4.3; this is different from the Hamiltonian symplectic case, where the level sets of H are preserved; ‚ that have no periodic orbits, Section 4.4; Weinstein conjecture in the contact setting and Arnol'd conjecture in the symplectic setting assert the existence of periodic orbits.This example emphasizes one difference between the CHF and the Reeb flows as well as the symplectic Hamiltonian flows.
We will give a 2-dimensional example of Lee flow that is minimal (Section 1.1.2),but we don't know if there is such an example in higher dimension.We will give in Part 5.2.1 of Section 5.2 an example of dissipative CHF, with one normally hyperbolic attractor that is a Lagrangian submanifold, one normally hyperbolic repulsor that is also a Lagrangian submanifold and the remaining part of the manifold that is filled with heteroclinic connections.This gives a C 1 -open set of CHF that are dissipative.See also Section 1.1.1.
There also exist C 1 -open sets of CHF such that both C `and D `have positive volume.This happens when there is a normally hyperbolic periodic attractor and one non degenerate local minimum of e θ H where θ is a local primitive of η.
Another feature of the conformally symplectic dynamics is that they preserve isotropy (this is even a characterization of these dynamics).This is a common point with symplectic dynamics and contact dynamics.Therefore, we extend or amend some classical results for the invariant submanifolds of Hamiltonian flows.
In Section 2.5, we prove that the CHF have a codimension 1 invariant foliation and explain in Section 6.2 the relation for a submanifold between being tangent to this foliation, being invariant and being isotropic (or coisotropic).This is reminiscent of Hamilton-Jacobi equation in the usual Hamiltonian setting.We deduce that on a conformal cotangent bundle (see Section 2.3), a Lagrangian invariant graph is necessarily contained in the zero level set, which is a major difference with the usual Hamiltonian setting.Motivated by the result of Herman in the exact symplectic setting, [6], which asserts that every invariant torus on which the dynamics is C 1 -conjugate to a minimal rotation is isotropic, we consider tori T that are invariant by a CHF and such that the restricted dynamics is topologically conjugate to a rotation.In Section 6.2, we recall the definition of the asymptotic cycle of an invariant measure and introduce in a similar way the asymptotic cycle for flows on tori that are C 0 -conjugate to a non necessarily minimal rotation.We prove that if the product of the cohomology class of the Lee form by the asymptotic cycle of T is non-zero, then T is isotropic.In particular, when the cohomology class of the Lee form is rational and when the rotation is minimal, the invariant torus is isotropic.
1.1.1.A dissipative example.Let us discuss a simple two-dimensional dissipative example that illustrates some of our results.Let pM, η, ωq " pT 2 , dx, dx ^dyq where T 2 denotes the 2-torus R 2 {Z 2 and let H : T 2 Ñ R be the Hamiltonian function Hpx, yq " sinp2πyq.We have pictured integral curves of the associated dynamics on Figure 1.In this figure, we see that the only level set of H that is preserved is tH " 0u and that it has two connected components: an attractive circle and a repelling one.Such a picture can be drawn in any dimension: if the Lee form is not exact, there exist Hamiltonian flows with attractive or repelling hyperbolic orbits (cf.Proposition 24).Attractors (or repellers) are not necessarily contained in tH " 0u: lifting this dynamics on the cover R{Z ˆR{2Z, one could see any of the two cylinders bounding the two attracting circles as attractors.However, as we will see attractors always intersect tH " 0u.On Figure 1, we see that the attractor is winding in the x's direction.In general, the Lee form is not exact in any neighborhood of the intersection of an attractor with tH " 0u (cf.Corollary 9).In particular, an attractor cannot be finite and must intersect tH " 0u.
1.1.2.A conservative example.In the opposite direction, let us point out the existence of conformal Hamiltonian dynamics that preserve the symplectic form ω but the behavior of which nonetheless differs from the symplectic Hamiltonian case.As a simple 2-dimensional example, let us consider the 2-torus T 2 endowed with its canonical area form ω " dx ^dy once again.Let us fix a, b P R and choose the Lee form η :" adx `bdy.The Hamiltonian flow of H " 1, which is called the Lee flow associated to the representative of the conformally symplectic structure (the gauge) pη, ωq, is ϕ t px, yq " px `bt, y ´atq.If a and b are rationally independent, this flow is minimal.This is a striking difference with autonomous Hamiltonian flows of symplectic manifolds, where trajectories are never dense and there usually are plenty of periodic orbits.In general, we prove that there exist topologically transitive Lee flows in any dimension and that there exist Lee flows without periodic orbit in any dimension (cf.Propositions 15 and 16).In both cases, the Lee form η is not completely resonant (i.e. the set of its integrals along the loops is a dense subgroup of R), which is necessary in order to have dense trajectories.One could ask whether there always is a periodic orbit when η is completely resonant, but this question is harder than solving the Weinstein conjecture (i.e. the existence of a periodic orbit for any Reeb flow of a closed contact manifold).

‚ In section 2, we introduce the notions of conformal manifold and conformal
Hamiltonian dynamics and prove some of their properties, Then we give some examples: the conformal cotangent bundle, the twisted conformal symplectization, and describe the invariant foliation.‚ In section 3, we characterize the global conservative-dissipative decomposition of the dynamics in term of recurrence.We prove the almost everywhere coincidence of the behaviours in the past and in the future.We also prove that the boundedness of the winding number implies the existence of invariant measures.We also provide an example of orbits that are conservative and have a strange oscillating behaviour.‚ In section 4, we give some topological conditions on tH " 0u that imply that the dynamics is conservative.We gave some examples of such dynamics that are transitive and some others that have no periodic orbit.We give an example of a conservative dynamics for which the topological condition for tH " 0u is not satisfied.‚ In section 5, we begin by studying some ergodic measures whose support is dissipative.Then we give examples of dissipative dynamics with Lagrangian attractors and repulsors, and also examples with periodic attractors and repulsors.We also give sufficient conditions implying that some connected component of tH " 0u cannot be an atttractor.‚ In section 6, we give some condition that implies that a component of tH " 0u is in the closure of a non-compact leaf of the invariant distribution.Then we study invariant submanifolds from different points of view: their position relatively to the invariant foliation, and when they are rotational tori, the relations between their asymptotic cycle and their isotropy.‚ Finally, there is an appendix dealing with isotropic submanifolds.1.3.Acknowledgements.The authors are grateful to Ana Rechtman for listening some preliminary versions of this work, discussing them and pointing out the link with classical results on foliations.The first author was supported by the postdoctoral fellowship of the Fondation Sciences Mathématiques de Paris.η " 0 and d η`df α " e f d η pe ´f αq.If d η α " 0, one says that α is η-closed.
Given an even dimensional manifold M , a conformal symplectic structure is an equivalence class of couples pη, ωq where η is a closed 1-form of M and ω is a non-degenerate 2-form that is η-closed, two such couples pη i , ω i q, i P t1, 2u, being equivalent if there exists a map f : M Ñ R such that η 2 " η 1 `df and ω 2 " e f ω 1 .A conformal symplectic manifold is an even dimensional manifold M endowed with a conformal symplectic structure, we will often work with a specific representative pη, ωq and write pM, η, ωq the conformal symplectic manifold.A notion that does not depend on the specific choice of representative pη, ωq is called gauge invariant or well defined up to gauge equivalence.The closed 1-form η is called the Lee form of pM, η, ωq, its cohomology class rηs P H 1 pM ; Rq is gauge invariant.A conformal symplectomorphism ϕ : pM 1 , η 1 , ω 1 q Ñ pM 2 , η 2 , ω 2 q is a diffeomorphism ϕ : M 1 Ñ M 2 such that ϕ ˚η2 " η 1 `df and ϕ ˚ω2 " e f ω 1 for some f : M 1 Ñ R (this notion is gauge invariant).When dim M ě 4, the second equality implies the first one.
Similarly to the symplectic case, a submanifold N of a conformal symplectic manifold pM, η, ωq is called isotropic if T N Ă T N ω , coistropic if T N ω Ă T N and lagrangian if T N " T N ω (where E ω denotes the ω-orthogonal bundle of the bundle E), this notion is gauge invariant.
A symplectic manifold pM, ωq has a natural conformal symplectic structure p0, ωq (which is the same as p0, λωq for λ P R ˚); conversely, a conformal symplectic structure pη, ωq comes from a symplectic structure if and only if η is exact.

