A connecting theorem for geodesic flows on the 2-torus
aa r X i v : . [ m a t h . D S ] F e b A connecting theorem for geodesic flows on the 2-torus
Stefan Klempnauer ∗ Faculty of Mathematics, Ruhr-University BochumFebruary 8, 2021
Abstract
We use a result of J. Mather on the existence of connecting orbits for compositions ofmonotone twist maps of the cylinder to prove the existence of connecting geodesicson the unit tangent bundle ST of the 2-torus in regions without invariant tori. The author thanks the SFB CRC/TRR 191
Symplectic Structures in Geometry, Algebraand Dynamics of the DFG and the Ruhr-University Bochum for the funding of hisresearch.
Let T ∼ = R / Z denote the 2-torus with universal covering π : R → T . The tangentbundle is given by T T ∼ = T × R . In the following we will recall the definition of aFinsler metric. For a general overview of Finsler geometry see [1]. Definition 1.1. A Finsler metric on T is a map F : T T → [0 , ∞ ) with the following properties1. (Regularity) F is C ∞ on T T −
2. (Positive homogeneity) F ( x, λy ) = λ · F ( x, y ) for λ >
3. (Strong convexity) The hessian ( g ij ( x, v )) = (cid:18) ∂ v i v j F ( x, v ) (cid:19) is positive-definite for every ( x, v ) ∈ T T − . ∗ [email protected] reversible if F ( x, v ) = F ( x, − v ) for every ( x, v ) ∈ T T . The unit tangent bundle ST ∼ = T × S is given by ST = F − ( { } ).We define the length l F ( c ) of a piecewise differentiable curve c : [ a, b ] → T via l F ( c ) = Z ba F ( ˙ c ( t )) dt Lifting the Finsler metric F to R allows us to define the length of a curve c : [ a, b ] → R in the universal covering in the same manner. The geodesic flow φ t : ST → ST is therestriction of the Euler-Lagrange flow of the Lagrangian L F , with L F = 12 F to the unit tangent bundle. A geodesic we call either a trajectory of the Euler-Lagrangeflow, or its projection to T , i.e. a geodesic is a curve t c ( t ) ⊂ T satisfying theEuler-Lagrange equation ∂ x L F ( c, ˙ c ) − ∂ t ( ∂ v L F ( c, ˙ c )) = 0In general we will assume a geodesic c to be parametrized by arclength, i.e. ˙ c ⊂ ST .Sometimes it can be convenient to consider lifts e c : R → R or ˙ e c : R → T R of a geodesicto the universal cover. These can be seen as the geodesics of the lifted Z -periodic Finslermetric e F on R . Definition 1.2.
A subset Λ ⊂ ST is called an invariant torus if Λ is the graph of acontinuous map X : T → S and φ t -invariant. An invariant torus (or more generallyany φ t -invariant set) Λ has bounded direction (with respect to v ∈ Z − { } ), if the lifts ˜ c : R → R of geodesics in Λ are graphs over the euclidean line R v . In analogy to a Birkhoff region of instability we define the following notion of an insta-bility region of the geodesic flow.
Definition 1.3. An instability region is a compact invariant subset U ⊂ ST withboundary being the disjoint union of two invariant tori Λ − , Λ + , such that every invarianttorus Λ ⊂ U is equal to Λ − or Λ + . The main result is the following.
Theorem 1.4.
Let F be a Finsler metric on T and let U ⊂ ST be an instability regionwith bounded direction (with respect to e ). Furthermore, we assume that there are noclosed geodesics in the boundary of U . Then there exists an F -geodesic c : R → T thatconnects the two boundary components of U (i.e. there exist sequences t n → ∞ and s n → −∞ with lim n →∞ ˙ c ( t n ) ∈ Λ + and lim n →∞ ˙ c ( s n ) ∈ Λ − , where Λ − and Λ + are thetwo boundary components of U ). T and twist maps Let F be a Finsler metric on the 2-torus T with e F denoting the lifted metric on theuniversal cover R . For a vector w = ( w , w ) ∈ R we will write w ⊥ for the vector w ⊥ = ( − w , w ). 2 efinition 2.1. For a Finsler metric F and a prime element v ∈ Z \{ } we define atime-dependent Lagrangian on R via e L ( t, x, r ) = e F ( tv + xv ⊥ , v + rv ⊥ ) which is 1-periodic in the time t and 1-periodic in the first component x . The periodicityis a direct consequence of the Z -periodicity of e F . Since e L is periodic in x it can beinterpreted as a time-periodic Lagrangian L on S . Note that e L is strictly convex, i.e. ∂ rr e L > because of the strict convexity of the Finsler metric e F . We have a correspondence of lifted geodesics, which are graphs over v R and solutions of L in the following theorem by J.P. Schr¨oder. A special case can be found in [5]. Theorem 2.2.
