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. The aim of this paper is to describe some recent results concerning the dynamics of Euler-Lagrange flows on prescribed energy levels. We show that if an Anosov energy level has a splitting of class C 1 then it must contain minimizing measures with non-zero homology. 1. Introduction The aim of this paper is to describe some recent results concerning the dynamics of Euler-Lagrange flows on prescribed energy levels. These results have been obtained using variational methods. Throughout this paper the Euler-Lagrange flows the we shall consider are generated by convex and superlinear Lagrangians on closed connected manifolds M . A very interesting aspect of the dynamics of the Euler-Lagrange flows is given by those orbits or invariant measures that satisfy some global variational properties, instead of the local ones that every orbit satisfy. Research on these special orbits goes back to M. Morse [39] and Hedlund [25] and has reappeared in recent years in the work of V. Bangert [2], M....

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... From (24) and lemma 3.5 we get that ...

... The following theorem is not explicitly stated in [24]. B.1. ...

... For if they were C 1 , the form λ defined by λ(X) ≡ 1 and λ| E ss ⊕E uu ≡ 0 is a contact form for H −1 {k} (c.f. U. Hamendstädt [11], G. Paternain [24,Th. 5.5]). ...

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma ne's critical value of the lift of the Lagrangian to the universal cover, almost all energy levels have conjugate points. We prove that if the energy level [E=k] is of contact type and M is not the 2-torus then the free time action functional of L+k satisfies the Palais-Smale condition.

. Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. Let be a smooth 1-form on M . Given 2 R small, let hEL () be the topological entropy of the Euler-Lagrange ow of the Lagrangian L (x; v) = 1 2 jvj 2 x x (v); and let hF () be the topological entropy of the geodesic ow of the Finsler metric, F (x; v) = jvj x x (v): We show that h 00 EL (0) + h 00 F (0) = h 2 Var(); where Var() is the variance of with respect to the measure of maximal entropy of and h is the topological entropy of . We derive various consequences from this formula. 1. Introduction Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. This happens for example, when all the sectional curvatures are negative. Let be a smooth 1-form on M . We think of as a function : TM ! R such that for each x 2 M , x is a linear functional of T x M . For 2 R consider the 1-parameter family of convex superlinear Lagrangians ...

We continue our study of the weak KAM theorem, which we obtained in our previous Note. We state the connection between this theorem and the Peierls's barriers as defined by Mather. From this connection one can deduce the fact, due to Mañé, that the dynamics on the set of M-minimizing extremals i.s chain transitive.

We give a weak version of the Kolmogorov-Arnold-Moser theorem In the framework of Mather's theory of Lagrangian systems. A particular case has already been established by Lions, Papanicolaou and Varadhan.

LetL be a convex superlinear Lagrangian on a closed connected man- ifold N. We consider critical values of Lagrangians as dened by R. Ma~ n e in (M3). We show that the critical value of the lift of L to a covering of N equals the inmum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotan- gent bundle of the covering. As a consequence, we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of L on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value cu(L) of the lift of L to the universal covering of N. It follows that given k

The central geometric objects associated with an Anosov dynamical system on a compact manifold are the invariant stable and unstable foliations. While each stable and unstable manifold is as smooth as the system itself, the foliations that they form are believed to have only a moderate degree of regularity for most systems. We will analyze the exact degree of regularity of codimension-one stable and unstable foliations for low dimensional systems. Our main results relate the regularity of these foliations to cohomology classes associated to the system: the Anosov class, a new invariant of the flow which we introduce in this paper, and the Godbillon-Vey class of the weak-stable foliations, which we show is a well-defined invariant of the system

Define the critical levelc(L) of a convex superlinear LagragianL as the infimum of thek such that the LagragianL+k has minimizers with fixed endpoints and free time interval. We provide proofs for Ma's statements [7] characterizingc(L) in termos of minimizing measures ofL, and also giving graph, recurrence covering and cohomology properties for minimizers ofL+c(L). It is also proven thatc(L) is the infimum of the energy levelsk such that the following for of Tonelli's theorem holds:There exists minimizers of the L+k-action joining any two points in the projection of E=k among curves with energy k.

On etudie les flots d'Anosov sur les varietes de dimension 3 dont les feuilletages stables et instables forts sont de classe C ∞ . Apres avoir montre qu'il existe de tels flots qui sont «exotiques», on montre comment il est possible de les decrire tous

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit. We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

We describe which Anosov flows on compact manifolds have C∞ stable and unstable distributions and a contact canonical 1-form: up to finite coverings and up to a C∞ change of parameters, each of them is isomorphic to the geodesic flow on (the unit tangent bundle of) a compact locally symmetric space of strictly negative curvature.

I - Differentiable Manifolds.- II - Foliations.- III - The Topology of the Leaves.- IV - Holonomy and the Stability Theorems.- V - Fiber Bundles and Foliations.- VI - Analytic Foliations of Codimension One.- VII - Novikov's Theorem.- VIII - Topological Aspects of the Theory of Group Actions.- Appendix - Frobenius' Theorem.- 1. Vector fields and the Lie bracket.- 2. Frobenius' theorem.- 3. Plane fields defined by differential forms.- Exercises.