Hamiltonian dynamics.
Given a map H : M Ñ R defined on a conformal symplectic manifold pM, η, ωq, we define its associated Hamiltonian vector field X by ι X ω " d η H, conversely H is the Hamiltonian of X.When the cohomology class of η is not 0, then H is unique.This matching Hamiltonian-vector field does depend on the choice of representative pη, ωq but not the algebra of Hamiltonian vector fields: the previous vector field X is the same as the one induced by e f H for the Lee form η `df .One can extend this definition to time-dependent Hamiltonian maps but we will focus on autonomous Hamiltonian in this paper.When H " 1, the associated vector field L η is called the Lee vector field of η and its flow is called the Lee flow.
Let us assume that the vector field X associated with H is complete.Let us denote pϕ t q its flow and r H t pxq :" ż t 0 ηpX ˝ϕs pxqqds, @x P M, @t P R.
When the choice of H is clear, we set r t :" r H t .Lemma 1.Given a complete Hamiltonian flow pϕ t q on pM, η, ωq associated with a Hamiltonian H, for all t P R, ϕ t ω " e rt ω, ϕ t d η H " e rt d η H, H ˝ϕt " e rt H and ϕ t η " η `dr t .
In particular, the level set tH " 0u is invariant under the flow and 1 H ω is an invariant 2-form on tH ‰ 0u.
Injecting X in ι X ω " d η H, one gets dH ¨X " ηpXqH, which implies that hptq :" H ˝ϕt pxq, for a fixed x P M , satisfies h 1 ptq " ηpX ˝ϕt pxqqhptq and the third statement follows.Finally, the last statement is due to L X η " dpηpXqq.
We remark that the relations ϕ t ω " e rt ω and ϕ t η " η `dr t are also satisfied in the time-dependent setting.This indeed implies that conformal Hamiltonian diffeomorphisms are conformal symplectomorphisms.
2.3.Conformal cotangent bundles.Given a manifold L endowed with a closed 1-form β, one can define a conformal symplectic structure on T ˚L denoted T β L in the following way.Let π : T ˚L Ñ L be the cotangent bundle map and λ the associated Liouville form: λ pq,pq ¨ξ :" ppdπ ¨ξq.The conformal structure defining T β L is pη, ωq :" pπ ˚β, ´dη λq.The neighborhood of the 0-section of T β L is a model of a neighborhood of a Lagrangian embedding of L pulling back the Lee form to β (see Section A.2).
Let us recall how one can canonically extend diffeomorphisms and flows of M to conformal symplectomorphisms and Hamiltonian flows of T β M .Let f : M Ñ N be a diffeomorphism, one can symplectically extend it to f : T ˚M Ñ T ˚N by the well-known formula: f pq, pq " `f pqq, p ˝df ´1 q ˘, @pq, pq P T ˚M.Now if the diffeomorphism f : M Ñ N satisfies f ˚β " α `dr, for closed 1-forms α, β and some map r : M Ñ R, the extension f : T α M Ñ T β N defined by f pq, pq " pf pqq, e rpqq p ˝df ´1 q q, @pq, pq P T ˚M is conformally symplectic.Indeed, let us denote by π M , π N the associated cotangent bundle maps, λ M , λ N the associated Liouville forms.Then f ˚λN " e r˝π M λ M : ´f ˚λN ¯pq,pq ¨ξ " e rpqq p ˝df ´1 ˝dπ N ˝d f ¨ξ " e rpqq p ˝dπ M ¨ξ Now given a flow f t : M Ñ M , with f 0 " id, of associated vector field X t , one has f t β " β`dr t with r t pqq :" ş t 0 βpX s ˝fs pqqqds so the associated conformal symplectic flow p ft q is well-defined and one checks that it corresponds to the Hamiltonian flow of H t pq, pq " ppX t pqqq.
Given a closed 1-form β of Y , we also define the β-twisted conformal symplectization of pV, αq by replacing η in the previous definition with η " π ˚β ´dθ, we denote it S conf β pY, αq.We check that ω is non-degenerate by showing that ω n`1 does not vanish: p´1q n`1 ω n`1 " pn`1qpdθ´π ˚βq^π ˚α^pdpπ ˚αqq n " pn`1qdθ^π ˚pα^pdαq n q ‰ 0, the second equality comes from the fact that β ^α ^pdαq n " 0 for a degree reason and the contact hypotheses implies the non-vanishing of the last expression.When the choice of the contact form α is clear, the couple pη, ωq " pπ ˚β ´dθ, ´dη pπ ˚αqq as well as the associated Lee vector field and Hamilton equations will be implicitly chosen or referred to as standard.
Let us show how the study of conformal Hamiltonian dynamics will also inform us about contact Hamiltonian dynamics , see also Proposition 23.We recall that the contact Hamiltonian vector field X associated with the contact Hamiltonian map H : where R is the Reeb vector field defined by αpRq " 1 and ι R dα " 0 (the Hamiltonian vector field associated with H " 1).
Lemma 2. Let H : Y Ñ R be a contact Hamiltonian map of the contact manifold pY, αq with fixed contact form α associated with the Reeb vector field R and let X be the associated Hamiltonian vector field.The conformal Hamiltonian vector field on S conf β pY, αq associated with r In particular, the standard Lee vector field is L :" R ' βpRqB θ .
Proof.Let us first derivate the expression of the Lee vector field L. Let η :" π ˚β ´dθ be the Lee form and ω :" ´dη pπ ˚αq be associated symplectic form.Let us write L " V ' f B θ , V being a vector field of Y and f : Y Ñ R. Since ι L ω " ´η, one has ηpLq " 0, that is f " βpV q.Developing the Lee equation, one then gets ι L dpπ ˚αq `αpV qpπ ˚β ´dθq " π ˚β ´dθ.
By identification, αpV q " 1 and ι V dα " β ´αpV qβ " 0, therefore V " R. The general case is also deduced by identification, once we have remarked that ηp r Xq " ωp r X, Lq " d r H ¨L ´r HηpLq " dH ¨R.
Therefore the conformal Hamiltonian flow pΦ t q of S conf β pY, αq lifting the contact Hamiltonian flow pϕ t q is Φ t px, θq " pϕ t pxq, θ `ρt pxq ´rt pxqq where The expression of r t is consistent with the following general fact for conformal Hamiltonian vector fields X of pM, η, ωq: ηpXq " ωpX, Lq " dH ¨L ´ηpLqH " dH ¨L, where L is the Lee vector field.An isotropic embedding i : L ãÑ pY, αq is by definition an embedding such that i ˚α " 0, it is Legendrian when the dimension of L is maximal: 2 dim L `1 " dim Y .One can associate to every isotropic submanifold L Ă Y the isotropic lift L ˆS1 Ă S conf β pY, αq.Therefore, dynamical properties of contact Hamiltonians can be deduced from properties of conformal Hamiltonians "by projection S conf β pY, αq Ñ Y ".See Part 5.2.1 of section 5.2.
2.5.The invariant distribution F. In the conformal setting, the Hamiltonian map H is not an integral of motion.But the (singular) distribution F :" ker d η H is still invariant since ϕ t d η H " e rt d η H (Lemma 1).Moreover, we have d `dη Hq " η ^dH " η ^dη H hence by Frobenius theorem, at every regular point the Pfaffian distribution ker d η H is integrable.
However, in dynamical systems with dissipative behaviors, its regular leaves are often non-compact (the important exception being tH " 0u).Let us describe the major properties of F.
Corollary 4. Every connected submanifold L Ă M tangent to F is either included in tH " 0u or in tH ‰ 0u.In the case where the pull-back of the Lee form to L is not exact, L is included in tH " 0u.
In the symplectic case, regular levels of H admit invariant volume forms (see e.g.[3, §I.8]), the following proposition generalizes this phenomenon.Proposition 5. Let pM, η, ωq be a 2n-dimensional closed conformal symplectic manifold and H : M Ñ R a Hamiltonian, the flow of which is pϕ t q.Let i : Σ ãÑ M be an embedded leaf tangent to F. Then there exists a volume form µ of Σ such that ϕ t µ " e pn´1qrt µ.Moreover, there exists a p2n ´1q-form µ 0 on M such that µ " i ˚µ0 and µ 0 ^dη H " ω n in the neighborhood of Σ.
Proof.Let µ 1 be a p2n ´1q-form on M such that i ˚µ1 is a volume form of Σ (which is oriented by d η H).By assumption, pd η Hq x ‰ 0 for x P Σ whereas i ˚dη H " 0 so pµ 1 ^dη Hq x ‰ 0 for every x P Σ.Since Σ is an embedded leaf, there exists an open neighborhood U of Σ on which µ 1 ^dη H does not vanish.There exists f : M Ñ R that does not vanish on U such that f µ 1 ^dη H " ω n restricted to U .Let us show that µ 0 :" f µ 1 and the volume form µ :" i ˚µ0 are the desired forms.
Let us recall that L X ω " ηpXqω and L X d η H " ηpXqd η H (Lemma 1).Let us apply L X to the equation µ 0 ^dη H " ω n : Therefore, pL X µ 0 q ^dη H " pn ´1qηpXqµ 0 ^dη H so that i ˚pL X µ 0 q " pn ´1qηpXqi ˚µ0 .Since the flow pϕ t q preserves Σ, i ˚pL X µ 0 q " L X pi ˚µ0 q and the conclusion follows.When the embedded leaf of F is not compact, this invariant volume can be unbounded.