Let F and L as above and let θ : R → R be a smooth function. Let γ : R → R be the curve given by γ ( t ) = tv + θ ( t ) v ⊥ Then γ is a reparametrization of an e F -geodesic if and only if θ is an Euler-Lagrangesolution of e L .Proof. Observe that we have the following relation between the Lagrangian action A e L and the Finsler length l e F . A e L ( θ | [ a,b ] ) = Z ba e L ( t, θ ( t ) , θ ′ ( t )) dt = Z ba e F ( tv + θ ( t ) v ⊥ , v + θ ′ ( t ) v ⊥ ) dt = Z ba e F ( γ ( t ) , ˙ γ ( t )) dt = l e F ( γ | [ a,b ] )Assume now that γ : [ a, b ] → R is the reparametrization of an e F -geodesic, i.e. ∂ s =0 l e F ( γ s ) =0 for any proper variation of γ . Let θ s : [ a, b ] → R be a proper variation of θ . From θ s we construct a proper variation γ s of γ via γ s ( t ) = tv + θ s ( t ) v ⊥ Then we have A e L ( θ s ) = l e F ( γ s ) for every s , and hence we have ∂ s | s =0 A e L ( θ s ) = ∂ s | s =0 l e F ( γ s ) = 0 . This proves one direction. To prove the other direction assume now that θ : [ a, b ] → R is critical with respect to the Lagrangian action and let X : [ a, b ] → R be a vector fieldalong γ with X ( a ) = X ( b ) = 0. Since ˙ γ ( t ) = v + θ ′ ( t ) v ⊥ the pair of vectors { ˙ γ ( t ) , v ⊥ } always forms a basis of R . Thus, we can rewrite the vector field X as X ( t ) = λ ( t ) ˙ γ ( t ) | {z } A ( t ) + µ ( t ) v ⊥ | {z } B ( t ) for functions λ, µ with λ ( a ) = λ ( b ) = µ ( a ) = µ ( b ) = 0. Let γ s be a proper variation of γ corresponding to the variational vector field X and let β s be a proper variation of γ corresponding to B , i.e. ∂ s | s =0 γ s ( t ) = X ( t ) and ∂ s | s =0 β s ( t ) = B ( t )3bserve that for small | s | the curve η s : t γ ( t + sλ ( t )) is a reparametrization of γ andhence has length independent of s . Thus we have0 = ∂ s | s =0 l e F ( η s )= ∂ s | s =0 Z ba e F ( η s ( t ) , ∂ t η s ( t )) dt = Z ba ∂ s | s =0 e F ( η s ( t ) , ∂ t η s ( t )) dt = Z ba ∂ e F ( η ( t ) , ∂ t η ( t )) ∂ s | s =0 η s ( t ) + ∂ e F ( η ( t ) , ∂ t η ( t )) ∂ s | s =0 ∂ t η s ( t ) dt = Z ba ∂ e F ( γ ( t ) , ∂ t γ ( t )) ∂ s | s =0 η s ( t ) + ∂ e F ( γ ( t ) , ∂ t γ ( t )) ∂ t ∂ s | s =0 η s ( t ) dt = Z ba ∂ e F ( γ ( t ) , ∂ t γ ( t )) A ( t ) + ∂ e F ( γ ( t ) , ∂ t γ ( t )) ˙ A ( t ) dt Hence it follows that ∂ s | s =0 l e F ( γ s ) = ∂ s | s =0 Z ba e F ( γ s ( t ) , ˙ γ s ( t )) dt = Z ba ∂ e F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 γ s ( t ) + ∂ e F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 ∂ t γ s ( t )= Z ba ∂ e F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 γ s ( t ) + ∂ e F ( γ ( t ) , ˙ γ ( t )) ∂ t ∂ s | s =0 γ s ( t )= Z ba ∂ e F ( γ ( t ) , ˙ γ ( t )) X ( t ) + ∂ e F ( γ ( t ) , ˙ γ ( t )) ˙ X ( t )= Z ba ∂ e F ( γ ( t ) , ˙ γ ( t ))( A ( t ) + B ( t )) + ∂ e F ( γ ( t ) , ˙ γ ( t ))( ˙ A ( t ) + ˙ B ( t ))= Z ba ∂ e F ( γ ( t ) , ˙ γ ( t )) B ( t ) + ∂ e F ( γ ( t ) , ˙ γ ( t )) ˙ B ( t )= ∂ s | s =0 l e F ( β s )Consequently, the curve γ is critical with respect to the Finsler length if ∂ s | s =0 l e F ( β s ) = 0for every proper variation β s , which varies γ only in v ⊥ direction, i.e. β s is of the form β s ( t ) = tv + θ s ( t ) v ⊥ For those variations we have already computed that if θ s is critical with respect to the e L -action then β s is critical with respect to the Finsler length.Remember that T R ∼ = R × R . To establish a connection between return maps of thegeodesic flow and the time-1 map of the Lagrangian e L we define the following sets e V = { ( x, w ) ∈ T R : h w, v i > , e F ( w ) = 1 } with sections e V i , where i ∈ Z , given by 4 V i = { ( iv + bv ⊥ , w ) ∈ T R : h w, v i > , e F ( w ) = 1 , b ∈ R } In general the Euler-Lagrange solutions of e L will not exist for all times. However, becauseof theorem 2.2 we have the following Proposition 2.3.
