Simplicity of the Lyapunov Spectrum for classes of Anosov flows
SSIMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS INNEGATIVE CURVATURE
DANIEL MITSUTANI
Abstract.
We study the Lyapunov spectrum of the geodesic flow of negatively curved1/4-pinched Riemannian manifolds. We show that the space of metrics with simple Lya-punov spectrum with respect to the equilibrium state of any H¨older continuous potentialis C k open and dense. The proof relies on a symbolic coding of the flow to apply thesimplicity criterion of Avila and Viana in [AV07] and on perturbational results for k -jetsof geodesic flows established in [KT72]. Introduction
The relationship between geometric data of a Riemannian manifold and dynamical ob-jects associated to its geodesic flow, such as entropy, marked length spectrum, and reg-ularity of splitting, has long been the object of study from the point of view of rigiditytheory.Among the invariants considered in more recent study are the Lyapunov exponentsassociated to the derivative cocycle of geodesic flows. In [Bu17], the Lyapunov exponentsover periodic measures are shown to be a rigid invariant of locally symmetric spaces, inthe sense that any Riemannian metric on a manifold homotopy equivalent to a locallysymmetric space with the same Lyapunov exponents over periodic measures as a locallysymmetric space is in fact locally symmetric.In this paper, we study a problem complementary to the one above: what can be saidabout the Lyapunov spectrum of the geodesic flow of most metrics? Lyapunov exponentswith multiplicity higher than 1 are expected to correspond to a positive codimension set,and conditions for simplicity have been well-studied in the setting of random products ofmatrices [GR86], dominated cocycles [BV04], and countable state shifts [AV07].Given such a wide range of criteria for simplicity, it is natural to ask whether any ofthem hold generically in the space of metrics. We give a positive answer to this question.
Date : February 18, 2021. a r X i v : . [ m a t h . D S ] F e b DANIEL MITSUTANI
Fix a closed smooth manifold M n +1 . For k ≥ G k the set of C k -Riemannianmetrics on M with sectional curvatures 1 ≤ − K < C k -topology. Theorem 1.1.
For fixed k ≥ there exists an open and dense set in G k of metrics suchthat with respect to the equilibrium state of any H¨older potential the derivative cocycleof the geodesic flow has simple Lyapunov spectrum, i.e., all its Lyapunov exponents havemultiplicity one. Remark 1.2.
From this point on we fix a k ≥ . The proof is based on the simplicity criterion of Avila and Viana [AV07], which is itselfan improvement of the criterion of Bonatti and Viana [BV04]. Originally, their result wasused to show that generically the spectrum is simple in the context of H¨older continuousdominated cocycles over a fixed hyperbolic base with respect to measures with local productstructure.Two main ingredients are used to adapt their result to geodesic flows: perturbationaltools for derivatives of geodesic flows, which are used in roughly unmodified form from thework of Klingenberg and Takens [KT72], and a carefully constructed symbolic coding ofthe geodesic flow as a suspension, given by a Markov partition.One main difficulty particular to the setting of R -cocycles which was already present in[BV04] arises in attempting to perturb the norms of pairs of complex eigenvalues generically.In [BV04], through the introduction of rotation numbers which vary continuously withthe perturbation for orbits near a periodic point, a small rotation on a periodic orbit ispropagated to an arbitrarily large one for a homoclinic point, which can then be made tohave real eigenvalues.While such rotation numbers are well defined for the particular perturbation of thecocycle introduced in [BV04], a general construction which allows for perturbations onthe base system has only been introduced recently in [Go20]. However, the constructionsin [Go20] do not apply directly to flows, and so we introduce new ideas to control theeigenvalues of the cocycle in the continuous time setting.The outline of the paper is as follows: in Section 2 we summarize the main results of[KT72] and [AV07] and introduce other necessary concepts for the proof; for a more basicintroduction to Lyapunov exponents and cocycles we refer the reader to [Vi14] and forbackground on geodesic flows [Pa99]. In Section 3 we prove genericity of the equivalentnotions of “pinching” and “twisting” that appear in [AV07] for the derivative cocycle of IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 3 the geodesic flow in the space of metrics. Subsection 3.2 contains the proof of the maintechnical result used about perturbing complex eigenvalues. Finally, in Section 4 the mainTheorem 1.1 is proved using a suspension construction.1.1.
Acknowledgements.
I would like to thank Amie Wilkinson for all her suggestionsand continued guidance in the process of research leading up to this paper, and also forher help in reviewing the text. 2.
Preliminaries
Throughout, all spaces of metrics are on a fixed closed smooth manifold M n +1 . Since themetric on M varies, it is useful to work on the sphere bundle over M of oriented directionsof the tangent space, which we denote by SM , rather than on the unit tangent bundle.When a metric g is fixed, T g M is canonically diffeomorphic to SM , and one can pullbackthe Sasaki metric from T g M to SM .2.1. Fiber Bunching and Holonomies.
We require the 1/4-pinching hypothesis to en-sure that the cocycle given by the restriction of the derivative to the unstable bundle isfiber bunched. This is crucial to apply the main results of Bonatti and Viana explainedSection 2.3.By well-known results on regularity of the splitting, which can be found in [HP75],any C Riemannian metric g of negatively 1/4-pinched curvature has an Anosov splitting T SM = E u ⊕ E s ⊕ E c for its geodesic flow which is C . Moreover, denoting by ϕ tg thegeodesic flow, for any v ∈ E u and u ∈ E s there exists C > Ce t (cid:107) v (cid:107) ≤ (cid:107) Dϕ tg ( v ) (cid:107) Sas ≤ Ce t (cid:107) v (cid:107) ,Ce t (cid:107) u (cid:107) ≤ (cid:107) Dϕ − tg ( u ) (cid:107) Sas ≤ Ce t (cid:107) u (cid:107) . From the above, it is clear that the cocycle given by A t := Dϕ tg | E u on the C bundle E u satisfies the following definition: Definition 2.1 (Fiber Bunching) . A β -H¨older continuous cocycle A : E × R → E , wherewe write A tx : E x → E Φ( x ) for a fixed t and fiber over x , over an Anosov flow Φ t : M → M ,is said to be α -fiber bunched if α ≤ β and there exists T > such that for all p ∈ M and t ≥ T : (cid:107) A tp (cid:107)(cid:107) A − tp (cid:107)(cid:107) D Φ t | E s (cid:107) α < DANIEL MITSUTANI (cid:107) A tp (cid:107)(cid:107) A − tp (cid:107)(cid:107) D Φ − t | E u (cid:107) α < C , we may take α = 1, and then the inequalities (1) abovefor || Dϕ tg ( v ) || Sas imply that Dϕ tg | E u is indeed 1-fiber bunched. Fiber bunching may beunderstood as a form of partial hyperbolicity on the projectivization of the fiber bundle,with the fibers composing the center direction and the base system the stable and unstabledirections. Existence of strong stable and unstable manifolds then translates into theexistence of holonomies: Theorem 2.2. [KS13]
Suppose A is β -H¨older and fiber bunched over a base system asin Definition 2.1. Then the cocycle admits holonomy maps h u , that is, a continuous map h u : ( x, y ) → h ux,y , x ∈ M , y ∈ W uloc ( x ) , such that: (1) h ux,y is a linear map E x → E y , (2) h ux,x = Id and h uy,z ◦ h ux,y = h ux,z , (3) h ux,y = ( A ty ) − ◦ h u Φ t ( x ) , Φ t ( y ) ◦ A tx for every t ∈ R .Moreover, the holonomy maps are unique, and, fixing a system of linear identifications I xy : E x → E y , see [KS13] , they satisfy: (cid:107) h ux,y − I x,y (cid:107) ≤ Cd ( x, y ) β . Using property (3), one may extend these holonomies for all y ∈ W cu ( x ) (as opposed to W uloc ( x )), and such holonomies are denoted by h cu .2.2. Generic Metrics.
We describe now the perturbational results of [KT72] that will beused to perturb the derivative cocycle by perturbing the metric.For a fixed embedded compact interval or closed loop γ ⊆ SM , the set of metrics forwhich γ is an orbit segment of the geodesic flow is denoted by G kγ ⊆ G k . For a fixed g ∈ G kγ ,pick local hypersurfaces Σ and Σ in SM that are transverse to ˙ γ ( t ) ∈ T SM at t = 0 and t = 1, respectively. This allows us to define a Poincar´e map P g : Σ ⊇ U → Σ , where U is a neighborhood of γ (0), by mapping ξ ∈ U to ϕ t g ( ξ ), where t is the smallestpositive time such that ϕ t g ( ξ ) ∈ Σ . By the Implicit Function Theorem and the fact that ϕ tg is C k − , the map P is C k − .By projecting the tangent spaces of Σ i =0 , to E u ⊕ E s one may give Σ i =0 , a symplecticstructure which is preserved by the Poincar´e map, since the symplectic form is invariant IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 5 by the geodesic flow [KT72]. With g fixed, we let G kg ,γ ⊆ G kγ be the set of metrics suchthat π ( γ (0)) , π ( γ (1)) / ∈ ( g − g ) ( π : SM → M is the canonical projection map) that is,metrics unperturbed at the ends of the fixed geodesic segment γ relative to g .We will repeatedly use the main result on generic metrics established by Klingenbergand Takens in [KT72] to perturb the metric g : Theorem 2.3. [KT72, Theorem 2]
Suppose g ∈ G ∞ γ , and let Q be some open dense subsetof the space of ( k − -jets of symplectic maps (Σ , γ (0)) → (Σ , γ (1)) .Then there is arbitrarily C k -close to g a g (cid:48) ∈ G kg ,γ such that P g (cid:48) ∈ Q , where P g (cid:48) :(Σ , γ (0)) → (Σ , γ (1)) is the Poincar´e map for the geodesic flow of g (cid:48) . Remark 2.4.