A global decomposition of the phase space: conservative versus dissipative
Let us introduce three notions of attractors that will be used in different parts of this article.
‚ An invariant compact subset A Ă M is a weak attractor if there exists an open subset U Ą A, called a basin of attraction of A, such that Ť xPU ωpxq Ă A where ωpxq is the omega-limit set of x.The basin of attraction is not necessarily unique.‚ A subset A Ă M is a strong attractor if there exists an open subset U Ą A, such that @t ą 0, ϕ t pU q Ă U and A " Ş tą0 ϕ t pU q (which implies that A is compact and invariant).‚ an invariant closed submanifold N Ă M is normally hyperbolically attractive if there exists a tubular neighbourhood V " jpN ˆr´ε 0 , ε 0 sq of N where j : N ˆr´ε 0 , ε 0 s ãÑ M is an embedding, τ ą 0 and a P p0, 1q such that ϕ H τ pV q Ă IntpV q and if we denote V " jpN ˆr´ε, εsq, then @ε P p0, ε 0 s, ϕ H τ pV ε q Ă V aε .Observe that a strong attractor is always a weak attractor.The second definition of C `is to be understood with p P N; the equality between both definitions is due to |B t r t pxq| ď }ηpXq} 8 ă `8 (see Lemma 1).
Proposition 7. Up to a set with zero Lebesgue measure, the set of positively recurrent points coincides with C `.The ω-limit set of every point in D `is in tH " 0u.Almost every point in D `is in tH ‰ 0u and if x P D `X tH ‰ 0u, r t pxq Ñ ´8 as t Ñ `8 and every neighbourhood of ωpxq " A contains a closed curve γ such that ş γ η ‰ 0. Hence A is infinite.Moreover, for every embedded leaf Σ included in tH ‰ 0u with a proper inclusion map, up to a set with zero pn ´1q-dimensional volume, C `X Σ coincides with the set of positively recurrent points in Σ.
Proof.Let us first remark that ω n -almost every point of tH " 0u is trivially recurrent and in C `: every point of the subset tH " 0u X tdH " 0u is fixed by the dynamics whereas tH " 0u X tdH ‰ 0u is negligible.Since H ˝ϕt " e rt H by Lemma 1, a point x P M satisfying r t pxq Ñ `8 must be in tH " 0u X tdH ‰ 0u which is a negligible set.Hence if x P D `X tH ‰ 0u, r t pxq Ñ ´8 as t Ñ `8 and then lim tÑ8 Hpϕ t xq " 0 and x is not positively recurrent.
Let us now show that almost every point of C 1 `:" C `X tH ‰ 0u is recurrent.According to Lemma 1, H ˝ϕt " e rt H, so For k P N ˚, let us define the following compact sets Then C 1 `is the increasing union of the C 1 k 's defined by For each k P N ˚, there is a well-defined first-return measurable map f k : C 1 k ý, f k pxq :" ϕ npxq pxq where npxq :" mintp P N ˚| ϕ p pxq P H k u.Since the 2-form ω H of tH ‰ 0u is preserved by ϕ p for all p P N (Lemma 1), the measurable maps f k 's are preserving the measure ν H n .Since, for k P N ˚, C 1 k has a countable basis of open sets and a measure νpC 1 k q ď νpH k q ď k n ω n pM q which is finite, the Poincaré's recurrence theorem implies that almost every point of C 1 k is recurrent for f k .Let us prove that if x P D `X tH ‰ 0u, every neighbourhood V of ωpxq contains a closed curve γ such that ş γ η ‰ 0. By compacity of ωpxq, one can assume that V is a finite union of path-connected contractible open sets V j .Let K ą 0 be such that | ş γ η| ă K for every γ : r0, 1s Ñ V j and every j (where η denotes the Lee form).Let T ą 0 be such that for all t ě T , ϕ t pxq P V and let j 0 be such that there exist arbitrarily large t's satisfying ϕ t pxq P V j0 .Let t 1 ą t 0 ą T be such that r t0 pxq ´rt1 pxq ą K and ϕ ti pxq P V j0 for i P t0, 1u.Then concatenating t Þ Ñ ϕ t pxq, t P rt 0 , t 1 s, with a path of V j0 connecting ϕ t1 pxq to ϕ t0 pxq, one gets a loop γ : I Ñ V satisfying ş γ η ‰ 0. The conclusion follows.Finally, let Σ Ă tH ‰ 0u be an embedded leaf of F with a proper inclusion map.We have seen that no point in D `XΣ is positively recurrent.Since C `XΣ " C 1 `XΣ, it is enough to prove that almost every point of C 1 k X Σ is recurrent, for all k P N ˚.
Let µ be the volume form associated with Σ by Proposition 5, then the first return maps k 's are compact, so the ν Σ pΣ X C 1 k q's are finite.The conclusion follows.
We recall that U Ă M is a wandering set if DT ą 0, @t ě T, ϕ t pU q X U " H. Corollary 8. Let U be a wandering set.Then almost every point of U belongs to D `X D ´and satisfies lim tÑ`8 Hpϕ H t pxqq " lim tÑ´8 Hpϕ H t pxqq " 0. Corollary 9. Let A be a weak attractor with basin U , then for almost every point x of U zA, r t pxq Ñ ´8 as t Ñ `8.In particular, the Lee form is not exact in any neighborhood of A X tH " 0u.
As a consequence, an attractor (or repeller) of pϕ t q is never a finite set.
Proof of Corollary 9.By definition, points of U zA are not recurrent so almost every point of U zA is in D `by Proposition 7. The same proposition implies the other results.