If the Euler-Lagrange flow of e L is complete then the set e V is invariantand every orbit starting in a section e V i will consecutively pass through the sections e V i + j for j ∈ N .Proof. Completeness of the Euler-Lagrange flow means that the Euler-Lagrange solu-tions of e L exist for all times. Let ( x, w ) ∈ e V . We write ( x, w ) = ( av + bv ⊥ , rv + hv ⊥ )for coefficients a, b, h, r ∈ R with r >
0. Let e c : R → R be the geodesic with e c ( a ) = x and ˙ e c ( a ) = r w . To show that e V is invariant we have to show that h ˙ e c ( t ) , v i > t ∈ R . Let θ : R → R be the Euler-Lagrange solution of e L with θ ( a ) = b and θ ′ ( a ) = hr .Since θ exists for all times it follows from Theorem 2.2 that the curve γ : R → R with γ ( t ) = tv + θ ( t ) v ⊥ is a reparametrization of a geodesic. We have γ ( a ) = av + bv ⊥ = x and ˙ γ ( a ) = v + hr v ⊥ = r w and hence γ is a reparametrization of the geodesic e c . Since h ˙ γ ( t ) , v i = k v k for every t , we have h ˙ e c ( t ) , v i > t . With the same constructionone sees that if ( x, w ) was chosen in e V i the constructed reparametrization γ will passthrough e V i + j at times i + j .By e R : e V → e V we denote the map that maps an element ( x, w ) ∈ e V to the point ofintersection of e V with the orbit passing through ( x, w ). We define the projected section V ⊂ ST via V := Π( V )Since the geodesic flow on T R is Z -invariant, the map e R descends to a map R : V → V ,which is the n -th return map of the section V , with respect to the geodesic flow on ST .Here, n − k v k − n − R v ⊥ , that lie in between R v ⊥ and v + R v ⊥ , which corresponds to the number of integerpoints in the rectangle spanned by v and v ⊥ . If v is a prime element in Z then this isequal to k v k − Proposition 2.4.
If the Euler-Lagrange flow of e L is complete the return map R isconjugated to the time-1 map ϕ , L : Z → Z of the Lagrangian L .Proof. Let ( x, w ) ∈ e V . We can rewrite ( x, w ) as( x, w ) = (cid:18) h x, v ⊥ ik v k v ⊥ , h w, v ik v k v + h w, v ⊥ ik v k v ⊥ (cid:19) Define real numbers b, h ∈ R and r >
0, such that ( x, w ) = ( bv ⊥ , rv + hv ⊥ ). Since theEuler-Lagrange flow is complete there exists an Euler-Lagrange solution θ : R → R of e L with θ (0) = b and θ ′ (0) = hr From theorem 2.2 it follows that there exists a reparametrization γ ( t ) = tv + θ ( t ) v ⊥ of an e F -geodesic. It follows then that e R ( x, w ) = γ (1) , ˙ γ (1) e F ( γ (1) , ˙ γ (1)) !
5e define the scaled sections f W i ⊂ T R for i ∈ Z via f W = { ( bv ⊥ , v + dv ⊥ ) ∈ T R : b, d ∈ R } and f W i = iv + f W together with a map e g : T R − → T R − e g ( x, w ) = x, w e F ( x, w ) ! Observe that the restrictions e g | f W i : f W i → e V i are diffeomorphisms (check that ( x, w ) ( x, k v k h v,w i w ) is the inverse of e g | f W i and both are differentiable) and that (cid:16)e g | f W (cid:17) − ◦ e R ◦ e g | f W = e P for the map e P : f W → f W with e P ( γ (0) , ˙ γ (0)) = ( γ (1) , ˙ γ (1))Define diffeomorphisms e l i : f W i → R via e l i ( x, w ) = (cid:18) h x, v ⊥ ik v k , h w, v ⊥ ik v k (cid:19) Then we have e l ◦ e P ◦ (cid:16)e l (cid:17) − = ϕ , e L The conjugacy of the return map R and the time-1 map ϕ , L follows from the Z -invariance of the geodesic flow of e F . Example 2.5.
Let e F be the flat metric, i.e. e F ( x, w ) = k w k = p h w, w i and v ∈ Z \{ } prime. The Lagrangian e L (see definition 2.1) is then given by e L ( t, x, r ) = e F ( tv + xv ⊥ , v + rv ⊥ )= k v k p r We obtain that a function θ : R → R is an Euler-Lagrange solution if and only if ∂ x e L ( t, θ ( t ) , θ ′ ( t )) | {z } =0 − ∂ t ( ∂ r e L ( t, θ ( t ) , θ ′ ( t ))) = 0 which is equivalent to k v k θ ′′ √ θ ′ = 0 and thus θ is an Euler-Lagrange solution if and only if θ ′′ ( t ) = 0 for every t ∈ R .Consequently the solutions are of the form θ ( t ) = at + b for a, b ∈ R Thus, we have for the time-1 map ϕ , e L : R → R of the Lagrangian e Lϕ , e L ( x, y ) = ( x + y, y )6 o determine the map e R let ( x, w ) ∈ e V , i.e. ( x, w ) = ( bv ⊥ , rv + hv ⊥ ) for b, h ∈ R and r > , such that k rv + hv ⊥ k = 1 . The Euler-Lagrange solution θ with θ (0) = b and θ ′ (0) = hr is given by θ ( t ) = hr t + b This yields the following reparametrization γ of an e F -geodesic γ ( t ) = tv + (cid:18) hr t + b (cid:19) v ⊥ We calculate e R ( x, w ) = (cid:18) γ (1) , ˙ γ (1) k ˙ γ (1) k (cid:19) = v + ( hr + b ) v ⊥ , v + hr v ⊥ k v + hr v ⊥ k ! = (cid:18) x + 1 r w, w k w k (cid:19) = (cid:18) x + k v k h w, v i w, w (cid:19) To determine the map e P recall that e P = (cid:16)e g | f W (cid:17) − ◦ e R ◦ e g | f W with e g | f W i ( x, w ) = (cid:16) x, w k w k (cid:17) and (cid:16)e g | f W i (cid:17) − ( x, w ) = (cid:16) x, k v k h w,v i w (cid:17) For ( x, w ) ∈ f W we obtain e P ( x, w ) = (cid:16)e g | f W (cid:17) − ◦ e R (cid:16) x, w k w k (cid:17) = (cid:16)e g | f W (cid:17) − (cid:16) x + k v k h w,v i w, w k w k (cid:17) = (cid:18) x + k v k h w, v i w, k v k h w, v i w (cid:19) = ( x + w, w ) with the last equation following from ( x, w ) ∈ f W . To conclude the example we calculatefurther e l ◦ e P ◦ ( e l ) − ( x, y ) = e l ◦ e P ( xv ⊥ , v + yv ⊥ )= e l ( xv ⊥ + v + yv ⊥ , v + yv ⊥ )= ( x + y, y ) with the last line being equal to the time-1 map of e L . In this example the time-1 map of e L is a twist map (the so-called shear map ). We willsee later that in general the strict convexity of L guarantees that the time-1 map φ , L is conjugated to a finite composition of positive twist maps. This will allow us to applythe theory of monotone twist maps to return maps of the geodesic flow.7 .2 Twist maps of the cylinder and Mather’s theorem In this section we will briefly recall some of the theory on finite compositions of monotonetwist maps of the cylinder and Mather’s theorem about connecting orbits in Birkhoffregions of instability.
Definition 2.6.
Let f : Z → Z be a map of the cylinder with a lift F : R → R , suchthat F ( x, y ) = ( X ( x, y ) , Y ( x, y )) is a diffeomorphism with1. F is isotopic to the identity2. Twist condition: The map ( x, y ) ( x, X ( x, y )) is a diffeomorphism for every y
3. Exact symplectic: F ∗ ( λ ) − λ = dS for some function S : Z → R .Then f : Z → Z is called a monotone twist map of the cylinder. The map f is called positive if ∂ y X > everywhere. Here, λ = ydx is the Liouville form on R . For further reading on twist maps we refer the reader to [3]. The following definitionsgo along the lines of [4]. The following theorem (originally due to Birkhoff [2]) can befound in the Appendix of [4] for maps f ∈ P . Theorem 2.7.
Let f ∈ P . Any f -invariant homotopically non-trivial Jordan curve(homeomorphic to S ) Γ in the infinite cylinder is the graph of a Lipschitz function u : S → R . By P we denote the set of finite compositions of positive monotone twist maps of thecylinder. Definition 2.8.
Let f ∈ P . A Birkhoff region of instability B ⊂ Z is a compact f -invariant subset, such that1. The boundary of B consists of two components Γ − and Γ + , each homeomorphic toa circle and each non-contractible in Z
2. If Γ ⊂ B is any f -invariant subset, homeomorphic to a circle, non-contractible,then Γ = Γ − or Γ = Γ + For every f ∈ P there is a variational principle h (cf [4] chapter 1) and consequentlyone can define the notion of a minimal orbit of f . Much like if f is a twist map one canprove that every minimal orbit has a rotation number. This let’s us define the set M f,ω of minimal orbits of f with rotation number ω . Let Γ ⊂ Z be an f -invariant circle (andconsequently after theorem 2.7 a Lipschitz graph) with rotation number ω . It followsfrom [4] proposition 2.8 that Γ ⊂ M f,ω . If ω is irrational it follows from [4] proposition2.6 that the projection π : Z → S restricted to M f,ω is injective, and hence we haveΓ = M f,ω .A special case of theorem 4.1 in [4] is the following connecting theorem by J. Mather Theorem 2.9.
Let f ∈ P . If f has a Birkhoff region of instability B , such that therotation numbers ω − < ω + corresponding to the boundary graphs Γ − (respectively Γ + )are irrational, then there exists an f -orbit O that is α -asymptotic to Γ − and ω -asymptoticto Γ + . x ∈ S × R is called ω -asymptotic to Γ + if d ( f n ( x ) , Γ + ) convergesto zero as n → ∞ . It is called α -asymptotic to Γ − if d ( f − n ( x ) , Γ − ) converges to zero as n → ∞ .The next theorem can be found as theorem 39.1 in [3] in a more general setting. Itallows us to determine when the time-( s, t ) maps of a time-periodic Hamiltonian on Z are twist maps. Theorem 2.10.