It is shown that for a volume preserving Anosov flow on a compact manifold the closed orbits span the first homology (with real coefficients). The proof uses the notion of asymptotic cycles and results concerning the space of invariant measures for hyperbolic flows.

We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C ∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C ∞ affine connection ∇ on V such that ∇ R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V ) and for any invariant ergodic measure μ, there exists χ μ > 0 such that the Lyapunov exponents are −2χ μ , −χ μ , 0, χ μ , and 2χ μ , μ-almost everywhere.
As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C ∞. Then, this flow is C ∞ conjugate to the geodesic flow on a manifold of constant negative curvature.

Motivated by the close relation between Aubry-Mather theory and minimal geodesies on a 2-torus we study the existence and properties of minimal geodesics in compact Riemannian manifolds of dimension ≥3. We prove that there exist minimal geodesics with certain rotation vectors and that there are restrictions on the rotation vectors of arbitrary minimal geodesics. A detailed analysis of the minimal geodesics of the ‘Hedlund examples’ shows that – to a certain extent – our results are optimal.

We consider Anosov flows on closed 3-manifolds which are circle bundles. Our main result is that, up to a finite covering, these flows are topologically equivalent to the geodesic flow of a suface of constant negative curvature. The same method shows that, if M is a closed hyperbolic manifold of any dimension, all the geodesic flows which correspond to different metrics on M and which are of Anosov type are topologically equivalent.

We prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up to C2−ε. Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.

We are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

It is proved here that minimizing measures of a Lagrangian flow are invariant and the Lagrangian is cohomologous to a constant on the support of their ergodic components. Moreover, it is shown that generic Lagrangians have a unique minimizing measure which is uniquely ergodic and is a limit of invariant probabilities supported on periodic orbits of the Lagrangian flows.

Let M denote a closed Riemannian manifold whose geodesic flow is Anosov. Given a real number λ and a smooth one form θ, consider the twisted geodesic flow obtained by twisting the canonical symplectic structure by the lift of λdθ to the tangent bundle of M . For λ in a certain open interval around the origin the twisted flow remains Anosov. We show that the Anosov splitting of the twisted geodesic flow is never of class C 1 unless λ = 0.

LetT
* M denote the cotangent bundle of a manifoldM endowed with a twisted symplectic structure [1]. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by a convex HamiltonianH: T
* M, and we consider a compact regular energy level ofH, on which this flow admits a continuous invariant Lagrangian subbundleE. When dimM3, it is known [9] that such energy level projects onto the whole manifoldM, and thatE is transversal to the vertical subbundle. Here we study the case dimM=2, proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case whenE is the stable or unstable subbundle of an Anosov flow, both properties hold.

We show that any expansive flow on a 3-manifold which is a Seifert fibration or a torus bundle overS
1 is topologically equivalent to a transitive Anosov flow. This is achieved by analyzing the trace of the stable foliation (with singularities) of the flow on incompressible tori embedded in such a manifold.

In this paper we construct stable and unstable foliations for expansive flows operating on 3-manifolds. We also prove that the fundamental group of the manifold has exponential growth.

The objective of this note is to present some results, to be proved in a forthcoming paper, about certain special solutions of the Euler-Lagrange equations on closed manifolds. Our main results extend to time dependent periodic Lagrangians with minor modifications.
We have chosen the autonomous case because this formally simpler framework allows to reach more easily the core of our concepts and results. Moreover the autonomous case exhibits certain special features involving the energy as a first integral that deserve special attention. They are closely related to the link found by Carneiro [C] between the energy and Mather's action function [Ma].

Let M be a closed manifold and a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let c
u
(L) denote the critical value of the lift of L to the universal covering of M and let c
a
(L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, . Let c
0
(L) denote the strict critical value of L defined as the smallest critical value of where ranges among all possible closed 1-forms. We show that c
a
(L) = c
0
(L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all , then . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy , thus answering in the negative a question raised by Mañé. This example also shows that the inequality could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures.
Finally, we describe a large class of Lagrangians for which c
u
(L) is strictly bigger than the maximum of the energy restricted to the zero section of TM.

Letf be an expansive homeomorphism of a compact oriented surfaceM. We show thatS
2 does not support such anf, and thatf is conjugate to an Anosov diffeomorphism ifM=T
2, and to a pseudo-Anosov map ifM has genus ≥2. These results are consequences of our description of local stable (unstable) sets: everyx∈M has a local stable (unstable) set that consists of the union ofr arcs that meet only atx. For eachx∈M
r=2, except for a finite number of points, wherer≥3.

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We improve a result due to M. Kanai on the rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable (horospheric) foliation is smooth. More precisely, the main result proven here is: Let M be a closed [...] Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow [...] on the unit tangent bundle V of M is [...]. Assume moreover that either (a) the sectional curvature of M satisfies [...] or (b) the dimension of M is odd. Then the geodesic flow of M is [...]-isomorphic (i. e., conjugate under a [...] diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature.