Lyapunov exponents for conservative twisting dynamics: a survey
LLyapunov exponents for conservative twisting dynamics:a survey
M.-C. ARNAUD ∗†‡
October 6, 2018
Abstract
Finding special orbits (as periodic orbits) of dynamical systems by variational methodsand especially by minimization methods is an old method (just think of the geodesicflow). More recently, new results concerning the existence of minimizing sets andminimizing measures were proved in the setting of conservative twisting dynamics.These twisting dynamics include geodesic flows as well as the dynamics close to acompletely elliptic periodic point of a symplectic diffeomorphism where the torsion ispositive definite (this implies the existence of a normal form ( θ, r ) (cid:55)→ ( θ + βr + o ( r ) , r + o ( r )) with β positive definite). Two aspects of this theory are called the Aubry-Mathertheory and the weak KAM theory. They were built by Aubry & Mather in the ’80s inthe 2-dimensional case and by Mather, Ma˜n´e and Fathi in the ’90s in higher dimension.We will explain what are the conservative twisting dynamics and summarize theexistence results of minimizing measures. Then we will explain more recent resultsconcerning the link between different notions for minimizing measures for twistingdynamics: • their Lyapunov exponents; • their Oseledets splitting; • the shape of the support of the measure.The main question in which we are interested is: given some minimizing measure ofa conservative twisting dynamics, is there a link between the geometric shape of its ∗ ANR-12-BLAN-WKBHJ † Avignon Universit´e , Laboratoire de Math´ematiques d’Avignon (EA 2151), F-84 018 Avignon, France.e-mail: [email protected] ‡ membre de l’Institut universitaire de France a r X i v : . [ m a t h . D S ] J a n upport and its Lyapunov exponents? Or : can we deduce the Lyapunov exponents ofthe measure from the “shape” of the support of this measure?Some proofs but not all of them will be provided. Some questions are raised in the lastsection. Key words:
Twist maps, Hamiltonian dynamics, Tonelli Hamiltonians, Lagrangianfunctions, Lyapunov exponents, Minimizing orbits and measures, Green bundles, weakKAM theory, contingent and paratangent cones. ontents G − ∩ G + Twisting conservative dynamics
All the dynamics we study here are defined on the cotangent bundle T ∗ M of someclosed manifold M , endowed with its usual symplectic form ω . More precisely, if q =( q , . . . , q n ) are some coordinates on M , we complete them with their dual coordinates p = ( p , . . . , p n ) to obtain some coordinates on T ∗ M : if λ ∈ T ∗ M is a 1-form on M , thenits coordinates p , . . . , p n are given by λ = n (cid:88) i =1 p i dq i . The expression of the symplecticform in these coordinates is ω = dq ∧ dp = n (cid:88) i =1 dq i ∧ dp i . A change of coordinates of M doesn’t change the symplectic form ω and then the definition is correct. We willgenerally use the notation ( q, p ) for such coordinates.When M = T n , we will identify T ∗ M with the 2 n -dimensional annulus A n = T n × R n .Let us recall that a diffeomorphism f of T ∗ M is symplectic if it preserves the symplecticform: f ∗ ω = ω . Notations . We denote by π : T ∗ M → M the canonical projection ( q, p ) (cid:55)→ q .At every x = ( q, p ) ∈ T ∗ M , we define the vertical subspace V ( x ) = ker Dπ ( x ) ⊂ T x ( T ∗ M ) as being the tangent subspace at x to the fiber T ∗ q M . Example 1.
A symplectic C diffeomorphism f : A → A of the 2-dimensionalannulus is a positive symplectic twist map if • it is homotopic to identity; • it twists the vertical to the positive side: ∀ x ∈ A , D ( π ◦ f )( x ) . (cid:18) (cid:19) > . D π( f(x) ) Df(x)
There exists of course a notion of negative symplectic twist map . he notion of twist map that we introduce is local, but the (local) twist condition impliesa global property: when we unfold the cylinder (i.e. we are in the universal covering R of A and we consider a lift of the twist map), the image of a fiber { q } × R by thelift of a symplectic twist map is then a graph above a part of of R × { } : f For example, the map f : ( q, p ) (cid:55)→ ( q + p, p ) is a symplectic twist map of A . When thinking of a possible extension of the notion of twisting dynamics to higherdimension, the first possibility is to ask that the image of any vertical V ( x ) by thetangent dynamics Df is transverse to the vertical V ( f ( x )). If we express Df in twocharts of coordinates ( q, p ): Df = (cid:18) a bc d (cid:19) this is equivalent to ask that ∀ x, det b ( x ) (cid:54) = 0. When M = T n , such diffeomorphimswere introduced and studied by M. Herman in [16], where he called them monotone .When dim M ≥
2, the monotonicity condition doesn’t imply that the image of afiber is a graph above the zero section, even if M = T n and if we unfold the 2 n -dimensional annulus. For example, the map f : A → A defined for ( q, p ) ∈ T × R = C / ( Z + i Z ) × R by f ( q, p ) = ( q + e p − ip , p ) is monotone but the projection of therestriction of its lift to any fiber is not injective.If f is a twist map of the 2-dimensional annulus, f is not necessarily a twist map:the twist condition is just valuable for “small times” (here time 1).Hence for Hamiltonians, we will translate the twist condition for small times. Incoordinates ( q, p ), the Hamilton equations for H ∈ C ( T ∗ M, R ) are:˙ q = ∂H∂p ( q, p ); ˙ p = − ∂H∂q ( q, p ) . Let us denote the Hamiltonian flow of H by ( ϕ Ht ) and let ( δq, δp ) be an infinitesimalsolution, i.e. (cid:18) δq ( t ) δp ( t ) (cid:19) = Dϕ t ( q (0) , p (0)) . (cid:18) δq (0) δp (0) (cid:19) . By differentiating the Hamilton quations, we obtain δ ˙ q = ∂ H∂q∂p δq + ∂ H∂p δp and then D ( π ◦ ϕ t )( q (0) , p (0)) . (cid:18) δp (cid:19) = t ∂ H∂p ( q (0) , p (0)) δp + o ( t ) . We will say that the Hamiltonian H satisfies the twist condition if at every point ∂ H∂p is non-degenerate. In this case, even for small times, the Hamiltonian flow is notnecessarily a twist map; indeed, the o ( t ) above is not uniform in ( q, p ). Unfortunately, we are able to do nothing with the local definition of twisting dynamicsthat we gave in the above subsection.There are two problems:1. we need to find some special invariant subsets for the dynamics;2. we want to say something about the Lyapunov exponents along these invariantsubsets.In general there are two main ways to find invariant subsets for those dynamics: pertur-bative methods and variational methods. Perturbative methods, as K.A.M. theoremsare, are valuable close to completely integrable dynamics (see [16] for the definition inthe case of the 2 n -dimensional annulus). C. Gol´e gives in [14], section 27.B a similarcondition, that he calls “asymptotic linearity”, that makes possible the use of varia-tional methods in this perturbative case. But we won’t explain the perturbative casein this survey. We will only work in the so-called coercive case (see section 27.B of [14]for example).More precisely, we will make some assumptions such that • we can associate a function F to the dynamics, such that the critical points of F are in some sense the orbits for the dynamics; • the function F admits some minima, and then some “minimizing orbits”. For thesymplectic twist maps of the 2-dimensional annulus, these minimizing orbits arethe heart of the theory that S. Aubry and J. Mather independently developpedat the beginning of the ’80s (see [6] and [22]).That is why we introduce the following definitions. The first one comes from [19]and [14], the second one is very classical. .2.1 Globally positive diffeomorphisms Definition . A globally positive diffeomorphism of A n is a symplectic C diffeomor-phism f : A n → A n that is homotopic to Id A n and that has a lift F : R n × R n → R n × R n that admits a C generating function S : R n × R n → R such that: • ∀ k ∈ Z n , S ( q + k, Q + k ) = S ( q, Q ); • there exists α > ∂ S∂q∂Q ( q, Q )( v, v ) ≤ − α (cid:107) v (cid:107) ; • F is implicitly given by: F ( q, p ) = ( Q, P ) ⇐⇒ (cid:40) p = − ∂S∂q ( q, Q ) P = ∂S∂Q ( q, Q ) Example 2.
The diffeomorphism F : ( q, p ) ∈ R n × R n (cid:55)→ ( q + p, p ) ∈ R n × R n is thelift of a globally positive diffeomorphism f of A n and a generating function associatedto F is defined by S ( q, Q ) = (cid:107) q − Q (cid:107) . If f , F satisfy the above hypotheses, the restriction to any fiber { q } × R n of π ◦ F and π ◦ F − are diffeomorphisms (a proof is given in [14]). In particular, this impliesthat f is (locally) monotone.Moreover, for every k ≥ q , q k ∈ R n , the function F : ( R n ) k − → R defined by F ( q , . . . , q k − ) = k (cid:88) j =1 S ( q j − , q j ) has a minimum, and at every critical point for F , thefollowing sequence is a piece of orbit for F :( q , − ∂S∂q ( q , q )) , ( q , ∂S∂Q ( q , q )) , ( q , ∂S∂Q ( q , q )) , . . . , ( q k , ∂S∂Q ( q k − , q k )) . For the map F defined in Example 2, the function F defined by F ( q , . . . , q k − ) = 12 k (cid:88) j =1 (cid:107) q j − − q j (cid:107) attains its minimum at its unique critical point( q , . . . , q k − ) = ( q + q k − q k , q + 2 q k − q k , . . . , q + ( k − q k − q k ) and the correspondingpiece of orbit is: ( q , q k − q k ) , ( q , q k − q k ) , . . . , ( q k , q k − q k ) . Let us discuss a little the condition on ∂ S∂q∂Q . If the matrix of Df in coordinates( q, p ) is Df = (cid:18) a bc d (cid:19) , we have( b ( q, p )) − = − ∂ S∂q∂Q ( q, Q ) . ence the condition that we gave for the partial derivatives of S can be rewritten interms of matrices: b − + t b − ≥ α where is the identity matrix and we use the usualorder for the symmetric matrices.The reader could think of some other possible notions of global twist, for which t b − + b − is indefinite. But in this case, very pathological phenomena can occur;M. Herman showed very strange phenomena in the case of a “normal indefinite torsion”in [17] (the torsion is t b + b and it has the same signature as t b − + b − = t b − ( t b + b ) b − ). A C function H : T ∗ M → R is a Tonelli Hamiltonian if it is: • superlinear in the fiber, i.e. ∀ A ∈ R , ∃ B ∈ R , ∀ ( q, p ) ∈ T ∗ M, (cid:107) p (cid:107) ≥ B ⇒ H ( q, p ) ≥ A (cid:107) p (cid:107) ; • C -convex in the fiber i.e. for every ( q, p ) ∈ T ∗ M , the Hessian ∂ H∂p of H in thefiber direction is positive definite as a quadratic form.We denote the Hamiltonian flow of H by ( ϕ Ht ) and the Hamiltonian vector-field by X H .Note that the flow of a Tonelli Hamiltonian defined on A n is not necessarily a globallypositive diffeomorphism. A geodesic flow is an example of a Tonelli flow. For example,the flat metric on T n corresponds to the Tonelli Hamiltonian H ( q, p ) = (cid:107) p (cid:107) and itstime-one flow is nothing but the map f that we defined in example 2.At the end of the ’80s, J. Mather extended Aubry-Mather theory to the TonelliHamiltonians, introducing the concept of globally minimizing orbits and minimizingmeasures (see [23] and [21]).To explain that, we associate to any Tonelli Hamiltonian H : T ∗ M → R its Lagrangian L : T M → R that is dual to H via the formula: ∀ ( q, v ) ∈ T M, L ( q, v ) = sup p ∈ T ∗ q M ( p.v − H ( q, p )) . Then L is as regular as H is and is superlinear and C -convex in the fiber direction(see for example [11]). Moreover, we have: L ( q, v ) + H ( q, p ) = p.v ⇐⇒ v = ∂H∂p ( q, p ) ⇐⇒ p = ∂L∂v ( q, v ) . If γ : [ α, β ] → M is an absolutely continuous arc, its Lagrangian action is then: A L ( γ ) = (cid:90) βα L ( γ ( t ) , ˙ γ ( t )) dt. .3 Minimizing measures We use the notations that were introduced in subsection 1.2.1. In the 2-dimensionalcase, J. Mather and Aubry & Le Daeron proved in [6] and [22] the existence of orbits( q i , p i ) i ∈ Z for F that are globally minimizing . This means that for every (cid:96) ∈ Z andevery k ≥
2, ( q (cid:96) +1 , . . . , q (cid:96) + k − ) is minimizing the function F defined by: F ( q (cid:96) +1 , . . . , q (cid:96) + k − ) = k (cid:88) i = (cid:96) +1 S ( q i − , q i ) . Then each of these orbits ( q i , p i ) i ∈ Z is supported in the graph of a Lipschitz mapdefined on a closed subset of T , and there exists a bi-Lipschitz orientation preservinghomeomorphism h : T → T such that ( q i ) i ∈ Z = ( h i ( q )) i ∈ Z . Hence each of these orbitshas a rotation number .Moreover, for each rotation number ρ ∈ R , there exists a minimizing orbit that has thisrotation number and there even exists a minimizing measure , i.e. an invariant measurewhose support is compact and filled by globally minimizing orbit, such that all theorbits contained in the support have the same rotation number ρ . These supports,which are Lipschitz graphs above a subset of T , are sometimes called Aubry-Matherset .In the following picture that concerns the so-called standard twist map, you can observesome invariant curves, some Cantor subsets and some periodic islands that must containone periodic point. ifferent kinds of Aubry-Mather sets can occur in this setting:1. some of them are invariant loops that are the graphs of some Lipschitz maps η : T → R ;2. some other ones are just periodic orbits;3. some of these Aubry-Mather sets are Cantor sets.In the case 1, it can happen that the dynamics restricted to the curve is bi-Lipschitzconjugate to a rotation; in this case the Lyapunov exponents of the invariant measuresupported in the curve are zero. This is the case for the KAM curves. But P. LeCalvez proved in [18] that in general (i.e. for a dense and G δ subset of the set of thesymplectic twist maps), there exists an open and dense subset U of R such that anyAubry-Mather set that has its rotation number in U is uniformly hyperbolic. For globally positive diffeomorphism in higher dimension, Garibaldi & Thieullen provethe existence of globally minimizing orbits and measures in [13]. The results thatthey obtain are very similar to the ones that we recall in subsection 1.3.3 for TonelliHamiltonians.
Remarks . There exists too an Aubry-Mather theory for time-one maps of time-dependent Tonelli Hamiltonians (see for example [7]). Even when the manifold M is T n ,the time-one map is not necessarily a globally positive diffeomorphism of A n . Moreover,except for the 2-dimensional annulus (see [24]), it is unknown if a globally positivediffeomorphism is always the time-one map of a time-dependent Tonelli Hamiltonian(see Theorem 41.1 in [14] for some partial results). In this survey, we won’t speakabout these time-one maps. It can be proved that is q b , q e ∈ M are two points of M and β > α two real numbers,if Γ( q b , q e ; α, β ) is the set of the C -arcs γ : [ α, β ] → M that join q b to q e endowedwith the C -topology, then γ is a critical point of the restriction of A L to Γ( q b , q e ; α, β )if and only if γ is the projection of an arc of orbit for H . This arc of orbit is then( γ ( t ) , ∂L∂v ( γ ( t ) , ˙ γ ( t ))) t ∈ [ α,β ] .In [23], J. Mather proves the existence of complete orbits ( ϕ Ht ( q, p )) t ∈ R = ( q ( t ) , p ( t )) t ∈ R that are globally minimizing , i.e. such that every arc ( π ◦ ϕ Ht ( x )) t ∈ [ α,β ] = ( q ( t )) t ∈ [ α,β ] isminimizing for the restriction of A L to Γ( q ( α ) , q ( β ); α, β ). He proves too the existenceof minimizing measures , i.e. invariant measures whose support is filled by globallyminimizing orbit. hen replacing L by L + λ where λ is any closed 1-form on M , we obtain the samecritical points for the Lagrangian action A L + λ as for the Lagrangian action A L . But theminima for those two functions are not the same. Hence, adding different closed 1-form λ to L is a way to find other invariant measures supported in graphs, these measuresbeing minimizing for L + λ . The supports of these measures are the generalizationof the Aubry-Mather sets. A rotation number can be associated to any minimizingmeasure (see [23]) and it can be proved that there exists a minimizing measure forany rotation number. But this doesn’t give the existence of minimizing orbits of anyrotation number (indeed the considered measures have not to be ergodic). Here we are interested in the Lyapunov exponents of the minimizing measures for glob-ally positive diffeomorphisms or Tonelli Hamiltonian flows. In the case of symplectictwist maps, we noticed at the end of subsection 1.3.1 that these exponents may benon-zero or zero.Let ( D t ) ( t in Z or R ) be either the Z -action generated by a globally positivediffeomorphism or the R -action generated by a Tonelli Hamiltonian. Let µ be anergodic minimizing measure. A general fact for ergodic measures and C -boundeddynamics is that the closer the stable and unstable bundles are (this means that thereis an orthogonal basis of the stable bundle that is close to a orthonormal basis of theunstable bundle), the closer to zero the Lyapunov exponents are (see Proposition 1 insubsection 2.1.1 for a more precise statement and [4] for a proof).But in general, the converse assertion is false. We will see that it is true in the caseof a twisting dynamics. • In subsection 2.1 , we will prove these two statements for the Dirac measures andeven give a precise statement for the first assertion in the general case in 2.1.1. • In subsection 2.2 , we will explain that there is link between the number of non-zero Lyapunov exponents for a minimizing measure and the dimension of theintersection of the so-called Green bundles. • In subsection 2.3, we will explain the second assertion (and in fact a more precisestatement): for a twisting dynamics with an hyperbolic minimizing measure, ifthe stable and unstable bundles are not close together (this means that all theunit vectors of the stable bundle are far from every unit vector of the unstablebundle), then all the positive Lyapunov exponents are large. .1 Some simple remarks for Dirac masses Before looking at the Lyapunov exponents of any invariant measure, let us have a lookto what happens for a Dirac mass in dimension 2.More precisely, let us assume that x is a fixed point of a 2-dimensional diffeomorphism.We assume that sup {(cid:107) Df ( x ) (cid:107) , (cid:107) ( Df ( x )) − (cid:107)} ≤ C where C is some constant. Let λ , λ be the two (complex) eigenvalues for Df ( x ); then the Lyapunov exponents for theDirac mass δ x are log( | λ | ) and log( | λ | ). Let us assume that λ and λ are real andlet us denote by E , E the corresponding eigenspaces. We have
Simple principle: if the eigenspaces E and E are close together, then the twoeigenvalues λ and λ have to be close together too.More precisely, if e i is a unit vector on E i , we have: | λ − λ | ≤ C inf { (cid:107) e − e (cid:107)(cid:107) e + e (cid:107) , (cid:107) e + e (cid:107)(cid:107) e − e (cid:107) } . Proof of the simple principle.
We just compute Df ( x ) . e − e (cid:107) e − e (cid:107) = λ + λ . e − e (cid:107) e − e (cid:107) + λ − λ e + e (cid:107) e − e (cid:107) . As e − e and e + e are orthogonal, we deduce | λ − λ | (cid:107) e + e (cid:107)(cid:107) e − e (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Df ( x ) . e − e (cid:107) e − e (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:107) Df ( x ) (cid:107) ≤ C. Changing e into − e , we obtain the second inequality.If f is symplectic, we have λ = λ . In this case, if the eigenspaces are closetogether, the two eigenvalues have to be close to 1 and then the Lyapunov exponentsare close to 0.This simple remark for fixed point can be generalized to any dimension and anyinvariant measure in the following way. For a proof, see [4]. Notations . We endow a compact manifold N with a Riemannian metric. If E , F aretwo linear subspaces of T x N that are d -dimensional with d ≥
1, the distance between E and F is: dist( E, F ) = inf ( e i ) , ( f i ) max {(cid:107) e − f (cid:107) , . . . , (cid:107) e d − f d (cid:107)} where the infimum is taken over all the orthonormal basis ( e i ) of E , ( f i ) of F . roposition 1. Let K be a compact subset of a manifold N , let C > be a realnumber. Then, for any f ∈ Diff ( M ) so that max {(cid:107) Df | K (cid:107) , (cid:107) Df − | K (cid:107)} ≤ C , if f hasan invariant ergodic measure µ with support in K such that the Oseledets stable andunstable bundles E s and E u of µ have the same dimension d , if we denote by Λ u thesum of the positive Lyapunov exponents and by Λ s the sum of the negative Lyapunovexponents, then: < Λ u − Λ s ≤ d log (cid:18) C + 1) (cid:90) dist( E u , E s ) dµ (cid:19) where dist is the distance. If for example f is a symplectic diffeomorphism of T ∗ M , then E u and E s havesame dimension (see for example [9]). We deduce from the above proposition that ifthe stable and unstable Oseledets bundles are close together, then all the Lyapunovexponents are close to 0. This result is not very surprising and not specific to thetwisting dynamics. What is more surprising and specific to the twisting dynamics willcome in the next section. For general symplectic dynamics, we can have simultaneously two eigenvalues that areclose together and two eigenspaces that are not close together. See for example thelinear isomorphism of R with matrix in the usual basis: M = (cid:18) ε ε (cid:19) with ε > x corresponding to such a minimizing Dirac mass δ x , the eigenvalues of Df ( x ) are real.We denote them by λ , λ and by E , E the two corresponding eigenspaces and by M = (cid:18) a bc d (cid:19) the matrix of Df ( x ) in coordinates ( q, p ). Then we have Simple result
If the torsion b is bounded from below by a positive number, if E and E are far from each other, then | λ − λ | cannot be to small.More precisely, if θ is the angle between E and E , then we have | b | ≤ sup { , (cotan( θ )) }| λ − λ | . roof of the simple result. The angle between E and E being θ , there existsa matrix R of rotation such that if P := R (cid:18) θ (cid:19) , then P (cid:18) (cid:19) = e and P (cid:18) (cid:19) ∈ R .e . As R is a matrix of rotation, the modulus of all the coefficients of P = (cid:18) α βδ γ (cid:19) is less that sup { , | cotan θ |} . Moreover, we have: M = (cid:18) γ − β − δ α (cid:19) . (cid:18) λ λ (cid:19) . (cid:18) α βδ γ (cid:19) = (cid:18) ∗ δ.γ ( λ − λ ) ∗ ∗ (cid:19) . We deduce that b = δβ ( λ − λ ) and the wanted result. Before giving an estimation of the non-zero Lyapunov exponents, we will try to findhow many they are. As the dynamics is symplectic, we know that the number ofnegative Lyapunov exponents is equal to the number of positive Lyapunov exponentsand then the number of zero Lyapunov exponents is even (see [9] for a proof).
The Green bundles are two Lagrangian bundles that are defined along the minimizingorbits. In general, they are measurable but not continuous. Let us recall that asubspace H of the symplectic space T x ( T ∗ M ) is Lagrangian if its dimension is n andif the restriction of the symplectic form to H vanishes: ω | H × H = 0.The Green bundles were introduced in the ’50s by L. W. Green to give a proof of the2-dimensional version of Hopf conjecture: a Riemannian metric of T n with no conjugatepoints is flat. Then P. Foulon extended the construction to the Finsler metrics in [12]and G. Contreras & R. Iturriaga built them for any Tonelli Hamiltonian in [10]. Theconstruction for the twist maps of the annulus, and more generally for the twist mapsof T n × R n is due to M. Bialy & R. MacKay (see [8]).We will recall here their precise definition and we will give their main properties. Be-fore this, let us recall that there exists a way to compare different Lagrangian subspacesof T x ( T ∗ M ) that are transverse to the vertical V ( x ). We choose some coordinates ( q, p )as explained at the beginning of section 1 and we denote the linearized coordinates by( δq, δp ) of T x ( T ∗ M ). If H , H are two Lagrangian subspaces of T x ( T ∗ M ) that aretransverse to the vertical V ( x ), we can write them in coordinates ( δq, δp ) as the graphof some symmetric matrices S , S . We say that L is under L and write L ≤ L when S − S is a positive semi-definite matrix. We say that L is strictly under L and write L < L if L ≤ L and L and L are transverse. This is equivalent to say hat S − S is positive definite. It can be proved that this definition doesn’t dependon the chart that we choose. For an equivalent but more intrinsic definition, see [1].Along every minimizing orbit of a globally positive diffeomorphism F : T n × R n → T n × R n or a Tonelli Hamiltonian flow H : T ∗ M → R that we will denote by ( D t ) (with t in Z or R ), we can define two Lagrangian bundles G − and G + . Definition . If the orbit of x is minimizing, then the familly ( D D t .V ( D − t x )) t> isa decreasing family of Lagrangian subspaces that converges to G + ( x ) and the familly( D D − t .V ( D t x )) t> is an increasing family of Lagrangian subspaces that converges to G − ( x ).We recall now some properties of the Green bundles. • they are transverse to the vertical and G − ≤ G + ; • G − and G + are invariant by the linearized dynamics, i.e. D D t .G ± = G ± ◦ D t ; • for every compact K such that the orbit of every point of K is minimizing, thetwo Green bundles restricted to K are uniformly far from the vertical; • (dynamical criterion) if the orbit of x is minimizing and relatively compact in T ∗ M , if lim inf t → + ∞ (cid:107) D ( π ◦ D t )( x ) v (cid:107) ≤ + ∞ then v ∈ G − ( x ).If lim inf t → + ∞ (cid:107) D ( π ◦ D − t )( x ) v (cid:107) ≤ + ∞ then v ∈ G + ( x ).The bundles G − and G + are the Green bundles . The proof of the results that wementioned before can be found in [1] for the Tonelli Hamiltonians and in [4] for theglobally positive diffeomorphisms.An easy consequence of the dynamical criterion and the fact that the Green bundlesare Lagrangian is that when there is a splitting of T x ( T ∗ M ) into the sum of a stable,a center and a unstable bundles T x ( T ∗ M ) = E s ( x ) ⊕ E c ( x ) ⊕ E u ( x ), for example anOseledets splitting or a partially hyperbolic splitting, then we have E s ⊂ G − ⊂ E s ⊕ E c and E u ⊂ G + ⊂ E u ⊕ E c . Let us give the argument of the proof. Because of the dynamical criterion, we have E s ⊂ G − . Because the dynamical system is symplectic, the symplectic orthogonalsubspace to E s is ( E s ) ⊥ = E s ⊕ E c (see e.g. [9]). Because G − is Lagrangian, we have G ⊥− = G − . We obtain then G ⊥− = G − ⊂ E s ⊥ = E s ⊕ E c .Let us note the following straightforward consequence: for a minimizing measure, thewhole information concerning the positive (resp. negative) Lyapunov exponents iscontained in the restricted linearized dynamics D D t | G + (resp. D D t | G − ). In particular,when the measure is weakly hyperbolic, we have almost everywhere G + = E u and G − = E s . Notations . Using a Riemannian metric on M , we define the horizontal subspace H as the kernel of the connection map. Then, for every Lagrangian subspace G of x ( T ∗ M ), there exists a linear map G : H ( x ) → V ( x ) whose graph is G . That is themeaning of graph in what follows. When M = T n , we choose of course H = R n × { } .We denote by s + (resp. s − ) the linear map H → V with graph G + (resp. G − ). Whenwe use symplectic coordinates, their matrices are symmetric.Along a minimizing orbit in the case of a globally positive diffeomorphism, G k = Df k ( V ◦ f − k ) (resp. G − k = Df − k ( V ◦ f k )) is the graph of s k (resp. s − k ). G − ∩ G + From E s ⊂ G − ⊂ E s ⊕ E c and E u ⊂ G + ⊂ E u ⊕ E c , we deduce that G − ∩ G + ⊂ E c .Hence G − ∩ G + is an isotropic subspace (for ω ) of the symplectic space E c . We deducethat dim( E c ) ≥ G − ∩ G + ). When E s ⊕ E c ⊕ E u designates the Oseledets splittingof some minimizing measure µ , what is proved in [3] is that this inequality is an equality µ almost everywhere for the Tonelli Hamiltonian flows and the same result is provedfor the globally positive diffeomorphisms in [5]. Theorem 1.