The technical assumption that g is C ∞ needed in [KT72] is virtually harm-less, since by smooth approximation G ∞ ⊆ G k is dense for all k . We will need two additional facts about how these perturbations can be made, both ofwhich follow directly from the proof of Theorem 2.3 in [KT72]:
Proposition 2.5.
Let h := g (cid:48) − g ∈ S T ∗ SM , where g (cid:48) and g given as in the statementof Theorem 2.3. For any tubular neighborhood V of γ , h can be taken to satisfy: (1) supp ( h ) ⊆ V ; (2) For a system of coordinates { x , ..., x n − } on V where ∂ x is parallel to the geodesicflow, the k -jets of h (where h = h ij dx i dx j ) vanish identically along { x = 0 } .In particular, this implies that the parametrization of γ by arc-length in g is thesame as that in g (cid:48) , i.e., the geodesic flow for both metrics agree along γ . Let J k − s denote the Lie group of ( k − C k − symplectic maps ( R n , → ( R n , (cid:80) i dx i ∧ dy i . If O is a closed orbit, we may take v := γ (0) = γ (1) ∈ O and fix Σ := Σ = Σ , so by Darboux’s theorem we may choosecoordinates that identify the space of ( k − C k − symplectic maps (Σ , v ) → (Σ , v )with J k − s . Corollary 2.6. If O is a closed geodesic for g ∈ G ∞O and Q ⊆ J k − s is an open denseinvariant ( Q satisfies σQσ − = Q for any σ ∈ J k − s ) set then there is arbitrarily C k -closeto g a g (cid:48) ∈ G k O such that for any v ∈ O and any Σ a transverse at v , P g (cid:48) ∈ Q , where P g (cid:48) = P ( v, Σ) is the Poincar´e return map for the geodesic flow of g (cid:48) . DANIEL MITSUTANI
Proof.
Choice of a different section Σ or a different point v of the orbit changes P g (cid:48) byconjugation, so the property that P g (cid:48) ∈ Q needs only be assured at one fixed point andone fixed section, which is done by Theorem 2.3. (cid:3) Remark 2.7.
Since the map π k − : J k − s → J s ∼ = Sp (2 n ) is a submersion, for Q an opendense invariant subset of Sp (2 n ) , ( π k − ) − ( Q ) is an open dense invariant subset of J k − s ,so in the statement of Corollary 2.6 we may take an open dense invariant Q ⊆ Sp (2 n ) instead, while the approximation is still in G k .In the context of Theorem 2.3, the analogous observation holds; that is, one may take Q to be an open dense subset of -jets of symplectic maps (Σ , γ (0)) → (Σ , γ (1)) , andapproximate in G k . Simplicity Criteria.
In the symbolic setting, Bonatti and Viana established in[BV04] a criterion for dominated linear cocycles over a subshift of finite type to havesimple Lyapunov spectrum, i.e. all Lyapunov exponents of multiplicity one, with respectto a large class of measures. The criterion requires that one has “pinching” along a periodicorbit with “twisting” of the exterior powers of the holonomy map along a homoclinic orbit.It was later extended by Avila and Viana to countable state shifts in [AV07]; moreover,their criterion is also sharper in that it only requires a condition on the holonomy map andnot its exterior powers. This sharpening turns out to be useful for us as we work in thespace of symplectic transformations.Following the notation in [AV07], we let ˆ f be the shift on the space ˆΣ = N Z . Letˆ A : ˆΣ T → GL ( d, R ) a measurable cocycle over ˆ f . In [AV07] it is also explained how theresults extend to subshifts of countable (and hence finite as well) type. Definition 2.8 (Domination) . ˆ A is dominated if there exists a distance d in ˆΣ and con-stants θ < and ν ∈ (0 , such that, up to replacing ˆ A by some power ˆ A N : (1) d ( ˆ f (ˆ x ) , ˆ f (ˆ y )) ≤ θd (ˆ x, ˆ y ) and d ( ˆ f − (ˆ x ) , ˆ f − (ˆ y )) ≤ θd (ˆ x, ˆ y ) for every ˆ y ∈ W sloc (ˆ x ) and ˆ z ∈ W uloc (ˆ x )(2) ˆ x → ˆ A (ˆ x ) is ν -H¨older continuous and (cid:107) ˆ A (ˆ x ) (cid:107)(cid:107) ˆ A − (ˆ x ) (cid:107) θ ν < for every ˆ x ∈ ˆΣ . Domination is the analogue of fiber-bunching for symbolic systems used in [BV04], andagain is used to establish the existence of stable and unstable holonomies:
IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 7
Theorem 2.9. [BV04, Proposition 1.2] If ˆ A is either dominated or constant on eachcylinder, there exists a family φ u ˆ x, ˆ y for each ˆ x, ˆ y ∈ ˆΣ in the same local unstable manifold of ˆ f and there exists C > such that: (1) φ u ˆ x, ˆ x = id and φ u ˆ x, ˆ y = φ u ˆ x, ˆ z ◦ φ u ˆ z, ˆ y , (2) the holonomies are equivariant with respect to the cocycle: ˆ A ( ˆ f − (ˆ y )) ◦ φ u ˆ f − ( x ) , ˆ f − ( y ) ◦ ˆ A − (ˆ x ) = φ u ˆ x, ˆ y , (3) (cid:107) φ u ˆ x, ˆ y − id (cid:107) ≤ C d (ˆ x, ˆ y ) ν . Remark 2.10.
With the appropriate definition of a “hyperbolic system” which includesboth Anosov maps and shifts it is possible to reduce the definition of fiber-bunching anddomination to one.However, for the sake of using the results in their original forms from the references, weuse both notions separately.
Remark 2.11.
Throughout the paper we use h to denote holonomies over an Anosov flow,in our setting a geodesic flow, and φ to denote holonomies over a symbolic system, as inTheorem 2.2 and Theorem 2.9. Finally, we define the “simple” cocycles for which simplicity holds; note that (P) and (T)below roughly correspond to pinching and twisting, respectively, on the monoid generatedby the cocycle using the dynamics of orbits related to ˆ p , and so we adopt such terminologyfor them: Definition 2.12 (Simple cocycles) . Suppose ˆ A : ˆΣ → GL ( d, R ) is either dominated orconstant on each cylinder of ˆΣ . We say that ˆ A is simple if there exists a periodic point ˆ p and a homoclinic point ˆ z associated to ˆ p such that: (P) the eigenvalues of ˆ A on the orbit of ˆ p have multiplicity 1 and distinct norms – let ω j ∈ R P d − represent the eigenspaces, for ≤ j ≤ d ; and (T) { ψ ˆ p, ˆ z ( ω i ) : i ∈ I } ∪ { ω j : j ∈ J } is linearly independent, for all subsets I and J of , ..., d with I + J ≤ d . An invariant probability measure ˆ µ has local product structure if for every cylinder[0 : i ]: ˆ µ | [0 : i ] = ψ · ( µ + × µ − ) DANIEL MITSUTANI where ψ : [0 : i ] → R is continuous and µ + and µ − are the projections of ˆ µ | [0 : i ] to spacesof one-sided sequences indexed by positive and negative indices respectively. For instance,this property holds for every equilibrium state of ˆ f associated to a H¨older potential [Bo75]. Theorem 2.13. [AV07, Theorem A] If ˆ A is a simple cocycle then it has Lyapunov expo-nents of multiplicity one with respect to any ˆ µ with local product structure. In [BV04] this criterion is used to prove that C α -generic fiber bunched cocycles over asymbolic hyperbolic system have simple Lyapunov spectrum. Here, by associating to thegeodesic flow of a negatively curved pinched a symbolic system through a Markov partitionwe will apply the criterion to prove the main Theorem 1.1.2.4. Rotation Numbers.