3.2.
Almost everywhere coincidence of past and future.In the remaining of the article, we will say that a property is almost everywhere satisfied with reference to every volume form.We have of course that C ´coincides with the set of negatively recurrent points.What is surprising is that the set of negatively recurrent points coincide with the set of positively recurrent points up to a set with zero volume.
Proposition 10.The sets C `and C ´coincide up to a set with zero volume.Moreover, for every embedded leaf Σ included in tH ‰ 0u with a proper inclusion map, up to a set with zero pn ´1q-dimensional volume, C `X Σ coincides C ´X Σ.
Proof.We will prove that up to a set with zero volume C `Ă C ´and we will deduce the first part of Proposition 10.We keep the notation of the proof of Proposition 7. The first return map f k : C 1 k ý preserves the finite volume 1 H n ω n , hence almost every point of C 1 k is negatively recurrent for f k .This implies that up to a set with zero volume, C 1 `and hence C `is a subset of C ´.The proof of the last part is similar.
3.3.An example where C `‰ C ´. Adapting the construction made in Section 3.5 and the shadowing lemma, it is not hard to obtain an orbit that is negatively dissipative and positively conservative, i.e. such that C ´‰ C `.
3.4.Boundedness of r t and invariant measures.Let us assume that L Ă M is an invariant measurable set of the dynamics on which pt, xq Þ Ñ r t pxq is a bounded map R ˆL Ñ R (in particular L Ă C `X C ´). Inspired by the proof of [7, Theorem 5.1.13],let us define the bounded measurable map h : L Ñ R, (4) hpxq :" sup tPR r t pxq.
Then h ˝ϕt " h ´rt , so that for instance Lemma 1 implies that, restricted to L, ϕ t pe h ωq " e h ω, ϕ t pe h d η Hq " e h d η H and pe h Hq ˝ϕt " e h H, @t P R.
Corollary 11.An invariant measurable set L Ă M of positive measure on which pt, xq Þ Ñ r t pxq is bounded admits an invariant Borel measure of positive density.
Proof.The measure A Þ Ñ ş A e nh ω n is positive and invariant.Corollary 12.If L is an embedded leaf of F on which pt, xq Þ Ñ r t pxq is bounded, then it admits an invariant Borel measure of positive density.
Proof.The desired measure is A Þ Ñ ş A e pn´1qh µ, where µ is given by Proposition 5.
We will see in Section 4.2 that when η is exact in the neighborhood of tH " 0u, the flow is conservative and Corollaries 11 and 12 apply.
One of the dynamical importance of these corollaries is signified by Poincaré's recurrence theorem: if the invariant measures in question are also finite, almost every point of the invariant set is recurrent.However, with the exception of regular leafs of F included in tH " 0u, the recurrence can also be deduced from Proposition 7.
3.5.An oscillating behavior.In this subsection, we give an example of a flow possessing orbits included in C `that are in tH ‰ 0u, positively recurrent and whose ω-limit set intersects tH " 0u.Hence they have an unbounded associated winding t P r0, `8q Þ Ñ r t pxq.In order to prove this proposition, let us briefly recall the statement of the Shadowing Lemma for flows (see e.g.[7,Theorem 18.1.6]).Let pϕ t q be a smooth flow on a Riemannian manifold M , the infinitesimal generator of which is X t .A differentiable curve c : cptq Xt pcptqq} ď ε for all t P I.A differentiable curve c : I Ñ R is said to be δshadowed by the orbit pϕ t pxqq tPJ if there exists s : J Ñ I with |s 1 ´1| ă δ such that dpcpsptqq, ϕ t pxqq ă δ for all t P J (d denoting the Riemannian distance).The Shadowing Lemma states that, given a hyperbolic set Λ of pϕ t q, there is a neighborhood U Ą Λ, so that for every δ ą 0 there is an ε ą 0 such that every ε-orbit included in U is δ-shadowed by an orbit of pϕ t q.
Proof.Let Σ be a closed hyperbolic surface, let us denote π : T 1 Σ Ñ Σ the associated unit tangent bundle and let β be a non-exact closed 1-form of Σ.Let us denote pG t q the geodesic flow on T 1 Σ and X the associated vector field.Let pM, η, ωq be a conformal symplectic closed manifold associated with pT 1 Σ, π ˚βq by Lemma 35 in Appendix A: that is one may assume that L :" T 1 Σ ˆS1 is a Lagrangian submanifold of M and that the restriction of η to this submanifold is α :" π ˚β ´dθ (we identify π ˚β with its pull-back by the projection by a slight abuse of notation).Let W be a Weinstein neighborhood of L: identifying the 0-section of T α L with L, one can see W as a neighborhood of the 0-section of T α L (see [2,Theorem 2.11] or Section A.2). Let us identify the vector field X of T 1 Σ with the vector field X ' 0 of L and let H : M Ñ R be a Hamiltonian function satisfying Hpq, pq " ppXpqqq on W Ă T α L (shrinking W if necessary).Let us prove that H satisfies the statement of the proposition.
Let us first find an orbit pγ, 9 γq : R `Ñ T 1 Σ of the geodesic flow pG t q such that ( # DK ą 0, @t ą 0, ş γ|r0,ts β ď K, lim sup tÑ`8 ş γ|r0,ts β ą lim inf tÑ`8 ş γ|r0,ts β " ´8.Such an orbit can be found applying the Shadowing Lemma to pG t q.Indeed, let us fix δ P p0, 1q and take an ε ą 0 associated by the Shadowing Lemma.Let a : R{T Z Ñ Σ be a closed geodesic of unit speed such that ş a β ą 0 (up to reparametrization, such an a can be obtained as a minimizer of the energy functional among loops homotopic to a loop b satisfying ş b β ą 0).By topological transitivity of pG t q, there exists an ε{2-orbit pc, 9 cq : r0, T 1 s Ñ T 1 Σ such that 9 cp0q " 9 ap0q and 9 cpT 1 q " ´9 ap0q.By successively concatenated c or c ´1 with higher and higher iterations of a and a ´1, one can thus build an ε-orbit pγ, 9 γq : R `Ñ T 1 Σ satisfying conditions (5) where γ is replaced with γ.The Shadowing Lemma applied to this ε-orbit gives us the desired γ.
According to Section 2.3, on W Ă T α L, the Hamiltonian flow pϕ t q of H takes the form ϕ t pq, p; zq " pG t pqq, e rtpqq p ˝pdG t pqqq ´1; zq, where pq, pq P T ˚pT 1 Σq, z P T ˚S1 and r t pqq :" as long as ϕ s pq, p; zq stays inside W for s P r0, ts.As pG t q is Anosov, one has the bundle decomposition T pT 1 Σq " E s ' RX ' E u which is preserved by pG t q with dG t ¨X " X ˝Gt .Let q Þ Ñ P q be the section of T ˚pT 1 Σq vanishing on E s ' E u and such that P pXq " 1; it satisfies P q ˝pdG t pqqq ´1 " P Gtpqq for all q.For fixed z P T ˚S1 and λ ą 0, let us consider the R `-orbit generated by p 9 γp0q, λP 9 γp0q ; zq (where γ satisfies ( 5)).By the first condition of (5), r t p 9 γp0qq is bounded from above so this orbit keeps inside W for a sufficiently small λ.The second condition of (5) implies the statement for x " p 9 γp0q, λP 9 γp0q ; zq (the orbit is in tH ‰ 0u since P pXq " 1).

Global conservative behaviors
As we have seen in Corollary 9, a necessary condition for attractors to appear is the non-exactness of the Lee form in the neighborhood of tH " 0u.Here, we study the opposite case: Hamiltonian flow pϕ t q of H on a closed conformal symplectic manifold pM 2n , η, ωq in the case where η is exact in the neighborhood of the invariant set tH " 0u.That is, we assume that there exists an open set U containing tH " 0u such that rη| U s " 0 in H 1 pU ; Rq.This hypotheses is thus gauge invariant.4.1.When H does not vanish.Let us first assume that H does not vanish and denote X η H its associated vector field for the Lee form η. Possibly reversing time, we will assume that H is positive.Since X η H " X η`df e f H , by setting f " ´log ˝H, we see that X η H is the Lee vector field of η 1 " η ´dpln ˝Hq.Therefore, Hamiltonian flows of non-vanishing H are Lee flows.
We now assume that H " 1 for the choice of gauge pη, ωq, so that the vector field is L η .Since ηpL η q " 0, r t " 0 and C `" C ´" M , i.e. the flow is positively and negatively conservative with the terminology given in the introduction.Thus almost every point is positively and negatively recurrent.
Lemma 1 implies that ω is preserved by the flow.Let us point out that this flow is not conjugated to a symplectic flow in general since one can have H 2 pM ; Rq " 0 (e.g. the conformal symplectization of the contact sphere pS 2n´1 , 1  2 pxdy ´ydxqq).The volume form ω n is preserved so almost every point is recurrent according to Poincaré's recurrence theorem.More precisely, almost every point of a proper embedded leaf of F is recurrent according to Proposition 7 and Corollary 12. Let us remark that in the case where η is completely resonant, (i.e. the subgroup t ş γ η; γ : S 1 Ñ M u is discrete), there exists k P R ˚and a map θ : M Ñ R{kZ such that η " dθ and the invariant foliation pθ ´1psqq sPR{kZ has compact leafs.

4.2.
When η is exact in the neighborhood of tH " 0u.Let us move on to the general case: there exists an open neighborhood U of tH " 0u on which η| U " dθ for some θ : U Ñ R. Proposition 14.Under the hypotheses of this section, the map pt, xq Þ Ñ r t pxq is bounded on R ˆM .
Let pt, xq P R ˆM , if ϕ s pxq P V 0 for all s P r0, ts (rt, 0s if t ă 0), then |r t pxq| ď A. If x P M zV and ϕ t pxq P M zV , then b{B ď |Hpϕ t pxqq{Hpxq| ď B{b so |r t pxq| ď logpB{bq according to Lemma 1.If x P V and ϕ t pxq R V , we assume t ą 0, let t 0 P r0, ts such that ϕ s pxq P V 0 for all s P r0, t 0 s and ϕ t0 pxq P M zV , then |r t0 pxq| ď A whereas |r t´t0 pϕ t0 pxqq| ď logpB{bq by the above case, implying |r t pxq| ď A `logpB{bq.The same is symmetrically true for t ă 0 and with x and ϕ t pxq intertwined.The last case is when x P V and ϕ t pxq P V , we assume t ą 0 (the other case is symmetrical), and ϕ s0 pxq R V 0 for some s 0 P r0, ts.One can find t 1 ă s 0 ă t 2 such that Hpϕ t1 pxqq " Hpϕ t2 pxqq " ε and ϕ s pxq P V for s P r0, t 1 s Y rt 2 , ts.By Lemma 1, r t2 pxq " r t1 pxq and the conclusion follows from the first case treated.
Therefore, according to the conservative-dissipative decomposition of Section 3.1, M " C ´" C `and almost every points of M or an embedded leaf of F in tH ‰ 0u is positively and negatively recurrent.Moreover, according to Corollaries 11 and 12, M and every embedded leaf of F in tH ‰ 0u admits an invariant Borel measure of positive density.In particular, almost every point of a closed regular leaf of F in tH " 0u is also positively and negatively recurrent.