Let H : S × S × R → R be a smooth Hamiltonian, such that the maps h t,x : R → R with h t,x : p ∂ p H ( t, x, p ) are diffeomorphisms for every ( t, x ) ∈ S × S (global Legendre condition). Then, givenany compact set K ⊂ S × R and starting time a , there exists an ǫ > , such that, forall < ǫ < ǫ the time- ( a, a + ǫ ) map of H is a twist map on K . To prove the main theorem we will define a time-periodic Lagrangian L on S , suchthat the Euler-Lagrange solutions of L correspond to reparametrizations of F -geodesics,whose lifts are graphs over the euclidean line e · R ⊂ R . After a perturbation of theLagrangian outside a tube S × ( − R, R ) we will be able to work with the conjugatedHamiltonian H . Theorem 2.10 will guarantee that the time-1 map ψ of H is in P ,i.e. a composition of finitely many positive monotone twist maps. We will then seethat the instability region of the geodesic flow corresponds to a Birkhoff region of insta-bility of ψ and hence Mather’s connecting theorem 2.9 will guarantee the existence ofan orbit connecting the boundary graphs of the Hamiltonian (respectively Lagrangian)system. Using the correspondence of Euler-Lagrange solutions and reparametrizationsof geodesics we obtain a connecting geodesic with the required asymptotic behaviour.Let v ∈ Z \{ } . Like in the previous section we associate a smooth and strictly convextime-periodic Lagrangian L on S to the Finsler metric F via L ( t, x, r ) = F (cid:16) tv + xv ⊥ , v + rv ⊥ (cid:17) We perturb the Lagrangian for large values of | r | : For R >
D >
0, such that there is an extension L R of L | S × S × ( − R,R ) to S × S × R with1. L R ( t, x, r ) = D r for large values of | r | L R is strictly convex with second partial derivative bounded and bounded awayfrom zero, i.e. 0 < C < ∂ rr L R < C for a constant C > Remark 3.1.
Observe that the Euler-Lagrange solutions of L R for large values of | r | arejust straight lines, and hence the integral curves of the Euler-Lagrange vector field existfor all times. To see this lift L R to a 1-periodic Lagrangian on R and observe that forlarge values of | r | the Euler-Lagrange equation reduces to ¨ γ = 0 for functions γ : R → R .Thus, for large | r | the solutions are straight lines. Consequently, for large C > we havethat the compact sets S × S × [ − C, C ] are invariant under the Euler-Lagrange flow,and hence every solution exists for all times, i.e. the flow is complete. The following theorem is a corollary of theorem 2.2.9 heorem 3.2.
Let F be a Finsler metric on T and let L R be the above Lagrangian foran R > . Let θ : R → R be a smooth function with − R < θ ′ < R and let γ : R → R bethe curve given by γ ( t ) = tv + θ ( t ) v ⊥ Then γ is a reparametrization of a lift of an e F -geodesic if and only if θ is an Euler-Lagrange solution of e L R .Proof. The theorem follows immediately for a curve θ : R → R with − R < θ ′ < R since the perturbed Lagrangian L R agrees with the unperturbed Lagrangian L in aneighbourhood of the curve. Consequently, the Euler-Lagrange equations agree on thisneighbourhood and hence θ is an Euler-Lagrange solution for L R if and only if θ is anEuler-Lagrange solution for L .To define the associated Hamiltonian H R : S × S × R → R of the Lagrangian L R weneed to consider the Legendre-transform L t : S × R → S × R with L t : ( x, r ) ( x, ∂ r L ( t, x, r ))For a moment we will omit the dependence on R and just write L instead of L R . Notethat the Legendre-transform is a diffeomorphism because ∂ rr L is bounded away fromzero. For the above Lagrangian L we can define the associated Hamiltonian H : S × S × R → R via H ( t, x, ∂ r L ( t, x, r )) = ∂ r L ( t, x, r ) r − L ( t, x, r ) (1)The time-( s, t ) maps of the Lagrangian and associated Hamiltonian for s < t are conju-gated via the Legendre transform ψ s,t = L t ◦ ϕ s,t ◦ L − s It follows from (1) and the strict convexity of L that C < ∂ pp H < C for a constant
C > Proposition 3.3.
The time-1 map ψ : Z → Z of the associated Hamiltonian H R : Z → R of the Lagrangian L R is in P .Proof. Since ∂ pp H ( t, x, p ) = 0 the maps h t,x are local diffeomorphisms. Injectivity followsfrom ∂ pp H > ∂ pp H being bounded away from zero.Consequently, for every large tube S × [ − P, P ] (those tubes are actually invariant underthe time-dependent flow of H if we chose P to be large enough) there is an ǫ > ψ ,ǫH to the tube is a twist map. Outside those tubes theLagrangian L reduces to L ( t, x, r ) = D r and hence with the Legendre-transform beingequal to L t ( x, r ) = ( x, Dr ) we find that for the Hamiltonian H it holds that H ( t, x, y ) = y D for large | y | Thus, the Hamiltonian vector field is identical to X H ( t, x, y ) = ( yD ,
0) outside large tubesand hence the time-1 map outside large tubes is identical to a shear-map of twist D .Since the starting time a in theorem 2.10 is arbitrary we can piece the time-1 map ψ = ψ , H : S × R → S × R together from finitely many twist maps, i.e. there exist realnumbers 0 = ǫ < · · · < ǫ n = 1, such that ψ = ψ ǫ n − ,ǫ n H ◦ · · · ◦ ψ ǫ ,ǫ H . and ψ ǫ i ,ǫ i +1 H is a twist map for every i . 10 emma 3.4. If Λ ⊂ ST is an invariant torus with bounded direction (with respect to v ) and R > is chosen large enough, then the time-1 map of the Lagrangian L R hasa corresponding invariant circle Γ . If there are no closed geodesics on Λ the invariantcircle has irrational rotation number. Remark 3.5.