Let ( D t ) ( t in Z or R ) be either the Z -action generated by a globallypositive diffeomorphism or the R -action generated by a Tonelli Hamiltonian. Let µ bea minimizing measure and let us denote by p the µ -almost everywhere dimension of G − ∩ G + . Then µ has exactly p zero Lyapunov exponents, n − p positive Lyapunovexponents and n − p negative Lyapunov exponents. The idea is the following one. Firstly, let us notice that we have nothing to provewhen dim( G − ∩ G + ) = n because we know that dim( E c ) ≥ G − ∩ G + ) = 2 n ; inthis case, dim( E c ) = 2 n and all the Lyapunov exponents are zero.In the other case, we consider the following restricted-reduced linearized dynamics.Let µ be an ergodic minimizing measure. Then the quantity dim( G − ∩ G + ) is µ -almosteverywhere constant. We denote this dimension by p and we assume that p < n . Notations . We introduce the following linear spaces (see [3]): E = G − + G + , R = G − ∩ G + , F is the reduced space F = E/R . As E is coisotropic for ω with E ⊥ ω = R , then F is the symplectic reduction of E . As E and R are invariant by thelinearized dynamics, then we can define a cocycle M t on F as the reduced linearizeddynamics. This cocycle is then symplectic for the reduced symplectic form Ω.In [3] and [5], we define for the cocycle ( M t ) a vertical subspace, some reduced Greenbundles g − and g + that have properties similar to the ones of G ± , and we prove that g − and g + are transverse µ -almost everywhere. As we will explain in next subsection, thetransversality of the Green bundles implies the (weak) hyperbolicity of the measure.Here we have only the transversality of the reduced Green bundles, but this imply thatthe cocycle ( M t ) is (weakly) hyperbolic and then that the linearized dynamics has atleast 2( n − p ) non-zero Lyapunov exponents. This gives the conclusion. .2.3 The transversality of the two Green bundles implies some hy-perbolicity We will explain here why a minimizing measure µ is weakly hyperbolic when theGreen bundles are transverse almost everywhere. We will deal with the discrete case(i.e. globally positive diffeomorphisms of A n ). The diffeomorphism is denoted by f and we assume that µ -almost everywhere we have: T x A n = G − ( x ) ⊕ G + ( x ). We wantto prove that f has at least n positive Lyapunov exponents; in this case, because f issymplectic, µ has also n negative Lyapunov exponents (see [9]).The idea is to use a bounded (but non continuous) symplectic change of linearizedcoordinates along the minimizing orbits where T x A n = G − ( x ) ⊕ G + ( x ) such that G + becomes the horizontal and that preserves the vertical space. Because G + is invariantby Df , the symplectic matrix of Df k is: M k ( x ) = (cid:18) a k ( x ) b k ( x )0 d k ( x ) (cid:19) .Because G k is transverse to the vertical, we have det b k (cid:54) = 0. Because of the defini-tion of G k , we have then d k ( x ) = s k ( f k x ) b k ( x ). As ( s k ( x )) k ≥ is decreasing and tendsto (because the horizontal is G + ), the symmetric matrix s k ( f k x ) is positive definite.Moreover, because the matrix M k ( x ) is symplectic, we have: (cid:16) M k ( x ) (cid:17) − = (cid:18) t d k ( x ) − t b k ( x )0 t a k ( x ) (cid:19) and by definition of G − k ( x ): t a k ( x ) = − s − k ( x ) t b k ( x ) and finally we have M k ( x ) = (cid:18) − b k ( x ) s − k ( x ) b k ( x )0 s k ( f k x ) b k ( x ) (cid:19) . The proof is then made of several lemmata. The first one is a consequence of Egorovtheorem and of the fact that µ -almost everywhere on supp µ , G + and G − are transverseand then − s − is positive definite. Lemma 1.
For every ε > , there exists a measurable subset J ε of supp µ such that: • µ ( J ε ) ≥ − ε ; • on J ε , ( s k ) k ≥ and ( s − k ) k ≥ converge uniformly ; • there exists a constant α = α ( ε ) > such that: ∀ x ∈ J ε , − s − ( x ) ≥ α . We deduce:
Lemma 2.
Let J ε be as in the previous lemma. On the set { ( k, x ) ∈ N × J ε , f k ( x ) ∈ J ε } , the sequence of conorms ( m ( b k ( x )) converges uniformly to + ∞ , where m ( b k ) = (cid:107) b − k (cid:107) − . roof Let k, x be as in the lemma.The matrix M k ( x ) = (cid:18) − b k ( x ) s − k ( x ) b k ( x )0 s k ( f k x ) b k ( x ) (cid:19) being symplectic, we have: − s − k ( x ) t b k ( x ) s k ( f k x ) b k ( x ) = and thus − b k ( x ) s − k ( x ) t b k ( x ) s k ( f k x ) = and: b k ( x ) s − k ( x ) t b k ( x ) = − (cid:0) s k ( f k x ) (cid:1) − .We know that on J ε , ( s k ) converges uniformly to zero. Hence, for every δ >
0, thereexists N = N ( δ ) such that: k ≥ N ⇒ (cid:107) s k ( f k x ) (cid:107) ≤ δ .Moreover, as G − ≤ G − k ≤ G and G − and G continuously depend on x in thecompact subset supp µ and because the linear change of coordinates that we use isbounded, there exists β > (cid:107) s ± k (cid:107) ≤ β uniformly in k on supp µ . Hence, if wechoose δ (cid:48) = δ β , for every k ≥ N = N ( δ (cid:48) ) and x ∈ J ε such that f k x ∈ J ε , we obtain: ∀ v ∈ R p , β (cid:107) t b k ( x ) v (cid:107) = t vb k ( x )( β ) t b k ( x ) v ≥ − t vb k ( x ) s − k ( x ) t b k ( x ) v = t v (cid:16) s k ( f k x ) (cid:17) − v and we have: t v (cid:0) s k ( f k x ) (cid:1) − v ≥ βδ (cid:107) v (cid:107) because s k ( f k x ) is a positive definite matrixthat is less than δ β . We finally obtain: (cid:107) t b k ( x ) v (cid:107) ≥ δ (cid:107) v (cid:107) and then the result that wewanted. Notations . We choose β > ∀ k ∈ Z \{ } , ∀ x ∈ supp µ, (cid:107) s k ( x ) (cid:107) ≤ β .From now we fix a small constant ε >
0, associate a set J ε with ε via Lemma 1 anda constant 0 < α < β ; then there exists N ≥ ∀ x ∈ J ε , ∀ k ≥ N, f k ( x ) ∈ J ε ⇒ m ( b k ( x )) ≥ α . Lemma 3.