As indicated in the introduction, in order to perturb awaycomplex eigenvalues by a small rotation, one needs the formalism of rotation numbers,which we introduce in complete form here. We roughly follow the discussion in Section 3of [Go20].As a brief introduction, recall that for an orientation preserving homeomorphism of thecircle f : S → S , the Poincar´e rotation number ρ ( f ) ∈ S = R / π of f is defined as: ρ ( f ) = lim n →∞ ˜ f n ( x ) − xn (mod 2 π ) , for a lift ˜ f : R → R of f . This limit always exists and is independent of the choice of x ∈ R and the lift ˜ f . For an orientation reversing homeomorphism we define ρ ( f ) = 0.The Poincar´e rotation number measures, on average, how much an element is rotatedby an application of f and is a conjugation invariant, i.e., ρ ( g − f g ) = ρ ( f ), for g alsoa homeomorphism of S . In what follows we extend this definition for cocycles on circlebundles.Throughout this section, let X a compact metric space and Φ t a continuous flow on X .Let M Φ ( X ) be the space of probability measures on X invariant under Φ t with the weak-*topology.For our purposes, it will suffice to work with trivial bundles E = X × S , and a continuouscocycle A : R × E → E over Φ t , that is, a continuous family of functions A tx := π ( A ( t, x, · )) : S → S , where π : E → S is the natural projection map, such that A s + tx = A t Φ s x ◦ A sx . IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 9
Then for ( x, θ ) ∈ E , the map t (cid:55)→ A tx ( θ ) is a continuous map from R → S , so it maylifted to some w x,θ : R → R . Let ˜ w x,θ ( t ) := w x,θ ( t ) − w x,θ (0), so that ˜ w does not dependon the lift w . Definition 2.14 (Pointwise Rotation Number) . The average rotation number ρ : X → R is defined by the limit: ρ ( x ) = lim t →∞ ˜ w x,θ ( t ) t , whenever it exists, and is independent of the choice of θ . Indeed, for any θ, θ (cid:48) ∈ S , we have | ˜ w x,θ ( t ) − ˜ w x,θ (cid:48) ( t ) | < π for any t so the limit doesnot depend on choice of θ ∈ S .Now define σ : X × R → R and τ : X × R → R by: σ t ( x ) := sup θ ∈ S ˜ w x,θ τ t ( x ) := inf θ ∈ S ˜ w x,θ , which, by continuity of A are evidently continuous in t and in x . Moreover, by the cocyleequation for A it is clear that σ is subadditive and τ is superadditive.By Kingman’s subadditive ergodic theorem for flows, for any µ ∈ M Φ ( X ):(1) The sequence t σ t converges µ -a.e. to a Φ invariant map, which agrees with ρ .(2) We may compute the integral of ρ by:(2) ρ µ := (cid:90) ρ dµ = inf t> t (cid:90) σ t dµ. The discussion above then implies:
Theorem 2.15.
The map M Φ ( X ) → R given by µ (cid:55)→ ρ µ is continuous.Proof. Note that by compactness of X and continuity of σ t : X → R , the map µ → (cid:82) σ t dµ is continuous, and hence by: (cid:90) ρ dµ = inf t> t (cid:90) σ t dµ and the analogous equation for τ , we obtain upper and lower semicontinuity of µ (cid:55)→ ρ µ . (cid:3) Remark 2.16.
When µ is supported on a periodic orbit O , we will often write ρ O for ρ µ . Next we consider perturbations of cocycles over a fixed base flow. The space of cocycles C Φ over the same Φ has a C -topology of uniform convergence defined by the property that A n → A if for each x ∈ X and | t | < A n ) tx → A tx in C ( S , S ) uniformly.Associated to the cocycles A are rotation numbers ρ µ ( A ) for invariant measures µ definedby Equation (2). Then: Proposition 2.17.
For a µ ∈ M Φ ( X ) , the map C Φ → R given by A (cid:55)→ ρ µ ( A ) is continuous.Proof. The proof is nearly identical to that of Theorem 2.15. Namely, one uses continuityof
A (cid:55)→ (cid:82) σ t ( A ) dµ and the subadditive ergodic theorems. (cid:3) Now we specialize to the case where X = O is a hyperbolic periodic orbit of a C flow Φ on a Riemannian manifold N , which will be N = SM with the Sasaki metric in the settingof this paper. We are interested in how ρ O varies as the flow Φ varies, for the derivativecocycle on certain circle bundles.By structural stability of the hyperbolic set O there exists U a C -neighborhood ofΦ and a continuous h : U × O → N such that the maps h Φ ( x ) := h (Φ , x ) are C -diffeomorphisms onto their images, and O Φ := h Φ ( O ) is a closed orbit of Φ. Moreover,since the maps h Φ are C there exists a continuous κ : U × O × R → R such that κ Φ ( x, t ) := κ (Φ , x, t ) is C and the flow ˜Φ (defined on O Φ ) given by:˜Φ t ( x ) = Φ κ Φ ( h Φ ( x ) ,t ) ( x ) , is in fact conjugated to Φ by h Φ , i.e., h Φ ◦ Φ = ˜Φ ◦ h Φ . .For any bundle E , we write P E for its projectivization. Let F be a 2-dimensional trivialsubbundle of T N | O which is part of a dominated splitting E ⊕ ≤ F ⊕ ≤ G of T N | O . Thederivative cocycle D Φ on P F then is a cocyle on a trivial S bundle, and it has a rotationnumber ρ O as before.Assuming U is taken sufficiently small, by persistence of dominated splittings for eachΦ ∈ U there is a splitting T N | O Φ = E Φ ⊕ ≤ F Φ ⊕ ≤ G Φ for Φ and the bundle F Φ is trivial.Moreover, the splitting is also dominated for the flow ˜Φ, which is simply a time change ofΦ. Hence, D ˜Φ and D Φ on P F Φ also have well defined rotation numbers ρ O ˜Φ , ρ O Φ , which IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 11 satisfy the relation: ρ O ˜Φ (cid:96) ( O Φ ) = ρ O Φ (cid:96) ( O ) , as they differ by a time change.With all the objects defined, we now state the continuity with respect to the parameters: Proposition 2.18.
The map
U → R given by Φ (cid:55)→ ρ O Φ , is continuous in some open V ⊆ U containing Φ .Proof. First, we would like to consider all cocycles D ˜Φ constructed on F Φ as existing onthe same bundle over the same base map.For x ∈ O there exists a unique length-minimizing geodesic segment (from the Riemann-ian structure on N ) from x to h Φ ( x ), as long as h Φ is close to the identity, which may beensured by passing to some V ⊆ U further if needed. By parallel transport of the bundle F Φ over O Φ along such segments, one then obtains a 2-dimensional trivial bundle F (cid:48) Φ over O . By shrinking V further if needed, the bundle F (cid:48) Φ obtained is a given by a graph over F with respect to the fixed Riemannian metric on N , and hence by orthogonal projectionthey may be identified.Since all maps above are continuous, the construction above describes a continuous map T h : U × F → T N , so that
T h Φ ( · ) := T h (Φ , · ) are bundle isomorphisms F → F Φ fiberingover h Φ . Hence, conjugating by T h Φ we may regard D ˜Φ on F Φ as a cocyle on F over Φ .By continuous dependence on Φ, this defines a continuous map Φ → D ˜Φ, where D ˜Φ arenow regarded as elements of the space of cocycles over Φ on F with the C -topology.Since all rotation numbers ρ O ˜Φ defined previously are preserved by conjugation, it sufficesto check continuity of the rotation numbers of the conjugated cocycles, which is given byProposition 2.17. Thus the map Φ (cid:55)→ ρ O ˜Φ is continuous, and finally since ρ O ˜Φ (cid:96) ( O Φ ) = ρ O Φ (cid:96) ( O ) , and the periods vary continuously, the map Φ (cid:55)→ ρ O Φ is continuous as well. (cid:3) Pinching & Twisting
In this section, we define pinching and twisting for orbits of the geodesic flow in analogywith Definition 2.12, and use the results on generic metrics to show that these are C k openand dense.We fix the following useful notation. For a metric g such that O ⊆ SM is a periodicorbit of its geodesic flow with period (cid:96) , let v ∈ O and let { λ , ..., λ n } be the generalizedeigenvalues of D v ϕ (cid:96)g | E u ⊕ E s , which do not depend on the choice of v , sorted so that | λ i | ≥ | λ j | whenever i < j . We write: (cid:126)λ u ( O , g ) := ( λ , ..., λ n ) , (cid:126)λ s ( O , g ) := ( λ n +1 , ..., λ n ) ∈ C n ,(cid:126)λ ( O , g ) := ( λ , ..., λ n ) ∈ C n . The i -th coordinates of the vectors above are written as (cid:126)λ u,s, · i ( O , g ) (where · means nosuperscript above).The following continuity lemma about these (cid:126)λ is the bread and butter of all “openness”arguments which follow: Lemma 3.1.