A topologically transitive Lee flow. Our goal is to provide examples of topologically transitive Lee flow in every dimension.
We have seen in Section 1.1.2that in dimension 2, there are very simple examples of minimal Lee flow.We recall it.Let T 2 " R 2 {Z 2 denote the 2-torus with canonical coordinates x, y P R{Z.Let us fix a, b P R and endow T 2 with the conformal symplectic structure pη, ωq " padx`bdy, dx^dyq, the Lee flow of which is ϕ t px, yq " px`bt, y´atq.This flow is minimal if and only if a and b are rationally independent.
One way to extend this example is to remark that in the case a " ´1, it corresponds to the β-twisted conformal symplectization of the contact manifold pS 1 , dyq with β " bdy.
Proposition 15.Let pY, αq be a closed connected contact manifold with a fixed contact form, the Reeb flow of which is Anosov and possess a periodic orbit γ such that ş γ β is irrational for some closed 1-form β.Then, the standard Lee flow of S conf β pY, αq is topologically transitive.
Such a pY, αq can be found in every dimension.Indeed, let N be a closed Riemannian manifold with negative sectional curvature and a non-trivial real homology group of degree 1: H 1 pN ; Rq ‰ 0. Let thus β 1 be a non-exact closed 1-form such that ş c β 1 is irrational for some loop c.Let π : T 1 N Ñ N be the unit sphere bundle of N endowed with its usual contact structure (the Reeb flow of which is the geodesic flow), the β-twisted conformal symplectization of Y :" T 1 N with β :" π ˚β1 follows the hypothesis of the statement.Indeed, by taking a minimum of the energy functional among loops homotopic to c, one gets a closed geodesic homotopic to c, so a periodic orbit γ of the geodesic flow such that ş γ β " ş c β 1 .However, Lee flows induced by these choices of pY, αq have a lot of periodic orbits in dimension 2n ě 4 so are not minimal.
Proof of Proposition 15.Let pϕ t q be the Reeb flow of Y .According to Lemma 2, the Lee flow pΦ t q of S conf β pY, αq " Y ˆS1 takes the following form: for all px, θq P Y ˆS1 , t P R, Φ t px, θq " pϕ t pxq, θ `ρt pxqq, where ρ t pxq :" ż t 0 βpB s ϕ s pxqqds.
In order to show topological transitivity, it is enough to prove that for every pair of product non-empty open sets U i ˆVi Ă Y ˆS1 , i P t1, 2u, there is some px, θq P U 1 ˆV1 and some t P R such that Φ t px, θq P U 2 ˆV2 (see e.g.[7,Lemma 1.4.2]).We can assume that the V i are arcs of length ą 0. By hypothesis, there exists a point y P Y such that ϕ t2 pyq " y for some t 2 ą 0 and ρ t2 pyq is irrational.We choose x i P U i for i " 1, 2. Let δ ą 0 be so small that if a curve c : ra, bs Ñ Y with cpaq " x 1 and cpbq " x 2 is δ-shadowed by an orbit ν : ra 1 , b 1 s Ñ Y , then νpa 1 q P U 1 , νpb 1 q P U 2 and | ş c β ´şν β| ă {3.By assumption, pϕ t q is a topologically transitive Anosov flow (contact Anosov flows on connected manifolds are topologically mixing [7,Theorem 18.3.6]).
Let ε ą 0 be associated with δ by the Shadowing Lemma.By transitivity, there exist ε{2-orbits c 1 : r0, t 1 s Ñ Y , c 3 : r0, t 3 s Ñ Y , with c 1 p0q " x 1 , c 1 pt 1 q " y and c 3 p0q " y, c 3 pt 3 q " x 2 .Then (up to a small deformation at the connecting points) the concatenated paths ν k :" c 1 ¨ck 2 ¨c3 , k P N, are ε-orbits for pϕ t q.Let R k :" ş ν k β mod 1 P S 1 .Since Let us fix k P N such that θ `rR k ´ {3, R k ` {3s P V 2 for some θ P V 1 (the length of the arcs V i 's being ą ).Applying the Shadowing Lemma to ν k , we find an orbit ν : r0, T s Ñ Y such that νp0q P U 1 , νpT q P U 2 and ş ν β mod 1 is {3-close to R k so that θ `şν β P V 2 .

4.4.
A Lee flow with no periodic orbit.Relaxing the transitivity hypothesis, one can easily produce a Lee flow without periodic orbit.
Proposition 16.Let T n be the flat n-torus with canonical coordinates x i P R{Z, and let β :" a 1 dx 1 `¨¨¨`a n dx n for some fixed a i P R. The standard Lee flow of S conf β pT 1 T n q does not have any periodic orbit if and only if the family p1, a 1 , . . ., a n q is rationally independent.This flow is clearly not minimal and, in general, there is not much hope for the standard Lee flow of a closed twisted conformal symplectization to be minimal in dimension 2n ě 4. Indeed, the Weinstein conjecture states that every Reeb flow of a closed contact manifold pY, αq should possess a closed orbit γ, so γ ˆS1 would be a closed invariant set of the standard Lee flow of the twisted conformal symplectizations of pY, αq.
Proof of Proposition 16.The Reeb flow of T 1 T n » T n ˆSn´1 is ϕ t px, vq :" px tv, vq.The associated ρ t px, vq :" ş t 0 βpB s ϕ s px, vqqds mod 1 satisfies ρ t px, vq " ř i a i tv i mod 1.Therefore, a point px, v, θq P T 1 T n ˆS1 is a τ -periodic point of the Lee flow if and only if τ v P Z n and ř i a i τ v i P Z. 4.5.A conservative behavior with η |tH"0u non exact.
Proof.The contact form is the restriction of the Liouville 1-form λ to T 1 T n and the Reeb vector field R at pq, pq is pp, 0q.Hence dH ¨R " 0 and the contact Hamiltonian flow X H satisfies ι X H dλ " ´dp 1 and X H " p1, 0, . . ., 0q.As dH ¨R " 0, we deduce from Lemma 2 that the conformal Hamiltonian vector field is p1, 0, . . ., 0q.