Let J ε be as in Lemma 1. For µ -almost point x in J ε , there exists asequence of integers ( j k ) = ( j k ( x )) tending to + ∞ such that: ∀ k ∈ N , m ( b j k ( x ) s − j k ( x )) ≥ (cid:16) − ε N (cid:17) j k . Proof As µ is ergodic for f , we deduce from Birkhoff ergodic theorem that for almostevery point x ∈ J ε , we have:lim (cid:96) → + ∞ (cid:96) (cid:93) { ≤ k ≤ (cid:96) − f k ( x ) ∈ J ε } = µ ( J ε ) ≥ − ε. We introduce the notation: N ( (cid:96) ) = (cid:93) { ≤ k ≤ (cid:96) − f k ( x ) ∈ J ε } .For such an x and every (cid:96) ∈ N , we find a number n ( (cid:96) ) of integers:0 = k ≤ k + N ≤ k ≤ k + N ≤ k ≤ k + N ≤ · · · ≤ k n ( (cid:96) ) ≤ (cid:96) uch that f k i ( x ) ∈ J ε and n ( (cid:96) ) ≥ [ N ( (cid:96) ) N ] ≥ N ( (cid:96) ) N −
1. In particular, we have: n ( (cid:96) ) (cid:96) ≥ N ( N ( (cid:96) ) (cid:96) − N(cid:96) ), the right term converging to µ ( J ε ) N ≥ − εN when (cid:96) tends to + ∞ . Hence,for (cid:96) large enough, we find: n ( (cid:96) ) ≥ (cid:96) − ε N .As f k i ( x ) ∈ J ε and k i +1 − k i ≥ N , we have: m ( b k i +1 − k i ( f k i ( x ))) ≥ α . Moreover, wehave: s − ( k i +1 − k i ) ( f k i x ) ≤ s − ( f k i x ) ≤ − α then m ( s − ( k i +1 − k i ) ( f k i x )) ≥ α ; hence: m ( b k i +1 − k i ( f k i x ) s − ( k i +1 − k i ) ( f k i x )) ≥ . But the matrix − b k n ( (cid:96) ) ( x ) s − k n ( (cid:96) ) ( x ) is the product of n ( (cid:96) ) − m ( b k n ( (cid:96) ) ( x ) s − k n ( (cid:96) ) ( x )) ≥ n ( (cid:96) ) − ≥ (cid:96) − ε N ≥ (cid:16) − ε N (cid:17) k n ( (cid:96) ) . This implies that all the Lyapunov exponents of the restriction of Df to G + aregreater than log (cid:16) − ε N (cid:17) > Notations . For a positive semi-definite symmetric matrix S that is not the zeromatrix, we denote by q + ( S ) its smallest positive eigenvalue. Theorem 2.
Let µ be an ergodic minimizing measure of a globally positive diffeomor-phism of A n that has at least one non-zero Lyapunov exponent.We denote the smallestpositive Lyapunov exponent of µ by λ ( µ ) and an upper bound for (cid:107) s − s − (cid:107) above supp µ by C . Then we have: λ ( µ ) ≥ (cid:90) log (cid:18) C q + (( s + − S )( x )) (cid:19) dµ ( x ) . The proof of this result is given in [5]. There is a similar result for Tonelli Hamil-tonians:
Theorem 3.
Let µ be an ergodic minimizing measure for a Tonelli Hamiltonian H : T ∗ M → R and with at least one non zero Lyapunov exponent; then its smallest positiveLyapunov exponent λ ( µ ) satisfies: λ ( µ ) ≥ (cid:82) m ( ∂ H∂p ) .q + ( s + − S ) dµ . The proof of Theorem 2 is a little long and involves some technical changes of bases.We prefer to give the proof of Theorem 3, that is simpler and shorter. The first pointis the following lemma: emma 4. Let H : T ∗ M → R be a Tonelli Hamiltonian. Let ( x t ) be a minimizing orbitand let U and S be two Lagrangian bundles along this orbit that are invariant by thelinearized Hamilton flow and transverse to the vertical. Let δx U ∈ U be an infinitesimalorbit contained in the bundle U and let us denote by δx S the unique vector of S suchthat δx U − δx S ∈ V (hence δx S is not an infinitesimal orbit). Then: ddt ( ω ( x t )( δx S ( t ) , δx U ( t ))) = t ( δx U ( t ) − δx S ( t )) ∂ H∂p ( x t )( δx U ( t ) − δx S ( t )) ≥ . Proof
As the result that we want to prove is local, we can assume that we are in thedomain of a dual chart and express all the things in the corresponding dual linearizedcoordinates.We consider an invariant Lagrangian linear bundle G that is transverse to the verticalalong the orbit of x = ( q, p ). We denote the symmetric matrix whose graph is G by G again. An infinitesimal orbit contained in this bundle satisfies: δp = Gδq . We deducefrom the linearized Hamilton equations (if we are along the orbit ( q ( t ) , p ( t )) = x ( t ), ˙ G designates ddt ( G ( x ( t )))) that: δ ˙ q = ( ∂ H∂q∂p + ∂ H∂p G ) δq ; δ ˙ p = ( ˙ G + G ∂ H∂q∂p + G ∂ H∂p G ) δq = − ( ∂ H∂q + ∂ H∂p∂q G ) δq. We deduce from these equations the classical Ricatti equation (it is given for examplein [10] for Tonelli Hamiltonians, but the reader can find the initial and simpler Ricattiequation given by Green in the case of geodesic flows in [15]):˙ G + G ∂ H∂p G + G ∂ H∂q∂p + ∂ H∂p∂q G + ∂ H∂p = 0 . Let us assume now that the graphs of the symmetric matrices U and S are invariantby the linearized flow along the same orbit. We denote by ( δq U , U δq U ) an infinitesimalorbit that is contained in the graph of U . Then we have: ddt ( t δq U ( U − S ) δq U ) = 2 t δq U ( U − S ) δ ˙ q U + t δq U ( ˙ U − ˙ S ) δq U = 2 t δq U ( U − S )( ∂ H∂q∂p + ∂ H∂p U ) δq U + t δq U ( S ∂ H∂p S − U ∂ H∂p U + S ∂ H∂q∂p + ∂ H∂p∂q S − U ∂ H∂q∂p − ∂ H∂p∂q U ) δq U = t δq U ( U ∂ H∂q∂p − S ∂ H∂q∂p + U ∂ H∂p s + − S ∂ H∂p U + S ∂ H∂p S + ∂ H∂p∂q S − ∂ H∂p∂q U ) δq U = t δq U ( U − S ) ∂ H∂p ( U − S ) δq U ≥ . o finish the proof, we just need to notice that in coordinates: ω ( δx S , δx U ) = ω ( δx U , δx U − δx S ) = t ( δq U , U δq U ) (cid:18) − (cid:19) (cid:18) U − S ) δq U (cid:19) = t δq U ( U − S ) δq U . Let µ be an ergodic minimizing Borel probability measure for a Tonelli Hamiltonian H : T ∗ M → R and with at least one non zero Lyapunov exponent; its support K iscompact and then, there exists a constant C > s + and s − are boundedby C above K . We choose a point ( q, p ) that is generic for µ and δx + = ( δq, s + δq ) inthe Oseledets bundle corresponding to the smallest positive Lyapunov exponent λ ( µ )of µ and we introduce δx − = ( δq, s − δq ). Using the linearized Hamilton equations (seeLemma 4), because ω ( x t )( δx − , δx + ) = t δq ( s + − s − ) δq , we obtain: ddt (( t δq ( s + − s − ) δq ) = t δq ( s + − s − ) ∂ H∂p ( q t , p t )( s + − s − ) δq. Let us notice that ( s + − s − ) δq is contained in the orthogonal space to the kernel of s + − s − . Hence: ddt (( t δq ( s + − s − ) δq ) ≥ m ( ∂ H∂p ) q + ( s + − s − ) t δq ( s + − s − ) δq. Moreover δq / ∈ ker( s + − s − ) because ( δq, s + δq ) corresponds to a positive Lyapunovexponent and then ( δq, s + δq ) / ∈ G − ∩ G + . Then : T log( (cid:107) δq ( T ) (cid:107) )+ log 2 CT ≥ T log( t δq ( T )( s + − s − )( q T , p T ) δq ( T )) ≥ T