Fix a metric g ∈ G k . Then there exists a neighborhood U ⊆ G k of g suchthat for any g ∈ U any orbit O of the geodesic flow of g has a hyperbolic continuation O g for the geodesic flow of g , and the maps U → C n given by g (cid:55)→ (cid:126)λ u,s ( O g , g ) are continuous.Proof. Let Σ be a smooth hypersurface parallel to E u ⊕ E s at v so that O ∩
Σ =: { v } .The return map for the geodesic flow ϕ g then defines a map P g : U → Σ, where U ⊆ Σis some neighborhood of v , for which v is a hyperbolic fixed point.For any g sufficiently close to g , we also obtain a map P g : U → Σ given by the returnmap of ϕ g , and by the standard hyperbolic theory, a fixed point v g such that g (cid:55)→ v g iscontinuous. The geodesic flow ϕ g varies in a C k − fashion as g varies in G k , and by theimplicit functon theorem so does P g . Then by fixing a coordinate system, since k ≥ D v g P g vary continuously, so their eigenvalues vary continuously as g varies in G k .Finally, the eigenvalues of the matrices D v g ϕ (cid:96) g g | E u ⊕ E s and D v g P g agree, so we obtain thedesired result. (cid:3) IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 13
Pinching.
Before moving to the definition of pinching, first we verify that genericallythere exists a periodic orbit with a dominated splitting of E u ⊕ E s into 1-dimensionalsubspaces and 2-dimensional subspaces corresponding to conjugate pairs of eigenvalues. Definition 3.2.
We let G kd ⊆ G k be the set of metrics g such that for some periodic orbit O of the geodesic flow all (cid:126)λ ui ( O , g ) ∈ C have distinct norms, except for pairs of complexconjugate eigenvalues. Proposition 3.3.
The set G kd is open and dense in G k .Proof. Openness follows directly from Lemma 3.1, since by continuity of (cid:126)λ u the continua-tions of O will satisfy the same condition defining G kd .For density, we start by assuming that g ∈ G ∞O , for some O , which is possible by densityof G ∞ in G k . It remains to check that the property defining G kd is indeed an open dense in J k − s (it is clearly invariant), so that we may apply Corollary 2.6 to finish the proof.By Remark 2.7, it suffices to check that having eigenvalues distinct with distinct norms,apart from complex conjugate pair, is an open dense (again, it is clearly invariant) in Sp (2 n ).Openness is clear, since the eigenvalues depend continuously on the matrix entries. Fordensity, we note that the condition of distinct eigenvalues is given by the complement ofthe equation ∆ = 0, where ∆ is the discriminant of the characteristic polynomial, whichis a non-empty Zariski open set in Sp (2 n ), and thus dense in the analytic topology. Inparticular, the set of diagonalizable matrices is dense. Since diagonalizable matrices aresymplectically diagonalizable, by the lemma following this proof, by a small perturbationon the norm of the diagonal blocks we obtain density of eigenvalues of distinct norms. (cid:3) We prove the linear algebra lemma used above, which will also be useful in what follows:
Lemma 3.4.
A matrix A ∈ Sp (2 n ) with all eigenvalues distinct is symplectically diagonal-izable in the sense that there exists P ∈ Sp (2 n ) such that P − AP is in real Jordan form(i.e., given by diagonal blocks which are either trivial or × conformal).Proof. Recall that eigenvalues of A ∈ Sp (2 n ) appear in 4-tuples { λ, λ, λ − , λ − } for λ / ∈ R and in pairs { λ, λ − } for λ ∈ R . For each λ we let E λ = E λ − be the 2-dimensionalsubspace spanned by the eigenspaces of λ and λ − .Extend ω and A to ω C and A C in the complexification C n = R n ⊗ C . By definition A C and ω C agree with A and ω on R n ⊗ v λ and v η : ω C ( v λ , v η ) = ω C ( A C v λ , A C v η ) = λη ω C ( v λ , v η ) , implies that, unless λη = 1, we have ω C ( v λ , v η ) = 0. Therefore E λ ⊗ C is symplecticallyorthogonal to E η ⊗ C for any λ (cid:54) = η, η − .In particular, this implies that the E λ ⊗ ω the real form, and symplectically orthogonal to each other. In each E λ , A can be put inJordan real form with respect to a symplectic basis. By orthogonality we may construct asymplectic basis for R n by taking the union of symplectic bases for the E λ . Then let P bethe matrix which sends the standard R n basis to the constructed symplectic basis. (cid:3) The next step is to construct a metric with a periodic orbit with simple real spectrumwith an arbitrarily small perturbation of the metric. Following [BV04], this is accomplishedby slightly perturbing a periodic orbit O rotating a complex eigenspace, and propagatingthe perturbation to a periodic orbit which shadows a homoclinic orbit of O that spends along time near O .Recall the definitions of pseudo-orbits and shadowing: Definition 3.5 (Pseudo-Orbit) . An ε -pseudo-orbit for a flow Φ on a space X is a (possiblydiscontinuous) function g : R → X such that: d ( γ ( t + τ ) , Φ τ ( γ ( t ))) < ε for t ∈ R and | τ | < . Definition 3.6 (Shadowing) . Let Φ be a flow on a metric space X and γ be an ε -pseudo-orbit for Φ . Then γ is said to be δ -shadowed if there exists a point p ∈ X and a homeo-morphism α : R → R such that α ( t ) − t has Lipschitz constant δ and d ( γ ( t ) , Φ α ( t ) ( p )) ≤ δ for all t ∈ R . The statement of shadowing is:
Theorem 3.7. [FH18] (Anosov Closing Lemma) If Λ is a hyperbolic set for a flow Φ thenthere are a neighborhood U of Λ and numbers ε , L > such that for ε ≤ ε any compact ε -pseudo-orbit in U is Lε -shadowed by a unique compact orbit for Φ . IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 15
We use it to prove the main result of this section:
Proposition 3.8.
The set of metrics g ∈ G k such that for some periodic orbit O of thegeodesic flow of g the exponents λ ui ( O , g ) are real for all ≤ i ≤ n and pairwise distinct isdense in G k .Proof. Fix an open set
U ⊆ G k . First, since G kd is C k -open and dense and G ∞ is C k -densein G k , we may fix some g ∈ U ∩ G ∞ d . Let O be as in Definition 3.2.Suppose that the vector (cid:126)λ u ( O , g ) has 2 c entries in C \ R , for some c >
0. It suffices toshow that there exists a metric g (cid:48) in U which has a periodic orbit O (cid:48) such that (cid:126)λ u ( O (cid:48) , g (cid:48) )has 2( c −
1) complex entries and all real entries distinct.Along O there is a dominated splitting E u = E − ⊕ · · · ⊕ E − k such that each E i is either1 or 2-dimensional. Fix the smallest index i ∈ { , ..., k } such that E ± i is 2-dimensional andlet P g denote the Poincar´e return map of the geodesic flow for a fixed section Σ transverseto the flow small enough so that O ∩
Σ =: { v } . By shrinking U further if needed we mayassume that U ⊆ G kd , i.e., that the dominated splitting for ϕ g along O persists for thecontinuation of O for all g ∈ U ; thus, by Lemma 3.1 the map g (cid:55)→ θ g := | arg( λ g ) | is welldefined and continuous, where λ g is an eigenvalue of Dϕ g on E − i on the continuation of O . Lemma 3.9.
There exists g ∈ U ∩ G k O such that θ g (cid:54) = θ g .Proof. The derivative of the Poincar´e map is conjugate to Dϕ g | E u ⊕ E s over the closed orbit O , so θ g agrees with the argument of the eigenvalue of DP g along the 2-dimensionalJordan block F − ⊆ T v Σ mapped to E − i under the conjugation aforementioned. Moreover,let F + be the Jordan block corresponding to E + i in the same manner.Identifying the space of symplectic maps T v Σ → T v Σ with Sp (2 n ) there exists someneighborhood V ⊆ Sp (2 n ) of the original map DP g , such that for A ∈ V the Jordan block F has a continuation for A , and we call the norm of the argument of the eigenvalue of A alongthis continuation θ A . Let W ⊆ V be the set of matrices A such that θ A (cid:54) = θ g . If W is openand dense in V then by Remark 2.7 we may apply Corollary 2.6 to W ∪ (( Sp (2 n ) \ Cl( V )),which will be open and dense in Sp (2 n ) to find that the set of metrics which has θ g (cid:54) = θ g is dense (and open) in U .It remains to check that W is open and dense in V . Openness is clear by continuousdependence of eigenvalues on matrix entries. For density, let R θ be given by rotation of anyangle of θ > F − , F + and the identity on the other subspaces, satisfies R θ Ω R Tθ = Ω, where Ω is the standard symplectic form. Then R θ DP g has θ R θ DP g (cid:54) = θ g ;since θ > (cid:3) Let g be given as in the lemma above, and for 0 ≤ s ≤ g s = sg + (1 − s ) g ,which, if g is taken sufficiently close to g , also satisfies { g s } ⊆ U ∩ G k O . Clearly, themap [0 , → G k given by s (cid:55)→ g s is continuous. Also note that, by Proposition 2.5 (2), O is not only a closed orbit of ϕ g s for all s ∈ [0 , g s .For the geodesic flow of g , fix w a transverse homoclinic point of v , i.e., w ∈ W u ( v ) ∩ W cs ( v ). Fix some ε > ε .Then there exists t , t > ϕ − t g ( w ) ∈ W uε ( v ), ϕ t g ( w ) ∈ W sε ( v ) and also a C > t > d ( ϕ − ( t + t ) g ( w ) , ϕ − tg ( v )) < Cεe − t ,d ( ϕ t + tg ( w ) , ϕ tg ( v )) < Cεe − t . Hence for n ∈ N the γ n : R → SM given by γ n ( t ) = ϕ ˜ t − ( t + n(cid:96) ) g ( w ) , where ˜ t = t mod ( t + t + 2 n(cid:96) )are ε n -pseudo-orbits where ε n < Cεe − n(cid:96) , by the inequalities (1). For n sufficiently large,there exist unique periodic w n ’s which Lε n -shadow γ n . Let w n,s be continuations of w n for0 ≤ s ≤ w n, = w n , by definition).Let w s be the hyperbolic continuations of w . By uniqueness of shadowing, note thatthe w n,s can also be constructed by shadowing segments of the orbit of w s . The followingproposition shows we can extend the dominated splitting of O to the new orbits we defined: Lemma 3.10.