Dissipative behaviors
5.1.Dissipative ergodic measures.Let ν be an ergodic measure and let us denote rpνq :" The following proposition is the ergodic counterpart of Corollary 4.
Proof.The first statement is obvious since tH " 0u is an invariant set.If rpνq ‰ 0 and supp ν Ć tH " 0u, there exists x P tH ‰ 0u such that r t pxq " trpνq as t Ñ ˘8.However H is bounded and H ˝ϕt " e rt H (Lemma 1), a contradiction.
Let us recall the result of Liverani-Wojtkowski about the Lyapunov spectrum of conformally symplectic cocycles [8].We state the results in the invertible case.Let pM, νq be a probability space with an inversible ergodic map T : M Ñ M and let A : M Ñ GLpR 2n q be a measurable map such that both measurable maps log `}A ˘1} are integrable defining the cocycle A m pxq :" ApT m´1 xq ¨¨¨Apxq for m P Z.According to Oseledets multiplicative ergodic theorem, there exists real numbers λ 1 ă ¨¨¨ă λ s called the Lyapunov exponents of A and an associated decomposition of R 2n (that we will call the Lyapunov decomposition of R 2n ) in linear subspaces F 1 pxq ' ¨¨¨' F s pxq defined for ν-almost every x P M , such that lim mÑ˘8 1 m log }A m pxqv} " λ k , @v P F k pxq, @k P t1, . . ., su.
The positive integer d k :" dim F k is well-defined and called the multiplicity of λ k , these multiplicities satisfy Liverani-Wojtkowski showed a symmetry of the Lyapunov spectrum in the case where A takes its values in the conformally symplectic linear group CSpp2nq :" CSppR 2n , ω 0 q.A conformally symplectic linear map S P CSppE, ωq is a linear map of a symplectic linear space pE, ωq satisfying S ˚ω " βω for some β P R ˚called the conformal factor of S.
Theorem 19 ([8, Theorem 1.4]).Let pM, νq be a probability space with an invertible ergodic map T : M Ñ M and let A : M Ñ CSpp2nq be a measurable cocycle such that log `}A ˘1} are integrable.Let β : M Ñ R ˚be such that Apxq ˚ω0 " βpxqω 0 for all x P M .Then we have the following symmetry of the Lyapunov spectrum λ 1 ă ¨¨¨ă λ s of A: for every k P t1, . . ., su.Moreover, the subspace Let us come back to our setting and consider an ergodic measure ν of M for the Hamiltonian flow pϕ t q.Let us fix a Riemannian metric g on M .By taking a measurable symplectic trivialization of T M , one naturally extends the Oseledets multiplicative ergodic theorem for measurable maps A : M Ñ GLpR 2n q such that log `}A ˘1} are integrable for ν to measurable section A : M Ñ GLpT M q of the fiber bundle GLpT M q such that log `}A ˘1} are integrable where } ¨} is the Riemannian operator norm associated with g.The section A : x Þ Ñ dϕ 1 pxq satisfies the integrability condition and the cocycle A m corresponds to dϕ m for m P Z.The associated Lyapunov exponents λ 1 ă ¨¨¨ă λ s define the Lyapunov exponents of the flow pϕ t q for the ergodic measure ν.By compactness of M , t Þ Ñ B t plog }dϕ t pxqv}q is bounded, x P M and v P T x M zt0u being fixed, so for every px, vq P T M for which one of the limit is defined, where m P Z ˚and t P R ˚.
Corollary 20.Let ν be an ergodic measure of the Hamiltonian flow pϕ t q of the closed conformal symplectic manifold M .Let λ 1 ă ¨¨¨ă λ s be the associated Lyapunov spectrum and F 1 , . . ., F s be the associated Lyapunov decomposition of T M .For every k P t1, . . ., su, λ k `λs´k`1 " rpνq and the subbundle Proof.We apply Theorem 19 to the section x Þ Ñ dϕ 1 pxq of the subbundle of conformally symplectic linear maps of T M , with associated conformal factor β : x Þ Ñ e r1pxq .We only need to prove that b :" ş M log |β|dν equals rpνq.By Fubini's theorem and invariance of ν, b " Let us remark that the fact that F 1 ' ¨¨¨' F s´k is the ω-orthogonal complement of F 1 ' ¨¨¨' F k for every k implies that (6) F ω k X F s´k`1 " 0, @k P t1, . . ., su, ν-almost everywhere.
Corollary 21.Let ν be an ergodic measure of a Hamiltonian flow pϕ t q of the closed conformal symplectic manifold M .There is a measurable sub-bundle F of the Lyapunov decomposition of T M which is transverse to F on which, for ν-almost every x P M , lim tÑ˘8 1 t log }dϕ t pxqv} " rpνq, @v P F pxqzt0u.
Proof.Let λ 1 ă ¨¨¨ă λ s be the associated Lyapunov spectrum and F 1 , . . ., F s be the associated decomposition of T M .Let X be the vector field of pϕ t q.Since RX is invariant with dϕ ¨X " X ˝ϕ, there is k P t1, . . ., su such that λ k " 0 and RX Ă F k ν-almost everywhere.Let us show that F :" F s´k`1 is the desired subbundle.According to Corollary 20, λ s´k`1 " rpνq.According to (6), ωpX, vq ‰ 0 for some v P F zt0u when X ‰ 0. Since d η H " ι X ω, the conclusion follows.
Let r P t1, . . ., su be the maximal integer such that λ r ă 0. According to the non-linear ergodic theorem of Ruelle [10, Theorem 6.3], for every k P t1, . . ., ru and for ν-almost every x P M , the set V k pxq :" where d denotes the Riemannian distance, is the image of F 1 pxq ' ¨¨¨' F k pxq by a smooth injective immersion tangent to identity at x. Therefore the last corollary implies the following proposition.
Corollary 22.Let ν be an ergodic measure of a Hamiltonian flow pϕ t q of the closed conformal symplectic manifold M such that rpνq ă 0 and such that supp ν is included in a connected component Σ of tH " 0u without critical point of H.For ν-almost every point x of Σ, there exists an immersed submanifold V Ă M transverse to Σ and containing x such that lim sup tÑ`8 dpϕ t pxq, ϕ t pyqq ď rpνq, @y P V.