log( t δq (0)( s + − s − )( q, p ) δq (0)) + T (cid:82) T m ( ∂ H∂p ( q t , p t )) q + (( s + − s − )( q t , p t )) dt. Using Birkhoff’s ergodic theorem, we obtain:lim T → + ∞ T log( (cid:107) δq ( T ) (cid:107) ) = λ ( µ ) ≥ (cid:90) m ( ∂ H∂p ) q + ( s + − s − ) dµ. For any subset A (cid:54) = ∅ of a manifold M and any point a ∈ A , different kinds of subsets of T a M can be defined, that are cones and also a generalizations of the notion of tangent pace to a submanifold. We introduce them here when M = R n , but by using somecharts, the definition can be extended to any manifold. Definition . Let A ⊂ R n a non-empty subset of R n and let a ∈ A be a point of A .Then • the contingent cone to A at a is defined as being the set of all the limit pointsof the sequences t k ( a k − a ) where ( t k ) is a sequence of real numbers and ( a k ) is asequence of elements of A that converges to a . This cone is denoted by C a A andit is a subset of T a R n ; • the limit contingent cone to A at a is the set of the limit points of sequences v k ∈ C a k A where ( a k ) is any sequence of points of A that converges to a . It isdenoted by (cid:101) C a A and it is a subset of T a R n ; • the paratangent cone to A at a is the set of the limit points of the sequenceslim k →∞ t k ( x k − y k )where ( x k ) and ( y y ) are sequences of elements of A converging to a and ( t k ) is asequence of elements of R . It is denoted by P a A and it is a subset of T a R n .The following inclusions are always satisfied C a A ⊂ (cid:101) C a A ⊂ P a A. Let us give an example of a contingent and paratangent cone at a point where A hasan angle.In the three last subsections of this survey, we will try to explain some relations betweenthe Green bundles and these tangent cones. Unfortunately, in some cases, we need to se some modified Green bundles (see subsection 3.4). In general, the tangent conesare not Lagrangian subspaces (they are neither subspaces nor isotropic). Because weneed to compare them to Lagrangian subspaces, we give a definition: Definition . Let L − ≤ L + be two Lagrangian subspaces of T x ( T ∗ M ) that aretransverse to the vertical. If v ∈ T x ( T ∗ M ) is a vector, we say that v is between L − and L + and write L − ≤ v ≤ L + if there exists a third Lagrangian subspace in T x ( T ∗ M )such that: • v ∈ L ; • L − ≤ L ≤ L + .A subset B of T x ( T ∗ M ) is between L − and L + if ∀ v ∈ B, L − ≤ v ≤ L + . Then wewrite L − ≤ B ≤ L + . Remarks . In the 2-dimensional case, v is between L − and L + if and only if the slopeof the line generated by v is between the slopes of L − and L + . In higher dimension, itis more complicated. Definition . • A subset A of R n × R n is C -isotropic at some point a ∈ A if (cid:101) C a A is containedin some Lagrangian subspace; • a subset A of R n × R n is C -regular at some point a ∈ A if P a A is contained insome Lagrangian subspace.Of course, the C -regularity of A at a point a implies the C -isotropy at the samepoint. But the converse implication is not true.Observe that a C Lagrangian submanifold is always C -regular. The results that we explain now for the symplectic twist maps of the 2-dimensionalannulus are proved in [2].
Theorem 4.
Let A be an Aubry-Mather set of a symplectic twist map of the 2-dimensional annulus A . Then we have ∀ a ∈ A, G − ( a ) ≤ P a A ≤ G + ( a ) . G − ( x ) G + ( x ) G − ( fx ) G + ( fx ) G − ( f − x ) G + ( f − x ) orollary 1. Let µ be a minimizing ergodic measure of a symplectic twist map ofthe 2-dimensional annulus. If the Lyapunov exponents of µ are zero, then the support supp( µ ) of µ is C -regular µ -almost everywhere. Question 1.
Is there an example of such an invariant measure with zero Lyapunovexponents such that supp µ is not C at every point of supp µ ? Question 2.
Is there an example of such an invariant measure with non-zero Lyapunovexponents such that supp µ is not uniformly hyperbolic? Moreover, the following result is also true.
Proposition 2.
Let µ be a minimizing ergodic measure of a symplectic twist map ofthe 2-dimensional annulus that has an irrational rotation number. If the Lyapunovexponents of µ are non-zero, then the support supp( µ ) of µ is C -irregular µ -almosteverywhere. We have even
Proposition 3.
Let µ be a minimizing ergodic measure of a symplectic twist map of the2-dimensional annulus that has an irrational rotation number. If the support supp( µ ) of µ is C -irregular everywhere, then supp µ is uniformly hyperbolic. Hence the size of the Lyapunov exponents can be read on the shape of supp( µ ). Buthow can we see in practice this irregularity? For example, if we want to “draw” (witha computer) our irregular (and hyperbolic) Aubry-Mather sets, we can use some se-quences of minimizing periodic orbits. But if we look at the pictures of Aubry-Mathersets that exist, we see Cantor sets or curves, but we never see angles of the tangentspaces. That’s why the following question was raised by X. Buff : Question 3. (X. Buff ) Is it possible (for example by using minimizing periodic orbits)to draw some Aubry-Mather sets with “corners”?
The proofs of the results we present in this section are given in [1]. We obtain astatement similar to Theorem 4 and Corollary 1 but no analogue to Proposition 2.Indeed, let us consider the following example: ( ψ t ) is a geodesic Anosov flow definedon the cotangent bundle T ∗ S of a closed surface S . Let N = T ∗ S be its unit cotangentbundle, which is a 3-manifold invariant by ( ψ t ). Then a method due to Ma˜n´e (see [20])allows us to define a Tonelli Hamiltonian H on T ∗ N such that the restriction of its ow ( ϕ t ) to the zero section N is ( ψ t ): the Lagrangian L associated with H is definedby: L ( q, v ) = (cid:107) ˙ ψ ( q ) − v (cid:107) where (cid:107) . (cid:107) is any Riemannian metric on N . In this case, thezero section is very regular (even C ∞ ), but the Lyapunov exponents of every invariantmeasure with support in N are non zero (except two, the one corresponding to the flowdirection and the one corresponding to the energy direction). Hence, it may happenthat some exponents are non zero and the support of the measure is very regular. Theorem 5.