There exists N large so that for each < s < the compact invariant set K N,s = (cid:91) n ≥ N O ( w n,s ) ∪ O ( w s ) ∪ O , for the geodesic flow ϕ g s of g s admits a dominated splitting for the bundle E u = E − s, ⊕· · · ⊕ E − s,k over K m,s coinciding with the dominated splitting of E u over O , and similarlyfor E s = E + s, ⊕ · · · ⊕ E + s,k (the s subscriptson the right hand side refer to the parameter s ∈ [0 , , whereas ± refers to the stable/unstable). IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 17
Proof. (See [BV04] ). We show the result for s = 0. Then since dominated splittings overcompact invariant sets persists under C small perturbations by an invariant cone argu-ment, this shows the result for all s ∈ [0 , w ∈ W u ( v ) ∩ W cs ( v ), one can extend the dominated splitting of O to O ( w )by using the holonomies along the local stable/unstable leaves. Then invariance of thesplitting follows from equivariance of holonomies with respect to the geodesic flow. For N sufficiently large we observe that K N, is contained in an arbitrarily small neighborhood of O ∪ O ( w ), so the dominated splitting extends by continuity. (cid:3) For each n , we let θ n : [0 , → S = R / π Z be defined by setting θ n ( s ) to be theargument of the eigenvalue of Dϕ g s along E − s,i on the closed orbit w n,s . By Lemma 3.1,the θ n are continuous so for each n they may be lifted to some ˜ θ n : [0 , → R .The main result about these rotation numbers, whose proof is postponed to the nextsection due to its length, is: Lemma 3.11.
There exists n ∈ N so that | ˜ θ n (1) − ˜ θ n (0) | > π . By continuity one then finds n, s such that ˜ θ n,s is an integer multiple of 2 π , i.e., suchthat the eigenvalues in (cid:126)λ u ( O ( w n,s ) , g s ) corresponding to the subspace E − i,s are real. Byanother perturbation using Corollary 2.6, there exists a metric such that these eigenvaluesbecome distinct.The proof then follows by induction on the other eigenspaces with complex eigenvalues. (cid:3) Finally, we define the set of metrics with a pinching periodic orbit, and obtain opennessand density.
Definition 3.12 (Pinching Property) . The set G kp ⊆ G k is defined as the set of metricswith at least one periodic orbit O such that (cid:126)λ ui ( O , g ) ∈ R and moreover the (cid:126)λ ui ( O , g ) are pairwise distinct.In this situation, we say O has the pinching property for g . Corollary 3.13.
The set G kp is open and dense in G k . Proof.
Openness follows again by Lemma 3.1, since the requirements on the products ofthe eigenvalues is an open condition.For density, Theorem 3.8 gives density of the set of metrics with real and distinct eigen-values. Again by an argument analogous to that of Proposition 3.3, i.e., perturbing alongthe diagonal after diagonalizing symplectically, obtaining the property defining G p fromreal distinct eigenvalues reduces to doing so in Sp (2 n ). (cid:3) Proof of Lemma 3.11.
We apply the notions introduced in Section 2.4 to give aproof of Lemma 3.11.
Proof of Lemma 3.11.
Fix N large enough so that K N,s satisfies the conclusion of Lemma3.10. We begin with:
Proposition 3.14.
There exists N (cid:48) > N , which we denote by N after this proposition, suchthat the bundles with total spaces E s defined by the fibers E s ( x ) := E − s,i ( x ) over x ∈ K N (cid:48) ,s are continuously trivializable, for each s ∈ [0 , .Proof. First, note that it suffices to prove that E s is trivializable for s = 0, since thebundles E s vary continuously in the ambient space T SM as s varies.We will construct a non-vanishing section of the frame bundle F associated to E oversome K N (cid:48) , for N (cid:48) large, which is equivalent to a continuous choice of basis for E , provingtriviality of the bundle.For δ > B δ ( O ) a δ -tubular neighborhood of O . If δ is sufficiently small relative tothe scale of local product structure of the Anosov flow, for all n ≥ N (cid:48) , and N (cid:48) sufficientlylarge, O ( w n ) δ := B δ ( O ) ∩ O ( w n ) consists of a connected segment of the embedded circle O ( w n ) and moreover, O ( w ) δ := B δ ( O ) ∩ O ( w ) consists of the complement of a connectedclosed interval in O ( w ), i.e., two immersed connected components (see Figure 1).Note that we may assume that the return map of Dϕ (cid:96) ( w n ) g is orientation preserving on E over any periodic orbit O ( w n ), since otherwise it would have real eigenvalues (any A ∈ GL (2 , R ) with negative determinant has real eigenvalues) and we would obtain a proofof Lemma 3.11. Hence, the bundle E is trivializable over any O ( w n ). It is also clearlyso over O ( w ), since it is an immersed real line, and we may assume it is too for O , sinceotherwise, again, we would have real eigenvalues. IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 19
Figure 1.
Proof of Proposition 3.14. The closed orbit O is schematicallyrepresented by the black dot.By shrinking δ further if necessary, there exists a well-defined closest point projection p : B δ ( O ) → O which is a surjective submersion. Fix a trivialization of E over O , i.e., anon-vanishing section S : O → F , which is possible by the previous paragraph.For x ∈ B δ ( O ) ∩ K N (cid:48) , =: K δ , again shrinking δ further if necessary, there exists aunique length-minimizing geodesic segment between x and p ( x ), and by parallel transport-ing E ( p ( x )) along such segments and then projecting orthogonally onto E ( x ) one obtainsa continuous bundle map E | K δ → E | O which is an isomorphism on fibers. This mapinduces a map F | K δ → F O and so by pulling back the non-vanishing section S : O → F weobtain a non-vanishing section, which we now denote by S : K δ → F , since its restrictionto O agrees with the previous S , of F over K δ .Recall that O ( w ) δ consists of two connected immersed components homeomorphic to R .Since O ( w ) is contractible, it is possible to define a determinant on F | O ( w ) ; up to scalar itis unique, and hence there is a well defined continuous sign function on each fiber. Thenwe claim that S | O ( w ) δ has the same determinant sign on both components, so that it maybe extended to a continuous section O ( w ) → F | O ( w ) . Suppose not for a contradiction.Define the line bundle L := (cid:86) E over K , which restricted to individual orbits istrivial since E is. At each point x ∈ K there is a natual map F ( x ) → L ( x ) given by ( e , e ) (cid:55)→ e ∧ e , which extends to a continuous global map W : F → L . Considering theimage of S | O ( w ) δ under W , we obtain a section O ( w ) δ → L , which has opposite signs inthe two connected components. Let B : O ( w ) → L be any extension of this section to allof O ( w ); by the previous remark, B must have an odd number of zeros.By shadowing O ( w n ) \ B δ/ ( O ) → O ( w ) \ B δ/ ( O )as n → ∞ , so by continuity of the bundle for n sufficiently large we can parallel transportthe section B on O ( w ) \ B δ/ ( O ) to O ( w n ) \ B δ/ ( O ) to obtain a section B n on O ( w n ) \ B δ/ ( O ) which has the same number of zeros as B on O ( w ) \ B δ/ ( O ), i.e., oddly many.On the other hand as n → ∞ , B n | O ( w n ) δ \ B δ/ ( O ) → ( W ◦ S ) | O ( w n ) δ \ B δ/ ( O ) , and hence for n large enough B n has constant sign on O ( w n ) δ \ B δ/ ( O ). Thus B n extendsto O ( w n ) δ without any zeros. Hence we obtain a global section B n on O ( w n ) with an oddnumber of zeros, contradicting the triviality of L over O ( w n ).Hence we may extend S | O ( w ) δ continuously to all of O ( w ). Since in K δ the section S is continuous, and again O ( w n ) \ B δ/ ( O ) → O ( w ) \ B δ/ ( O ) as n → ∞ , we can thencontinuously extend S | O ( w ) \ B δ/ ( O ) to O ( w n ) \ B δ/ ( O ) while agreeing with S in K δ . Since S | O ( w ) is non-vanishing, the section obtained in this way is also globally non-vanishing. (cid:3) The projectivization P E s of the bundle E s then defines a trivial circle bundle over K N,s ,and we fix a trivializing bundle isomorphism φ s : P E s → K N,s × S . By conjugating with φ s , the derivative of the geodesic flow then defines a continuous cocycle A s on K N,s × S over the geodesic flow, so we may apply the results of Section 2.4 for A s .Then the the rotation numbers have the following characterization over periodic orbits: Lemma 3.15.