By ”Corresponding” in lemma 3.4 we mean that every Euler-Lagrangesolution t ( θ s ( t ) , ∂ t θ s ( t )) of e L R with ( θ s (0) , ∂ t θ s (0)) = ( s, ∂ t θ s (0)) ∈ Γ corresponds to a reparametrization γ s of the geodesic in Λ with γ s (0) = sv ⊥ .Proof. (of lemma 3.4) It is useful to see Λ as a Z -periodic continuous graph in S R ∼ = R × S , which is invariant under the geodesic flow φ : S R × R → S R of the liftedFinsler metric. First we want to define a reparametrization of the restricted geodesicflow φ : Λ × R → Λ on Λ. Since the projections of the geodesics on Λ are graphs over v R and because of the periodicity of Λ, we have h π ◦ φ ( w, t ) , v i ≥ const > w, t ) ∈ Λ × R . We define a smooth map k : S R × R → S R × R via k ( w, t ) = (cid:18) w, Z t h π ◦ φ ( w, s ) , v i ds (cid:19) It follows from (2) that k is a local diffeomorphism in a neighbourhood of every ( w, t ) ∈ Λ × R . Furthermore, it follows also from (2) that the restriction of k to Λ × R is bijectiveand consequently the map k : Λ × R → Λ × R is a homeomorphism. Hence, we havethe inverse k − : Λ × R → Λ × R , which is continuous and differentiable in the secondcomponent. We define a reparametrization ˆ φ : Λ × R → Λ of the geodesic flow viaˆ φ ( w, t ) = φ ◦ k − ( w, k v k t )Note, that since the graph Λ is Z -periodic, we have for z ∈ Z π ◦ ˆ φ ( w ′ , t ) = π ◦ ˆ φ ( w, t ) + z if π ( w ′ ) = π ( w ) + z (3)From k ◦ k − = id we calculate that h ∂ t ( π ◦ ˆ φ )( w, t ) , v i = k v k (4)for every ( w, t ) ∈ Λ. Hence we can write the map π ◦ ˆ φ : Λ × R → R as π ◦ ˆ φ ( w, t ) = (cid:16) k v k h π ( w ) , v i + t (cid:17) v + θ ( w, t ) v ⊥ (5)where θ : Λ × R → R is continuous and differentiable in the second component. For i ∈ Z we define subsets Λ i ⊂ Λ viaΛ i = { w ∈ Λ | π ( w ) ∈ R v ⊥ + iv } Note, that for w ∈ Λ we get from (5) π ◦ ˆ φ ( w, t ) = tv + θ ( w, t ) v ⊥ (6)As an intermediate step we will show thatˆ φ ( w, t + t ′ ) = ˆ φ ( ˆ φ ( w, t ′ ) , t ) for every t, t ′ ∈ R (7)11o see this we define curves γ , γ : R → Λ via γ ( t ) = ˆ φ ( w, t ′ + t )and γ ( t ) = ˆ φ ( ˆ φ ( w, t ′ ) , t )Since the curves are both reparametrizations of a geodesic on Λ with γ (0) = γ (0) theyare reparametrizations of the same geodesic. From equation (4) it follows for every t that h ∂ t ( π ◦ γ )( t ) , v i = h ∂ t ( π ◦ γ )( t ) , v i together with γ (0) = γ (0) we obtain h π ◦ γ ( t ) , v i = h π ◦ γ ( t ) , v i for every t ∈ R . Since every projection of a geodesic in Λ is a graph over R v this impliesthat π ◦ γ ( t ) = π ◦ γ ( t )for every t ∈ R . Finally, since Λ is a graph over R we have γ = γ and thus we haveproven (7). Next we will prove that π ◦ ˆ φ ( w, ∈ Λ i +1 for every w ∈ Λ i (8)To see this note that from (5) and h π ( w ) , v i = i k v k it follows that h π ◦ ˆ φ ( w, , v i = ( i + 1) k v k We can now define a graph Γ ⊂ S × R , which we will later prove to be an invariantcircle of the time-1 map of L R . We setΓ = { ( θ ( w, , ∂ t θ ( w, | w ∈ Λ } To see that Γ is actually a graph in S × R note that for every w ∈ Λ we have π ( w ) = sv ⊥ for an s ∈ R . Consequently it follows from (6) that θ ( w,
0) = s if π ( w ) = sv ⊥ . To seethat Γ is periodic (i.e. a subset in S × R ) note that if w, w ′ ∈ Λ with π ( w ) = sv ⊥ and π ( w ′ ) = ( s + 1) v ⊥ we have π ◦ ˆ φ ( w ′ , t ) = π ◦ ˆ φ ( w, t ) + v ⊥ because of (3). Thus itfollows from (6) that ∂ t θ ( w,
0) = ∂ t θ ( w ′ ,
0) and hence Γ can be seen as the graph of the1-periodic function h : S → R with h ( s ) = ∂ t θ (Λ( sv ⊥ ) , h is a composition of continuous functions. Wecan thus pick an R > ⊂ S × ( − R, R ). To see now that Γ isinvariant under the time-1 map ϕ : S × R → S × R of the Lagrangian L R observe thatfor a w ∈ Λ it follows from (5) and (7) that θ ( w, t ) = 1 k v k h π ◦ ˆ φ ( w, t ) , v ⊥ i = 1 k v k h π ◦ ˆ φ ( ˆ φ ( w, , t ) , v ⊥ i Since ˆ φ ( w, ∈ Λ (see (8)) and because of the invariance (3) there exists a w ′ ∈ Λ ,such that the last line is equal to 12 1 k v k h π ◦ ˆ φ ( w ′ , t ) + v, v ⊥ i = 1 k v k h π ◦ ˆ φ ( w ′ , t ) , v ⊥ i = θ ( w ′ , t )for every t ∈ R . This implies that θ ( w,
1) = θ ( w ′ ,
0) and ∂ t θ ( w,
1) = ∂ t θ ( w ′ ,
0) and henceusing theorem 3.2 the time-1 map ϕ with ϕ ( θ ( w, , ∂ t θ ( w, θ ( w, , ∂ t θ ( w, ψ of the associated Hamiltonian H R . Assume now,that Γ has rational rotation number. Consequently there exists a periodic point x =( θ ( w, , ∂ t θ ( w, w is closed,which is a contradiction to Λ having no closed geodesics. Lemma 3.6.
Let U ⊂ ST be an instability region with boundary graphs Λ − and Λ + .Assume that U has bounded direction (with respect to v = e ). If R > is chosen largeenough, such that the time-1 map of the Lagrangian L R has two corresponding invariantcircles Λ − and Λ + , then the only invariant circles in the subset B ⊂ S × R enclosed by Λ − and Λ + are the boundary circles Λ − and Λ + .Proof. To prove this theorem we have to show that if we have an invariant circle Γ ofthe map ϕ , which is not equal to Γ − or Γ + then we also have an φ -invariant torus Λ notequal to Λ − or Λ + . Now, assume that Γ ⊂ S × R is such an invariant circle of ϕ . Itfollows from theorem 2.7 that Γ is equal to the graph { ( s, h ( s )) | s ∈ S } of a Lipschitzfunction h : S → R . For every fixed s ∈ S the map t ϕ ,t ( s, h ( s )) is a trajectory ofthe Euler-Lagrange flow on S × R . We define maps θ s : R → R via( θ s ( t ) , θ ′ s ( t )) := ϕ ,t ( s, h ( s ))Note that the maps ( s, t ) θ ( s, t ) and ( s, t ) θ ′ s ( t ) are continuous, since θ s and θ ′ s arethe compositions of ϕ ,t ( s, h ( s )) with the projections of S × R onto S respectively R .We extend θ to a map θ : R → R , ( s, t ) θ s ( t ) by seeing Γ as a 1-periodic graph in R and using the Euler-Lagrange flow ϕ of the lifted Lagrangian e L in the definition of θ . We then have ( θ s +1 ( t ) , θ ′ s +1 ( t )) = ( θ s ( t ) + 1 , θ ′ s ( t ))Observe that since ϕ , = id we have θ s (0) = s for every s ∈ R . Note that for r = p we have θ r ( t ) = θ p ( t ). To see this assume that we have θ r ( t ) = θ p ( t ). Since the set ϕ ,t (Γ) = { ( θ s ( t ) , θ ′ s ( t )) | s ∈ R } is invariant under the map ϕ t,t +1 it is a Lipschitz graph.Hence it also holds that θ ′ r ( t ) = θ ′ p ( t ). We have the associated reparametrizations ofgeodesics γ r ( t ) = tv + θ r ( t ) v ⊥ and γ p ( t ) = tv + θ p ( t ) v ⊥ . This would imply that thegeodesics meet at the point γ r ( t ) = γ p ( t ) with ˙ γ r ( t ) = ˙ γ p ( t ), which is not possiblebecause geodesics can not become tangent to each other. As mentioned above, since thesets Γ t := ϕ ,t (Γ) are Lipschitz graphs there exist periodic Lipschitz functions h t : R → R with Γ t = { ( s, h t ( s ) | s ∈ R } with h t ( θ s ( t )) = θ ′ s ( t ) . We define a function g : R → R via g : ( s, t ) h t ( s ) .
13e will now show that g is continuous. To see this take real sequences t n → t and s n → s . Since Γ t is a Lipschitz graph there exists exactly one r n ∈ R for every n ∈ N with θ r n ( t n ) = s n . The sequence r n is bounded. To see this observe that it follows from the continuity ofthe map ( s, t ) θ s ( t ) and the graph property of the sets Γ t that the map s θ s ( t ) isstrictly increasing for every fixed t . Now, assume that the sequence r n is unbounded.Without loss of generality we assume that r n is unbounded from above with r n ≥ n forevery n ∈ N . Then we have θ r n ( t n ) ≥ θ n ( t n ) = θ ( t n ) + n and thus the sequence θ r n ( t n ) is also unbounded. This contradicts the fact that s n converges to s and consequently the sequence r n has to be bounded. To see that r n actually converges assume that there are two limit points r = lim i →∞ r n i and p =lim j →∞ r n j . Since we have θ r ni ( t n i ) = s n i it follows from the continuity of θ that θ r ( t ) = s . Analogously we obtain θ p ( t ) = s . As above it follows that r = p . To see that g is continuous observe that h t n ( s n ) = h t n ( θ r n ( t n )) = θ ′ r n ( t n )and hence it follows from the continuity of ( s, t ) θ ′ s ( t ) thatlim n →∞ h t n ( s n ) = lim n →∞ θ ′ r n ( t n ) = θ ′ r ( t ) = h t ( s )Thus we have proven that g : ( s, t ) h t ( s ) is continuous. To construct an invariantgraph in ST , observe that since the reparametrized geodesics γ s with γ s ( t ) = tv + θ s ( t ) v ⊥ can not cross and since γ s +1 = γ s + v ⊥ the geodesics γ s foliate R . This allows us todefine a graph Λ ⊂ S R viaΛ = (cid:26)(cid:18) γ s ( t ) , ˙ γ s ( t ) F ( ˙ γ s ( t )) (cid:19) : ( s, t ) ∈ R (cid:27) Furthermore, Λ is φ -invariant since it only consists of images of geodesics. We define amap X : R → R via X : x v + h D x, v k v k E (cid:18)(cid:28) x, v ⊥ k v k (cid:29)(cid:19) v ⊥ Observe that X is continuous (because h is) and maps γ s ( t ) to ˙ γ s ( t ). Hence Λ is thegraph of the continuous function x X ( x ) F ( X ( x ))What is left to prove is that X is Z periodic, i.e. Λ is actually a graph in ST . We willshow that if we assume our instability region to have bounded direction with respect to v ∈ Z − { } then X is v Z × v ⊥ Z -periodic. As a special case we get that X is Z -periodicif v is chosen to be equal to e . To prove that X (and thus XF ( X ) ) is v Z × v ⊥ Z -periodicobserve that for every ( s, t ) ∈ R we have γ s ( t ) + v ⊥ = tv + θ s ( t ) v ⊥ + v ⊥ = tv + ( θ s ( t ) + 1) v ⊥ = tv + θ s +1 ( t ) v ⊥ = γ s +1 ( t )14ince θ ′ s ( t ) = θ ′ s +1 ( t ) we have ˙ γ s ( t ) = ˙ γ s +1 ( t ) and thus X is v Z -periodic. From theinvariance of Γ t under ϕ t,t +1 we obtain that for every s ∈ R there exists an r ∈ R with ϕ t,t +1 ( θ r ( t ) , θ ′ r ( t )) = ( θ s ( t ) , θ ′ s ( t ))Thus we have γ s ( t ) + v = ( t + 1) v + θ s ( t ) v ⊥ = ( t + 1) v + θ r ( t + 1) v ⊥ = γ r ( t + 1)And since θ ′ r ( t + 1) = θ ′ s ( t ) we have ˙ γ r ( t + 1) = ˙ γ s ( t ). Hence we have shown the v Z -periodicity of X .We can now prove the main theorem. Proof. (theorem 1.4)If
R > ϕ of L R has twoinvariant circles Γ − and Γ + in S × ( − R, R ), which correspond to the invariant bound-ary graphs. With lemma 3.6 it follows that the region bounded by the correspondinginvariant circles of the Hamiltonian time-1 map is a Birkhoff region of instability. Thus,after going back to the Lagrangian setting, we obtain from theorem 2.9 a point x in thearea between the boundary circles, and sequences t ′ n → ∞ , s ′ n → −∞ withlim n →∞ ϕ ,t ′ n ( x ) ∈ Γ + and lim n →∞ ϕ ,s ′ n ( x ) ∈ Γ − Using theorem 3.2 we obtain after reparametrization a geodesic c : R → T and sequences t n → ∞ , s n → −∞ with lim n →∞ ˙ c ( t n ) ∈ Λ + and lim n →∞ ˙ c ( s n ) ∈ Λ − References [1] D. Bao, S.-S. Chern, Z. Shen,
An Introduction to Riemann-Finsler Geometry ,Springer (2000)[2] G. D. Birkhoff,
Surface transformations and their dynamical applications , CollectedMath. Papers, vol. 2, (1920)[3] C. Gol´e,
Symplectic Twist Maps , World Scientific Publishing, 2001.[4] J. Mather,
Variational Construction of Orbits of Twist Diffeomorphisms , Journal ofthe American Math. Soc., 4 (1991), 207 - 263[5] J. P. Schr¨oder,