Let G be a Lipschitz Lagrangian graph that is invariant by the flow of aTonelli Hamiltonian H : T ∗ M → R . Then we have: ∀ x ∈ G , G − ( x ) ≤ P x G ≤ G + ( x ) . The following corollary is not proved in [1] but is an easy consequence of Theorems5 and 1.
Corollary 2.
Let µ be a minimizing ergodic measure for a Tonelli Hamiltonian of T ∗ M . If the Lyapunov exponents of µ are zero and if the support of µ is a graphabove the whole manifold M , then the support supp( µ ) of µ is C -regular µ -almosteverywhere. Question 4.
Is there an example where such a µ has zero Lyapunov and its supportis not C at least one point? With the hypotheses of Corollary 2, if we have further information about the re-stricted dynamics to supp( µ ), we can improve the result in the following way. Proposition 4.
Let G be a Lipschitz Lagrangian graph that is invariant by the flow ofa Tonelli Hamiltonian H : T ∗ T n → R . We assume that for some T > , the restrictedtime- T map ϕ T |G is Lipschitz conjugated to some rotation of T n . Then G is the graphof a C function. When µ is a minimizing measure with a support smaller than a Lagrangian graph,we don’t obtain such a result (even if we have the feeling that it could be true). Afundamental tool to prove the previous results is the following proposition (that isproved in [1]). Proposition 5.
Assume that the orbit of x ∈ T ∗ M is globally minimizing for theTonelli Hamiltonian H : T ∗ M → R and that L defined on R is such that • every L ( t ) is a Lagrangian subspace of T ϕ Ht ( x ) ( T ∗ M ) that is transverse to thevertical subbundle; • ∀ s, t ∈ R , Dϕ Ht − s L ( s ) = L ( t ) .Then we have ∀ t ∈ R , G − ( ϕ Ht ( x )) ≤ L ( t ) ≤ G + ( ϕ Ht ( x )) . sing Proposition 5 at any point where the invariant Lagrangian graph G is differen-tiable, we deduce a similar inequality for L being the tangent subspace at such a point.Then using a limit (and the notion of Clarke subdifferential), we deduce Theorem 5.If we could obtain a result similar to Proposition 5 for vectors (instead of Lagrangiansubspaces), we could deduce a similar statement for all minimizing measures. Hencewe raise the question Question 5.
Let ( D t ) ( t in Z or R ) be either the Z -action generated by a globallypositive diffeomorphism or the R -action generated by a Tonelli Hamiltonian. Assumethat the orbit of x ∈ T ∗ M is globally minimizing and that the vector v ∈ T x ( T ∗ M ) issuch that: ∀ t, Dϕ t ( v ) / ∈ V ( D t x ) . Is it true that: G − ( x ) ≤ v ≤ G + ( x )? Remarks . Without a lot of change, all the results of this subsection could beproved for any Lipschitz Lagrangian graph that is invariant by a globally positivediffeomorphism of T n × R n . The results contained in this subsection come from [3] and [5]. They use in a fundamen-tal way a recent theory called the weak KAM theory that was developped by A. Fathiin [11] in the case of the Tonelli Hamiltonians and by E. Garibaldi & P. Thieullen in[13] in the case of the globally positive diffeomorphisms.Let us now introduce the modified Green bundles that we will use in this section.We use the constant c = √ − . We identify T x ( T ∗ M ) to R n × R n in such a waythat { } × R n = V ( x ) is the vertical subspace and R n × { } is the horizontal subspace H . Definition . We denote by S ± ( x ) : R n → R n the linear operator such that G ± ( x ) isthe graph of S ± ( x ): G ± ( x ) = { ( v, S ± ( x ) v ); v ∈ R n } . Then the modified Green bundles G ± are defined by: (cid:101) G − ( x ) = { ( v, ( S − ( x ) − c ( S + ( x ) − S − ( x ))) v ); v ∈ R n } and (cid:101) G + ( x ) = { ( v, ( S + ( x ) + c ( S + ( x ) − S − ( x ))) v ); v ∈ R n } . emarks . We have: (cid:101) G − ≤ G − ≤ G + ≤ (cid:101) G + . Moreover, only the two following cases are possible • either (cid:101) G − ( x ), G − ( x ), G + ( x ), (cid:101) G + ( x ) are all distinct; • or (cid:101) G − = G − = G + = (cid:101) G + . Theorem 6.
Let µ be a minimizing measure for a Tonelli Hamiltonian of T ∗ M . Then ∀ x ∈ supp µ, (cid:101) G − ( x ) ≤ (cid:101) C x (supp µ ) ≤ (cid:101) G + ( x ) . Hence, the more irregular supp µ is, i.e. the bigger the limit contingent cone is, themore distant (cid:101) G − and (cid:101) G + (and thus G − and G + too) are from each other and the largerthe positive Lyapunov exponents are. Corollary 3.
Let H : T ∗ M → R be a Tonelli Hamiltonian and let µ be an ergodicminimizing probability all of whose Lyapunov exponents are zero. Then, at µ -almostevery point of the support supp( µ ) of µ , the set supp( µ ) is C -isotropic. There are two natural questions, that are related to question 5 and that concernalso the globally positive diffeomorphims.
Question 6.
Can we replace (cid:101) C x (supp µ ) by P x (supp µ ) in Theorem 6 and Theorem 7? Question 7.
Can we replace (cid:101) G ± ( x ) by G ± ( x ) in Theorem 6 and Theorem 7? If the answer to question 6 is positive, we can replace “ C -isotropic” by “ C -regular”in Corollary 3 and Corollary 4.For the globally positive diffeomorphisms, we obtain a result only for the so-calledstrongly minimizing measures (the point is that for Tonelli Hamiltonians, miniminizingmeasures are also strongly minimizing). Definition . Let F be a lift of a globally positive diffeomorphism f with generatingfunction S : R n × R n → R . A invariant Borel probability ν on T n × T n is stronglyminimizing if ν is a minimizer in the following formulainf µ (cid:90) R n × R n S ( x, y ) d ˜ µ ( x, y );where the infimum is taken on the set of the Borel probability measures that areinvariant by f and ˜ µ is any lift of µ to a fundamental domain of R n × R n for theprojection ( x, y ) (cid:55)→ ( x, − ∂S∂x ( x, y )) onto T n × R n . . Garibaldi & P. Thieullen proved in [13] that such strongly minimizing measuresexist. Moreover, they are minimizing. Theorem 7.
Let µ be a strongly minimizing measure of a globally positive diffeomor-phism of A n et let supp µ be its support. Then ∀ x ∈ supp µ, (cid:101) G − ( x ) ≤ (cid:101) C x (supp µ ) ≤ (cid:101) G + ( x ) . Corollary 4.
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