For a closed orbit O ( u ) of a point u ∈ K N,s , the argument θ ( u ) of theeigenvalue of the return map of the geodesic flow on E − s,i satisfies: θ ( u ) = (cid:96) ( u ) · ρ O ( u ) ( mod π ) where (cid:96) ( u ) is the period of u , and ρ O ( u ) is as in Remark 2.16 for the cocycle A s definedabove. IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 21
Proof.
On one hand, it follows from the definition of ρ that (cid:96) ( u ) · ρ O ( u ) agrees mod 2 π withthe Poincar´e rotation number for the map( A s ) (cid:96) ( u ) u : S → S . On the other, the projectivization of the derivative of the flow also defines on the fibera homeomorphism S → S with Poincar´e rotation number equal to the argument of theeigenvalue of the derivative.Since the two above differ by a conjugation given by π ◦ φ s ( u, · ) : S → S , where π : K N,s × S is the natural projection, by invariance we obtain the result. (cid:3) Applying Lemma 3.15 to the θ n ( s ), we obtain for 0 ≤ s ≤ θ n ( s ) = (cid:96) ( w n,s ) ρ O ( w n,s ) (mod 2 π ) , By continuity of the functions θ n we may lift them to ˜ θ n : [0 , → R satisfying ˜ θ n (0) = (cid:96) ( w n, ) ρ O ( w n, ) . By the continuity of ρ O ( w n,s ) in s , given by Proposition 2.18, our choice oflift then implies: ˜ θ n ( s ) = (cid:96) ( w n,s ) ρ O ( w n,s ) , for 0 ≤ s ≤ , Let θ ( s ) be the argument of the eigenvalue of the Dϕ g s on E − i,s on the periodic orbit O (recall O is a closed geodesic for all g s with (cid:96) ( O ) fixed), and repeat the constructionsabove to obtain ˜ θ ( s ) as well satisfying(3) ˜ θ ( s ) = (cid:96) ( O ) ρ O ( s ) , where ρ O ( s ) is ρ O of the geodesic flow of g s .Since µ O ( w n,s ) → µ O (where µ O is the invariant probability measure supported on theclosed orbit O ) we have ρ O ( w n,s ) → ρ O ( s ) as n → ∞ by Theorem 2.15. By hypothesis θ (1) (cid:54) = θ (0), and since (cid:96) ( O ) is constant as s varies, Equation (3) gives that ρ O (1) − ρ O (0) (cid:54) =0. Hence for n large enough there exists some δ > | ρ O ( w n, ) − ρ O ( w n, ) | ≥ δ .Finally, let δ n = | (cid:96) ( w n, ) − (cid:96) ( w n, ) | . Again, we defer the proof of the following finalproposition we need: Proposition 3.16.
There exists M > such that δ n < M for all n ∈ N . With Lemma 3.16, we complete the proof of Lemma 3.11: | ˜ θ n (1) − ˜ θ n (0) | = | (cid:96) ( w n, ) ˜ ρ O ( w n, ) − (cid:96) ( w n, ) ˜ ρ O ( w n, ) |≥ | (cid:96) ( w n, )( ˜ ρ O ( w n, ) − ˜ ρ O ( w n, ) ) |− | ( (cid:96) ( w n, ) − (cid:96) ( w n, )) ˜ ρ O ( w n, ) |≥ δ(cid:96) ( w n, ) − δ n | ˜ ρ O ( w n, ) | > δ(cid:96) ( w n, ) − M M > π for all n sufficiently large, since (cid:96) ( w n, ) → ∞ . (cid:3) At last, we prove Proposition 3.16.
Proof of Proposition 3.16.
To bound the variations δ n , we use exponential shadowing andH¨older continuity of the geodesic stretch, defined below. Since the geodesic flow is unper-turbed on O and the orbits O ( w n,s ) approximate O , the two mentioned properties give usthe bound on δ n .Recall that the w n are constructed by shadowing γ n : R → SM given by γ n ( t ) = ϕ ˜ t − ( t + n(cid:96) ) g ( w ) , where ˜ t = t mod ( t + t + 2 n(cid:96) ) , which is a ε n -pseudo-orbit, where t (resp. t ) is such that ϕ t g ( w ) (resp. φ − t g ( w )) is in W sε ( v ) (resp. W uε ( v )) and ε n < Cεe − n(cid:96) , by the inequalities 1.The following well-known theorem is an adaptation for flows of the usual “exponential”shadowing theorem, which uses the Bowen bracket in its proof. The statement gives asharper estimate on how well shadowing orbits approximate pseudo-orbits: Theorem 3.17. [FH18, Theorem 6.2.4]
For a hyperbolic set Λ of a flow Φ on a closedmanifold ∃ c, η > such that ∀ ε > , ∃ δ > so that: if x, y ∈ Λ , s : R → R continuous, s (0) = 0 and d (Φ t ( x ) , Φ s ( t ) ( y )) < δ for all | t | ≤ T , then (1) | t − s ( t ) | < ε for all | t | ≤ T , (2) there exists t ( x, y ) with | t ( x, y ) | < ε so that the ε -stable manifold Φ t ( x,y ) ( x ) inter-sects uniquely the ε -unstable manifold of y and: d (Φ t ( y ) , Φ t (Φ t ( x,y ) ( x ))) < ce η ( T −| t | ) for | t | < T. In the context of the current proof, we apply the above theorem as follows.
IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 23
Let T n = (cid:96) ( w n ), x = ϕ T n / g ( w n ) and y = ϕ τ n g ( w ) where τ n := T n / − ( t + n(cid:96) ). For n sufficiently large, d ( ϕ tg ( x ) , ϕ s ( t ) g ( y )) < δ is satisfied, by the statement of shadowing, for | t | < T n / δ given by the theorem for the ε > t n ∈ R such that: d ( ϕ t n + tg ( w n ) , ϕ τ n + tg ( w )) < ce η ( T n / −| t | ) , for | t | < T n / . Now we turn to computing the period of w n, using the facts established above. Bystructural stability, there exists h : SM → SM which conjugates the orbits of ϕ g tothose of ϕ g . This conjugacy can be taken to be H¨older continuous and C along the flowdirection. Thus, there exists some a : SM → R which is H¨older continuous with someexponent 1 ≥ β >
0, such that for u ∈ SM : dh ( u ) X g ( u ) = a ( u ) X g ( h ( u )) , where X g (resp. X g ) is the vector field generating the geodesic flow for g (resp. g ). Thefunction a is referred to as the geodesic stretch , and the proof of the facts above can befound, for instance, in [GKL19, p. 12-13]The period of w n, is given by the formula: (cid:96) ( w n, ) = (cid:90) T n a ( ϕ tg ( w n )) dt By Proposition 2.5 (2), since O is a closed geodesic, with same arclength parametrizationfor g and g , it is clear that a | O ≡
1. Therefore, we may compute the difference δ n = | (cid:96) ( w n, ) − (cid:96) ( w n, ) | as follows: | (cid:96) ( w n, ) − (cid:96) ( w n, ) | ≤ (cid:90) T n / − T n / | a ( ϕ tg ( w n )) − | dt ≤ M (cid:90) T n / − T n / d ( ϕ t n + tg ( w n ) , O ) β dt, since a is β -H¨older continuous and the distance between a point and a compact set is welldefined. To estimate the distance, note: d ( ϕ t n + tg ( w n ) , O ) ≤ d ( ϕ t n + tg ( w n ) , ϕ τ n + tg ( w )) + d ( ϕ τ n + tg ( w ) , O ) ≤ c ( e η ( T n / −| t | ) + e −| t | ) , for | t | < T n / , , since w is a homoclinic point of O so d ( ϕ τ n + tg ( w ) , O ) ≤ ce −| t | for some c > c . Substituting this inequality into the previous integral, we obtain: | (cid:96) ( w n, ) − (cid:96) ( w n, ) | ≤ M (cid:90) T n / − T n / ( e η ( T n / −| t | ) + e −| t | ) β dt < M < ∞ , for M independent of n , as an easy calculus exercise shows. (cid:3) Twisting.
Next we will show that the twisting condition in the definition of a simplecocycle is also generic, and in fact open and dense, for the geodesic flow of a metric in G k . Without the obstruction imposed by the distinction between real and complex-valuedcocycles, this reduces to an application of Theorem 2.3 presented above.Following the previous section, we fix a metric g ∈ G kp . Let O , v, l be the orbit with thepinching property, v ∈ O and l the period of O . We fix an arbitrary w ∈ W csg ( v ) ∩ W cug ( v )a transverse homoclinic point of the orbit of v , and consider the holonomy maps ψ g v,w = h csw,v ◦ h cuv,w , given from Theorem 2.2, for the unstable bundle E u .Recall that Dϕ (cid:96)g | E u has all eigenvalues distinct and real, so let { e i } be an (non-generalized,real) eigenbasis for E u . For all 1 ≤ j ≤ k the alternating powers Λ j E u ( v ) have a basisobtained as exterior products of the e i . We write e kI := e i ∧ · · · ∧ e i k , where I = { i , ..., i k } . Definition 3.18 (Twisting) . For g ∈ G kp as above we say g has the twisting propertyfor w ∈ SM with respect to v , and we write g ∈ G kp,t , if ψ g v,w : E u ( v ) → E u ( v ) has thefollowing property:For all e kI , e lI (cid:48) , with k + l = n : ( ∧ k ψ g v,w )( e kI ) ∧ e lI (cid:48) (cid:54) = 0 , which is to say that the image of any direct sums of eigenspaces intersects any direct sumof eigenspaces of complementary dimension only at the origin. Then we show:
Proposition 3.19.
Let
U ⊆ G k be any neighborhood of g . Then there exists a metric g ∈ U ∩ G k O ∩ G kp,t such that w has the twisting property with respect to v .Proof. Again, by density of G ∞ ⊆ G k and openness of G kp we may assume that g ∈ G ∞ sowe can apply Theorem 2.3. IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 25
For ε >
0, consider the geodesic segment γ = ϕ g [0 ,ε ] ( w ). Note that, since O ( w ) accumu-lates as | t | → ∞ on the compact set O , if we take ε > π ( γ ) tobe disjoint from π ( O ( w ) \ γ ) ∪ π ( O ).Then we apply Theorem 2.3 to γ (cid:48) ⊆ γ , where γ (cid:48) = ϕ g [ δ,ε − δ ] ( w ) for δ > D w ϕ εg by perturbing the metric only on a tubular neighborhood V γ (cid:48) of γ (cid:48) small enough sothat V γ (cid:48) ∩ Cl( π ( O ) ∪ π ( O ( w ) \ γ )) = ∅ . where Cl denotes closure. Again, this local perturbation is possible by Proposition 2.5 (1).Recall that by equivariance of holonomies: ψ g v,w = D v ϕ − εg | E u ◦ h csϕ εg ( w ) ,v ◦ D w ϕ εg | E u ◦ h cuv,w . By recalling the construction of center stable and unstable holonomies, we see that theperturbations of the form above vary only the D w ϕ εg | E u term in the composition above.Indeed, we recall that h cuw,v depends only on the values of the cocycle along the ( −∞ , ϕ tg ( w ), and h csϕ εg ( w ) ,v on the [ ε, ∞ ) part of the orbit ϕ tg ( w ), as well asthe values of the cocyle on O . By construction of V γ (cid:48) , the cocyle is not perturbed in anyof these sets.It remains to check that for an open dense set of k -jets of ϕ g [ δ,ε − δ ] ( w ) the map ψ g v,w hasthe twisting property. Since D v ϕ − εg | E u , h csϕ εg ( w ) ,v , h cuv,w are all linear isomorphisms, an open dense set in the set of matrices E u ( v ) → E u ( v ) whichare a restriction of a symplectic matrix is mapped under composition to an open dense setof the symplectic 1-jets of ϕ g [ δ,ε − δ ] ( w ).Again, observe that the condition defining twisting is given by a Zariski open subset ofthe matrices E u ( v ) → E u ( v ). Hence, as long this set is non-empty in the set of matriceswhich come from restricting of a matrix in Sp (2 n ) to a Lagrangian subspace the twistingset must also be open and dense in the analytic topology. Then by the paragraph above,this translates to an open and dense condition in the symplectic 1-jets of ϕ g [ δ,ε − δ ] ( w ), andas there is no condition imposed on higher jets, we obtain the desired result by Remark2.7.To finish the proof, it thus suffices to check that the Zariski open set defining twistingis non-empty in the symplectic group, which is done below. (cid:3) Remark 3.20.
Notice that after perturbing w is still a homoclinic point of v , and v is stilla periodic orbit with pinching, i.e., we obtain twisting for ψ v,w for the original fixed w . Lemma 3.21.
There exists a matrix A ∈ Sp (2 n ) , where R n is taken with standard sym-plectic basis { e i , f i } such that A preserves E u := span { e i } ni =1 and moreover for all e kI , e lI (cid:48) ,with k + l = n : ( ∧ k A )( e kI ) ∧ e lI (cid:48) (cid:54) = 0 . Proof.
Note that for fixed e kI , e lI (cid:48) the property that ( ∧ k A )( e kI ) ∧ e lI (cid:48) (cid:54) = 0 is open in Sp (2 n ).Thus suffices to show that by an arbitrarily small perturbation of A ∈ Sp (2 n ) one canarrange so that ( ∧ k A )( e kI ) ∧ e lI (cid:48) (cid:54) = 0 for some e kI , e lI (cid:48) and moreover A still preserves E u .Indeed, if the claim above holds, then starting at any matrix (e.g. the identity) preserving E u , inductively for each pair of basis elements e kI , e lI (cid:48) one may perturb the matrix so that( ∧ k A )( e kI ) ∧ e lI (cid:48) (cid:54) = 0, A still preserves E u , and moreover make the perturbation small enoughso that all the other pairs basis elements which have already been perturbed preseve thedesired property, which is possible by openness.To prove the claim, suppose ( ∧ k A )( e kI ) ∧ e lI (cid:48) = 0, and write:( ∧ k A )( e kI ) = (cid:88) J a J e kJ . Since A is invertible, there exists J such that a J (cid:54) = 0 and such that | J ∩ I (cid:48) | is minimal.Since | J | + | I (cid:48) | = n , we have | J ∩ I (cid:48) | = { , . . . , n } \ ( J ∪ I (cid:48) ), so we take an arbitrarilychosen bijection i (cid:55)→ j i from J ∩ I to { , . . . , n } \ ( J ∪ I (cid:48) ).For θ >
0, let R i,jθ given by rotating the (oriented) planes span( e i , e j ) and span( f i , f j ) by θ and preserving the other basis elements. Let A (cid:48) be obtained by composing A with each of R i,j i θ for i ∈ J ∩ I (in any order, since the rotation matrices commute). One checks directlythat R i,jθ Ω( R i,jθ ) T = Ω, where Ω is the standard symplectic form; hence R i,jθ preserves E u so A (cid:48) is symplectic and preserves E u .Writing (cid:89) i ∈ J ∩ I (cid:48) ( ∧ k R i,i j θ ) e kJ = (cid:88) L b L e kL , by a direct computation one checks that b { ,...,n }\ I (cid:48) (cid:54) = 0 if and only if J = J , which impliesthat ( ∧ k A (cid:48) )( e kI ) ∧ e lI (cid:48) (cid:54) = 0. (cid:3) The above then shows density in G kp of metrics with a periodic orbit with the pinchingproperty and with a homoclinic orbit with the twisting property. IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 27
Finally then we have density of “simplicity” of the derivative cocycle:
Theorem 3.22.
The set G kp,t is open and dense in G k .Proof. Density follows from 3.19. Openness follows from yet another argument like that ofLemma 3.1. (cid:3) Proof of Main Theorem
We finish the proof of the main theorem. In what follows, let σ : Σ → Σ be the shiftmap of an invertible subshift of finite type Σ. The suspension of Σ under a continuous f : Σ → R + is the compact metric space:Σ f := (Σ × R ) / (( x, s ) ∼ α n ( x, s ) , n ∈ Z ) , where α ( x, s ) := ( σ ( x ) , s − f ( x )). The shift σ lifts to a continuous-time system σ tf : Σ f → Σ f given by σ tf ( x, s ) = ( x, s + t ) for t ∈ R .First, we need to represent Anosov flows by the suspension of a shift. The following isthe standard statement of the construction of a Markov partition for an Anosov flow: Theorem 4.1. [FH18, Theorem 6.6.5]
Let
Φ : M → M be a C Anosov flow. There is asemiconjugacy from a hyperbolic symbolic flow to Φ that is finite-to-one and one-to-one ona residual set of points, where the roof function for the subshift of finite type correspondsto the travel times between the local sections for the smooth system. For a metric g ∈ G k let ( σ g f ) t : Σ g f → Σ g f be the suspension of a subshift of finite typeand P g a semi-conjugacy from ( σ g f ) t to ϕ tg , constructed by a Markov partition; we referto Σ g as the shift based at g and g as the base of Σ g . Then it is a standard fact [FH18]that both P g and f may be taken to be H¨older continuous as well. Proposition 4.2.
There exists a neighborhood U g of g in G k , such that for g ∈ U g thereexists f g : SM → R H¨older and a semi-conjugacy P g g : Σ g f g → SM between ( σ g f g ) t and ϕ tg .In other words, in a small neighborhood of g all geodesic flows may be represented throughsuspensions over the shift Σ g based at g with different H¨older-continuous roof functions.In particular, by construction, if an orbit O of ϕ tg has a unique lift to Σ g f by P g , thenthe hyperbolic continuation O g of O for the flow ϕ g also has a unique lift to Σ g f g under P g g . Proof.
In general, for a flow Φ (cid:48) sufficiently C close to Φ a C Anosov flow, by structuralstability for Anosov flows, there exists a homeomorphism ψ : M → M which is an orbitconjugacy between the two flows. The homeomorphism may be taken to be H¨older con-tinuous as well (Theorem 6.4.3 in [FH18]). Then ψ ◦ P is a H¨older orbit semi-conjugacyfrom σ tf to Φ (cid:48) . Hence, we obtain that Φ (cid:48) is semi-conjugate to some σ tf (cid:48) : Σ f (cid:48) → Σ f (cid:48) for f (cid:48) : Σ → R H¨older as well obtained by integrating the return times. The result then followssince if two metrics are C k close for k ≥ C close. (cid:3) Fix some g ∈ U g as above. Pulling back by P g g the derivative cocycle Dϕ tg | E ug on theunstable bundle E ug → SM to Σ g f g one obtains a cocycle ( A g g ) t on a bundle E g g → Σ g f g .Recall that σ g : Σ g → Σ g is the return map of the suspension flow on the shift based at g . Following the propositions in Section 2.1 of [BGV03] there exists a distance d g g on Σ g such that the return map of ( A g g ) t defines a dominated cocycle A g g on E g g → Σ g over thediscrete system with holonomies H s,ug,g ; in particular, we may speak of A g g being a simplecocycle.The following proposition now concludes that an open dense set of metrics has simplederivative cocycle when represented over some shift based at a nearby metric : Proposition 4.3.
There exists an open dense set
V ⊆ G k of metrics g ∈ V with thefollowing property: there exists a g ∈ G k such that g ∈ U g given by Proposition 4.2 andthe associated pullback cocycle A g g on E g g over the shift σ g : Σ g → Σ g based at g issimple.Proof. First we prove the following lemma which verifies agreement of holonomies of thegeodesic flow and its symbolic discrete representation:
Lemma 4.4.
Let g ∈ G k and g ∈ U g . The stable and unstable holonomies H s,ug,g of A g g on E g g are given by the center-stable and unstable h cs,cu holonomies of ϕ tg on the stable bundle.More precisely, let x = x × { } ∈ Σ g f , x ∈ Σ g , and y = y × { } ∈ Σ g f , y ∈ Σ g , where y ∈ W s ( x ) , so that y ∈ W cs ( x ) . Let v = P g g ( x ) and w = P g g ( y ) , then h csvw = ( H sg,g ) xy .The analogous result holds for unstable holonomies.In particular, if x is periodic and y is a homoclinic point of x that makes A g g a simplecocycle with x, y playing the role of ˆ p and ˆ z in Definition 2.12 (we say x (resp. y ) has thepinching (resp. twisting) property) then g ∈ G kp,t (see Definition 3.18) with w having thetwisting property with respect to v , and v has the twisting property, where v, w are as in IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 29 the previous paragraph. Conversely, if w has the twisting property with respect to v , andthere exist x, y as in the previous paragraph such that y is a homoclinic point of x , thenthe cocycle A g g is simple.Proof. Throughout the proof, we omit the g superscripts as the base of the shift is fixedand there is no ambiguity. By the proof of Theorem 2.9, as in [BV04], one obtains theholonomy map as a limit: H sx,y = lim n →∞ (( A ng ) x ) − ◦ I σ n xσ n y ◦ ( A ng ) y . As n → ∞ , note that σ n x × { } and σ n y × { } converge to the same stable manifold inΣ f . Hence, if we let T n := (cid:80) n − i =0 f ( σ i x ) so that ( A ng ) x = ( A T n g ) x ×{ } , then (cid:80) n − i =0 f ( σ i y ) − ( T n + r ) →
0, as n → ∞ , where r ∈ R is such that σ rf ( y ) ∈ W s ( x ).On the other hand, using the formula defining the holonomies and the definition of A tg as a pullback cocycle of Dϕ tg | E u : h csvw = lim T →∞ ( Dϕ g | TE u ) − v ◦ I ϕ Tg ( v ) ,ϕ T + rg ( w ) ◦ ( Dϕ g | TE u ) ϕ rg ( w ) ◦ ( Dϕ g | rE u ) w , = lim T →∞ ( A Tg ) − x ◦ I σ Tf y,σ T + rf y ◦ ( A Tg ) σ rf y ◦ ( A rg ) y , so letting T = T n we conclude that H sx,y = h csvw . The claims in the second paragraph about translating simplicity on the discrete systemto the geodesic flow then become immediate from the definitions. (cid:3)
Now we first verify openness of V . For g ∈ V and g as in the definition of V , let v, w be the images under P gg of the points with pinching and twisting p and z for A gg . Then v, w also have pinching and twisting for the holonomies of the geodesic flow by Lemma 4.4.Since by Proposition 3.8 and Proposition 3.19 there is a neighborhood W of g such thatthe hyperbolic continuations of v and w maintain pinching and twisting, for g (cid:48) ∈ W ∩ U g the periodic point x and its homoclinic point y are still lifts of the continuations of v, w and with pinching and twisting by Lemma 4.4, and so for such g (cid:48) the cocycles A g (cid:48) g are allsimple.For the density claim, fix some g ∈ G k . Note that since periodic orbits are dense in SM ,we may start with an orbit O in proof of Proposition 3.3 which intersects transversely the interior of a rectangle defining the Markov partition for the geodesic flow of g . Moreover,the periodic orbits w n constructed in Proposition 3.8 can be taken to cross any sectiontransverse to O as well for n large, and so by periodicity they must also intersect the interior of the rectangle infinitely often. Then for all n large enough there exist unique x n ∈ Σ g [FH18, Claim 6.6.9, Corollary 6.6.12] which lift the w n . Since the orbit withpinching is obtained as a hyperbolic continuation of one of the w n for some g , which maybe taken arbitrarily close to g and thus in particular in U g by density in Proposition 3.8,the orbit with pinching, which we henceforth call v ∈ SM , has a unique lift, which wehenceforth call x , to Σ g . The pinching property of Dϕ g on v immediately translates topinching of A g g over the orbit of x by Lemma 4.4.By Proposition 3.19, for an arbitrarily small perturbation of the metric there exists ahomoclinic point w to v with the twisting property. Again since we use the same base shiftfor different metrics, by [FH18, Claim 6.6.9, Corollary 6.6.12] w has a unique well-definedlift y ∈ Σ which is a homoclinic point of x . Without loss of generality, we may assumethat the original points v, w ∈ SM lift to Σ × { } ⊆ Σ f , so that by Proposition 4.4 abovethe holonomies for the discrete system are given by the center-stable and center-unstableholonomies for the flow, and finally by equivariance given by Proposition 3.19 we obtainthat the twisting property holds for the discrete system; thus, the derivative cocycle on theunstable bundle is simple. (cid:3) Finally, we finish the proof of Theorem 1 . Proof of Theorem 1.1.
Let g ∈ V and g be given as in Proposition 4.3 which is an opendense set of metrics. Let ρ : SM → R be a H¨older potential and µ ρ its associated equi-librium state for the geodesic flow of g . Again, we omit the g superscripts as g is fixed.Let ˜ ρ be the H¨older continuous potential on Σ f g given by ˜ ρ = ρ ◦ P g , and ˜ µ ρ its associatedequilibrium state for σ tf g : Σ f g → Σ f g .It is a well-known fact (see e.g. [BR75]) that P g is in fact a measurable isomorphismbetween (Σ f g , ˜ µ ρ ) and ( SM, µ ρ ). Hence the Lyapunov spectrum of A tg with respect to˜ µ ρ agrees with that of Dϕ tg with respect to µ ρ , and it suffices to show simplicity of thespectrum of the former.Since f g : Σ → R is H¨older, identifying Σ with Σ × { } ⊆ Σ f g , the H¨older continuousfunction: (cid:90) f g ( x )0 ˜ ρ ( x, t ) dt − P ( σ tf g , ˜ ρ ) f g ( x ) , IMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE 31 where P ( σ tf g , ˜ ρ ) is the pressure of σ tf g with respect to ˜ ρ , defines a potential on Σ = Σ ×{ } ⊆ Σ f g and has a unique equilibrium state µ which satisfies, for F ∈ C (Σ f g ): (cid:90) Σ fg F d ˜ µ ρ = (cid:82) Σ (cid:16)(cid:82) f g ( x )0 F ( x, t ) dt (cid:17) dµ (cid:82) Σ f g ( x ) dµ by [FH18, Proposition 4.3.17]. In particular, since µ is an equilibrium state it has localproduct structure.The product µ × dt defines a measure for the suspension flow σ t on Σ (where 1 is theconstant function 1) which has the same Lyapunov spectrum as µ . Since µ × dt and ˜ µ ρ are related by a time change, the Lyapunov spectrum of A tg with respect to µ ρ and theLyapunov spectrum of A g with respect to µ differ by a scalar, see e.g. [Bu17, Proposition2.15]. Hence applying Theorem 2 .
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Department of Mathematics, the University of Chicago, Chicago, IL, USA, 60637
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