Legendrian attractors. Contact Hamiltonian dynamical systems can provide
examples of conformal dynamical systems by taking their lift to the conformal symplectization (which is closed if the contact manifold is closed).
Proposition 23.Every Legendrian submanifold is a hyperbolic attractor for some autonomous contact Hamiltonian flow.
Proof.Let L be a Legendrian submanifold of a contact manifold.According to the contact Weinstein neighborhood theorem, one can assume that L is the 0-section of pT ˚L ˆR, dz ´ydxq, with local coordinates px, yq P T ˚L and z P R (see e.g. [4, Corollary 2.5.9 and Example 2.5.11]).Given H : T ˚L ˆR Ñ R, the contact Hamilton equations (1) takes the form x " ´By H, z ´y 9 x " H.
Let us give some explicit global examples.Let us first consider the standard contact sphere pS 2n´1 , 1 2 pxdy ´ydxqq.Since pC n z0, dx ^dyq is the symplectization of the standard sphere, every contact Hamiltonian flow can be obtained in the following way: let H : C n zt0u Ñ R be a positively 2-homogeneous Hamiltonian, the flow of which is pΦ t q, then ϕ t pzq :" Φ t pzq }Φ t pzq} , @z P S 2n´1 , @t P R, defines a contact Hamiltonian flow of S 2n´1 .Let Hpx, yq :" 1 2 p}x} 2 ´}y} 2 q, so that Φ t px, yq " pcoshptqx `sinhptqy, sinhptqx `coshptqyq.The associated contact flow has one Legendrian attractor L `:" tx " yu and one Legendrian repeller L ´:" tx " ´yu, every point outside of them having its α-limit set inside L ´and its ω-limit set inside L `.
Let us now consider a vector field X on some closed manifold M generating a flow pf t q.According to Section 2.3, this flow extends to a Hamiltonian flow p ft q (identifying the 0-section with M ) on T ˚M which is fiberwise homogeneous: ft pq, apq " a ft pq, pq, @pq, pq P T ˚M , @a P R. Let us endow M with a Riemannian metric, the flow p ft q induces a Contact Hamiltonian flow pϕ t q on the unit cotangent bundle pS ˚M, i ˚λq (λ being the Liouville form and i : S ˚M ãÑ T ˚M the inclusion) by ϕ t pq, pq :" ft pq, pq } ft pq, pq} , @pq, pq P S ˚M.
A hyperbolically attracting (resp.repelling) fixed point x P M of pf t q corresponds to a normally hyperbolically attracting (resp.repelling) Legendrian fiber S x M of p ft q.
Let us remark that in both examples, one can directly work in the conformal symplectization by taking the flow induced by pΦ t q (resp.p ft q) on the quotient space pC n zt0uq{pz " ezq (resp.pT ˚M zt0uq{ppq, pq " pq, epqq), where e :" expp1q.5.2.2.Hyperbolic attractive and repulsive closed orbit in every non-symplectic manifold.Here, by a non-symplectic manifold, we mean a conformally symplectic manifold, the conformal structure of which is not " p0, ωq.
Proposition 24.Let pM, η, ωq be a conformally symplectic manifold and let γ : S 1 ãÑ M be an embedded loop such that ş γ η ă 0. There exists a Hamiltonian H : M Ñ R admitting γ as a hyperbolic attracting periodic orbit.
In particular, every non-symplectic manifold admits a conformal Hamiltonian flow that has a hyperbolic attractive periodic orbit and a hyperbolic repulsive periodic orbit Hamiltonian.
Proof.Let us first remark that γ is included in an open Lagrangian submanifold.According to Theorem 36 in Appendix A, using a cut-off function to define H globally, one can replace M with the normal bundle N Ñ S 1 of γ » S 1 , identified with the 0-section.As a vector bundle N equals W 'T ˚S1 , where W is a symplectic vector bundle.Since the Lagrangian Grassmannian is connected, one can find a lagrangian subbundle L Ă W . Then L is a Lagrangian submanifold of N containing γ.
One can thus assume that M " T β L with γ included in the 0-section identified with L. The closed form β of L is the pull-back of the Lee form η to L, in particular r :" ş γ β ă 0. Let X be a complete vector field of L inducing a flow pf t q for which γ is a 1-periodic hyperbolic orbit such that the eigenvalues µ's of df 1 pγp0qq satisfy e r ă µ ă 1.Let p ft q be the lifted Hamiltonian flow of T β L properly cut-off outside a neighborhood of γ (see the end of Section 2.3).The differential of f1 at γp0q is equivalent to df 1 pγp0qq ' e r pdf 1 pγp0qqq ´1 so its eigenvalues are in p0, 1q.5.3.Connected components of tH " 0u and attraction.Here we wonder if a connected component of tH " 0u can be an attractor.In Section 1.1.1,we gave a 2-dimensional example where a connected component of tH " 0u is attractive.This is the only example that we know, and here we give conditions that ensure that such a component cannot be attractive.We will say that a subset Σ of M separates locally M in two connected components if in every neighbourhood V of Σ, there exists an open neighbourhood U Ă V of Σ such that U zΣ has exactly two connected components.The manifold M being connected, we say that Σ separates globally M if M zΣ is not connected.
Proposition 25.Assume that Σ is an isolated connected component of tH " 0u that separates globally and locally M in two connected components.Then Σ cannot be a strong attractor.
Proof.Let us assume that Σ is a strong attractor.Then there exists an open neighbourhood U 0 of Σ such that U 0 X tH " 0u " Σ.As Σ separates locally M in two connected components, there exists a neighbourhood U of Σ such that U Ă U 0 has two connected components, U ´and U `.We denote by ε ˘P t´1, 1u the sign of H |U˘.
We know that M zΣ is not connected, and the boundary of each of its connected components intersects Σ and thus contains U ´or U `.This implies that M zΣ has exactly two connected components, M ´that contains U ´and M `that contains U `.We denote by ε : M zΣ Ñ t´1, 1u the function such that ε |M˘" ε ˘.
We choose a smooth bump function χ : M Ñ r0, 1s such that χ is equal to 1 in a neighbourhood of Σ and the support of χ is contained in U .Then the Hamiltonian K : M Ñ R is defined by K " χH `p1 ´χqε.We have Σ " tK " 0u.The Hamiltonian flow of K coincides with the flow of H in a neighborhood of Σ and then Σ is also a strong attractor for pϕ K t q.If x is a generic point in the basin of attraction of Σ for K but not in Σ, x is wandering.Observe that a wandering point is wandering for pϕ K t q and pϕ ´K t q.We deduce from Corollary 8 that lim tÑ´8 Kpϕ K t pxqq " 0 since x was taken generically.But as Σ " tK " 0u is a strong attractor, this is not possible.
We do not know whether a similar statement is true without the separation assumption.We obtain the following result when we assume normal hyperbolic attraction.
Theorem 26.Let us assume pM, η, ωq has dimension 2n ě 4 and let H : M Ñ R be Hamiltonian.Let Σ be a closed connected component of tH " 0u without critical point of H. Then Σ cannot be hyperbolically normally attracting.
Proof.Let us assume that such a Σ is hyperbolically normally attracting and reach a contradiction.Let us restrict ourself to a neighborhood of Σ.One can assume that Σ " tH " 0u and that M " Σ ˆp´ε 0 , ε 0 q for some ε 0 ą 0 with Hpx, yq " y for all px, yq P Σ ˆp´ε 0 , ε 0 q by a change of variables in a tubular neighborhood of Σ.Let V ε :" Σ ˆp´ε, εq, then Σ being normally hyperbolic means that one can assume that there exists a P p0, 1q and τ ą 0 such that (7) ϕ τ pV ε q Ă V aε , @ε P p0, ε 0 q.
According to Proposition 5 applied to the leaf Σ, there exists a volume form µ of Σ such that ϕ t µ " e pn´1qrt µ and which is the pull-back of a form µ 0 of M such that µ 0 ^dy " ω n in the neighborhood of Σ.Let π : M Ñ Σ be the projection on the first factor.By decreasing ε 0 , one can assume that π ˚µ ^dy does not vanish so that there exists a non-vanishing map f : M Ñ R such that f π ˚µ ^dy " ω n .Since µ 0 ^dy " ω n and µ 0 and π ˚µ agree on T Σ, f | Σ " 1.By (7), (8) ω n pϕ τ pV ε qq ď ω n pV aε q, @ε P p0, ε 0 q.

Invariant distribution and submanifolds
We introduced in section 2.5 the invariant distribution F. Let us recall the definition of the holonomy π 1 pF q Ñ G of a leaf F of a foliation G of codimension p on a manifold N n .We refer to [5].A distinguished map f : V Ñ R p of G is a map on a trivialization neighborhood V » R p ˆRn´p that factors by the projection R p ˆRn´p Ñ R p .Given a point z P F , let G be the group of germs of local homeomorphisms of R p fixing 0 defined up to internal automorphisms (i.e. up to conjugacy by such germs).Given a loop γ : S 1 Ñ F based at z and a germ of distinguished map f sending z to 0, there is a unique continuous lift pf t q of γ in the space of germs of distinguished maps such that f 0 " f and f t sends γptq to 0. There exists a unique germ g : R p Ñ R p fixing the origin such that f 1 " g ˝f0 .This germ only depends on f and the homotopy class of γ.If one takes another germ f 1 of distinguished map at z, the same procedure will give a germ g 1 : R p Ñ R p fixing the origin that is conjugated to g.Therefore, one defines the holonomy of F (based at z) as the morphism π 1 pF, zq Ñ G sending the class of the loop γ to the class of the germ g.The holonomy group of F is the image of the holonomy.Up to isomorphisms, these notions do not depend on the base point z (a leaf being path-connected).
Proposition 27.The holonomy of an embedded leaf outside tH " 0u is trivial.Let Σ Ă tH " 0u be a connected component of tH " 0u without critical point of H, the holonomy of Σ is In particular, the holonomy group of Σ is isomorphic to the subgroup xrηs, π 1 pΣqy of R.
Proof.One can prove this proposition by considering the global distinguished map e ´θ H ˝p defined on the universal cover p : Ă M Ñ M with dθ " p ˚η.Let us give a more intrinsec proof.
If F is an embedded leaf outside tH " 0u, the pull-back of the Lee form η to F is exact according to Corollary 4, so η " dθ on a tubular neighborhood U of F .Therefore F is trivially fibered by e ´θ H in U and the holonomy is thus trivial.
Let Σ be a connected component of tH " 0u without critical point.Let i : Σ ãÑ M be the inclusion map.If i ˚η is exact, the holonomy is trivial, as above.Otherwise, let us fix z P Σ such that pi ˚ηq z ‰ 0, which implies that ker η z is transversed to T z Σ.Let T Ă M be an open connected 1-dimensional manifold containing z and tangent to ker η.By shrinking T , one can assume that H induces an isomorphism H| T : T Ñ p´ε, εq sending z to 0, for some ε ą 0. Let us remark that there exist a distinguished map f in the neighborhood of z such that f | T " H| T .Indeed, in the neighborhood of z, let θ be such that θpzq " 0 and dθ " η, then f :" e ´θ H is a distinguished map.Since T is tangent to ker η, θ| T » 0 so that f | T " H| T .
Let γ : r0, 1s Ñ Σ be a smooth loop based at z.According to [5, §2.5] for every x P W a connected neighborhood of z in T , there are smooth paths γ x : r0, 1s Ñ M tangent to F and C 0 -close to γ such that, γ x p0q " x, γ x p1q P T and the image of the holonomy π 1 pΣ, zq Ñ G at rγs is the class of the germ y Þ Ñ H ´γH| ´1 T pyq p1q ¯, (here, we used that H| T " f | T where f is a distinguished map).According to Lemma 3, H ´γH| ´1 T pyq p1q ¯" e ş γx η y, with x :" H| ´1 T pyq.
Since T is tangent to ker η, by concatenating γ x with the image of the segment rHpγ x p0qq, Hpγ x p1qqs under H| ´1 T , one gets a loop γx such that ş γx η " ş γx η.Since γ x is C 0 -close to γ, one can reparametrize γx such that this loop is C 0 -close to γ, so γx is homotopic to γ and the conclusion follows.
Corollary 28.Let Σ Ă tH " 0u be a connected component of tH " 0u without critical point of H.If the pull-back of the Lee form to Σ is not trivial, there exist leafs of F different from Σ, the closure of which contains Σ.
Examples 1.1.2and 4.3 show that a non-compact leaf can go far away from tH " 0u.

Invariance and isotropy.
Proposition 29.Let L be a submanifold of M .
(1) If L is isotropic and invariant, then it is tangent to F.
(2) If L is coisotropic and tangent to F, then it is invariant.
(3) If L is invariant and tangent to F, then the pull-back of ω to L is degenerate.
In particular, if L is of even dimension 2k, the pull-back of ω k to L is zero.An invariant surface tangent to F is thus isotropic.
Proof.If L is an invariant isotropic submanifold, then X is tangent to L so @v P T L, d η H ¨v " ωpX, vq " 0. Conversely, if L is coisotropic and tangent to F, for x P L, T x L Ą pT x Lq ω Ą pF x q ω " RXpxq so L is invariant.If i : L ãÑ M is invariant and tangent to F, then X| L is in the kernel of i ˚ω.
Combining Proposition 29 and Corollary 4, one gets the following result.
Corollary 30.An isotropic invariant submanifold on which the pull-back of the Lee form is not exact is included in tH " 0u.
Corollary 30 can be applied to Lagrangian graphs of T β Q for a non-exact closed 1-form β of Q.Indeed, for every β-closed 1-form α of Q, q Þ Ñ α q defines a Lagrangian section Q ãÑ T β Q pulling back the Lee form to the non-exact form β.
We now are interested in how the dynamics can force the isotropy.Following [11], we recall that a point x P M is quasi-regular if for every continuous map f : M Ñ R, the following limit exists lim tÑ`8 Then we can associate to every quasi-regular point its asymptotic cycle Apxq P H 1 pM, Rq that satisfies for every continuous closed 1-form ν on M xrνs, Apxqy " lim Moreover, if µ is an invariant Borel probability by pϕ t q, µ almost every point is quasi-regular and the asymptotic cycle Apµq P H 1 pM, Rq of µ is defined by xrνs, Apµqy " ż xrνs, Apxqydµpxq.
We have Proposition 31.Let pR tα q tPR the flow of rotations of T n with vector α P R n that is defined by R tα pθq " θ `tα.We identify H 1 pT n q with R n in the usual way.Then every point of T n is quasiregular, and the asymptotic cycle of every point of T n and of every invariant probability measure is α.
Observe that when two flows pf t q : M ý and pg t q : N ý are conjugated via some homeomorphism h : M Ñ N , then the quasi-regular points of pg t q are the h-images of the quasi-regular points of pf t q and that when x P M is quasi-regular, we have h ˚Apxq " Aphpxqq.This allows us to introduce a notion of rotational torus T for a flow pf t q : M ý.A rotational torus is a C 0 -embedded torus j : T m ãÑ M such that pj ´1 ˝ft ˝jq is a flow of rotation.When j is a C 1 -embedding, T is a C 1 -rotational torus.Thanks to Proposition 31, all the points of a rotational torus are quasi-regular with the same asymptotic cycle that we denote by ApT q P H 1 pM q and every measure with support in T has also the same asymptotic cycle.
Let us prove a result that is reminiscent of a result of Herman in the symplectic setting [6].
of the characteristic foliation G of j ˚ω.We assume that T is not isotropic.We denote the maximum rank of j ˚ω by r and by U the open set U " tx P T | rank pj ˚ωpxqq " ru As ϕ t pj ˚ωq " e rt j ˚ω, this set is invariant by the flow.A result of Proposition 31 is (12) r n pxq " n pxrηs, ApT qy `onÑ8 p1qq , @x P T .
There are two cases: ‚ either U " T is compact; we choose x P U and K :" U ; ‚ or U ‰ T .Then the closure of the orbit of a fixed point x P U is homeomorphic to a torus with dimension k ă m.As pϕ t | T q is conjugate to a flow of rotation, there exists a compact invariant neighbourhood K of x in U that is homeomorphic to T k ˆr´1, 1s m´k .In K, we consider the characteristic foliation G of ω.We now follow the arguments and notation of [1] (except that F and the F i 's are here denoted G and G i ).We use a finite covering of K by foliated charts W 1 , . . ., W I in U and denote by G i the foliation restricted to W i .Then there exists a constant µ ą 0 such that every pm ´rq-submanifold S of W i that intersects every leaf of G i at most once satisfies |ω r 2 pSq| ď µ.Moreover, we may assume that there exists ε ą 0 such that: (i) if x, y are in some W i , and such that d G px, yq ă ε, then x and y are in the same leaf of W i where d G is the distance along the leaves.We also have the existence of ν P p0, εq such that (13) d G px, yq ă ν ñ d G pϕ ´1pxq, ϕ ´1pyqq ă ε, @x, y P K.
We then use a decomposition pQ j q 1ďjďJ of K into submanifolds with corners that may intersect only along their boundary such that every Q j is contained in at least one W i that satisfies: (ii) if Q j Ă W i , then if x, y P Q j are in the same leaf of W i , we have d G px, yq ă ν.If S is a piece of r-dimensional submanifold contained in some Q j0 Ă W i0 that is transverse to G and intersects every leaf of G i0 at most once, let us consider S 1 " ϕ 1 pS X Q j0 q X Q j1 for some j 1 .Then S 1 is also transverse to G. Let W i1 that contains Q j1 and let us assume that x, y P S 1 are in a same leaf of G i1 .Because of (ii) and ( 13), d G pϕ ´1pxq, ϕ ´1pyqq ă ε and by (i), we have ϕ ´1pxq " ϕ ´1pyq and x " y.Iterating this argument, we deduce that all the sets S 1 " ϕ k pS X Q j0 q X ϕ k´1 pQ j1 q X ¨¨¨X Q j k are such that if Q j k Ă W i k , S 1 intersects every leaf of F i k at most once and then |ω r 2 pS 1 q| ď µ.If now N k is the number of k-uples pj 1 , . . ., j k q such that ϕ k pS XQ j0 qXϕ k´1 pQ j1 qX ¨¨¨X Q j k ‰ H, then we have Theorem 36 (Weinstein neighborhood).Let Q Ă pM, η, ωq be an isotropic submanifold, there exist a neighborhood U 1 Ă M of Q, a neighborhood U 2 Ă νQ of the 0-section and a conformal symplectomorphism ϕ : pU 1 , η, ωq Ñ pU 2 , η νQ , ω νQ q sending Q to the 0-section canonically.
The proof is an adaptation of the symplectic case.We will use the following stability theorem proven by Chantraine-Murphy.
Let us first prove the following conformal extension of [9, Lemma 3.14].
Lemma 38.Let pM, ηq be a manifold endowed with a closed 1-form and Q Ă M be a compact submanifold.Let us assume that there exist two η-conformal symplectic forms ω 0 and ω 1 agreeing on T q M for all q P Q.There exist two neighborhoods U 0 and U 1 of Q, a diffeomorphism ψ : U 0 Ñ U 1 and a map f : U 0 Ñ R vanishing on Q such that ψ| Q " id, ψ ˚ω1 " e f ω 0 and ψ ˚η " η `df .
Proof of Lemma 38.Let us endow M with a Riemannian metric and let us define a tubular neighborhood U of Q as the image under the diffeomorphism pq, vq Þ Ñ exp q pvq of a neighborhood of the 0-section of the normal bundle of Q.Let π : U Ñ Q be the orthogonal projection.Since π is a retraction by deformation, η| U is cohomologous to π ˚β where β is the pull-back of η to Q.One can assume η " π ˚β: indeed, η " π ˚β `dg with g : U Ñ R vanishing on Q so one can do a gauge transformation.

Corollary 6 .
Every embedded leaf of F outside tH " 0u admits an invariant volume form Proof. Let µ be the volume form associated with Σ by Proposition 5.The volume form µ H n´1 is invariant.

3. 1 . 8 |r
The conservative-dissipative decomposition.We defined in the introduction the partition in invariant sets M " C `\ D `with(2) C `:""x P M | lim inf tÑ`

Proposition 13 . 8 r t pxq ą lim inf tÑ` 8 r
There exists a Hamiltonian map H : M Ñ R on some conformal symplectic manifold, the flow of which satisfying lim sup tÑ`t pxq " ´8, for some point x P tH ‰ 0u.

6. 1 .
Holonomy of embedded leafs of F. Let us study the holonomy of a regular leaf F of F. By definition, one can find an open neighborhood U of F on which F defines a non-singular foliation.The holonomy of F is well-defined as the holonomy of F in U for this foliation.