Density of positive Lyapunov exponents for symplectic cocycles
aa r X i v : . [ m a t h . D S ] J a n DENSITY OF POSITIVE LYAPUNOV EXPONENTS FORSYMPLECTIC COCYCLES
DISHENG XU
Abstract.
We prove that Sp (2 d, R ), HSp (2 d ) and pseudo unitary cocycleswith at least one non-zero Lyapunov exponent are dense in all usual regularityclasses for non periodic dynamical systems. For Schr¨odinger operators on thestrip, we prove a similar result for density of positive Lyapunov exponents. Itgeneralizes a result of A.Avila in [2] to higher dimensions. Introduction and main result
Let f : X → X be a homeomorphism of a compact metric space, and µ a f − invariant probability measure on X , and F be either R or C . Suppose A : X → SL ( n, F ) or GL ( n, F ) is a bounded measurable map, we can define the linear cocycle( f, A ) acting on X × F n ) as the following:( x, y ) ( f ( x ) , A ( x ) · y )Some examples of linear cocycles are the derivative cocycle ( f, Df ) of a C − map of arbitrary dimensional torus, the random products of matrices, Schr¨odingercocycles, etc.The main object of interest of linear cocycles is the asymptotic behavior of theproducts of A along the orbits of f , especially the Lyapunov exponents. We considerthe following definition.The iterates of ( f, A ) have the form ( f n , A n ), where A n ( x ) := A ( f n − ( x )) · · · A ( x ) , n ≥ , n = 0 A ( f n ( x )) − · · · A ( f − ( x )) − , n ≤ − f, A ) is defined by(1.1) L ( A ) = L ( A ) = L ( f, µ, A ) = lim n →∞ n Z ln k A n ( x ) k dµ ( x )The k − th Lyapunov exponent is defined as,(1.2) L k ( A ) := lim n →∞ n Z ln σ k ( A n ( x )) dµ ( x )where σ k ( A ) is the k − th singular value of A . We denote L k ( A ) := P kj =1 L j ( A ).The following remark gives the well-definedness of all the Lyapunov exponents: Remark . For A ∈ GL ( n, F ) , we can define its natural action, Λ k ( A ) on the space Λ k ( F n ) . Λ k ( A ) · v ∧ · · · ∧ v k := Av ∧ · · · ∧ Av k Date : January 25, 2016.
As a result, for a cocycle ( f, A ) acting on X × F n we can define a cocycle ( f, Λ k ( A )) on ( X, Λ k ( F n )) . By Oseledec theorem the top Lyapunov exponent of cocycle ( f, Λ k ( A )) is L k ( A ) . We say the Lyapunov exponent of linear cocycle A is positive if L ( A ) >
0, theLyapunov spectrum of A is simple if L ( A ) > · · · > L n ( A ) . An important problem in the study of dynamical system is whether we can approx-imate a system by one with hyperbolic behavior. In the setting of linear cocycles,we always assume the base dynamics ( f, µ ) is fixed and only the fiber dynamics A should be allowed to vary. Then we ask whether the given linear cocycle can beapproximated by one with positive Lyapunov exponents (or simple spectrum) insome regularity classes.Basically, the study of Lyapunov exponents of linear cocycles depends on theregularity classes and base dynamics. When the base dynamics has hyperbolicity,and the cocycles are in general position of some higher regularity spaces, they canin most cases ”borrow” some hyperbolicity from the base dynamics. When ( f, µ )is a random system, for example for products of random matrices, simplicity ofLyapunov spectrum was investigated in H.Furstenberg [21], H.Furstenberg and H.Kesten [22], Y.Guivarc’h and A.Raugi [25], etc. In particular, when the support ofthe distribution of random matrices is Zariski dense, we have simple Lyapunov spec-trum, see I.Y.Gol’dsheid, G.A.Margulis’s result in [24] for example. And A.Avilaand M.Viana [12] gave a criterion of simplicity of Lyapunov spectrum for linearcocycles over Markov map and proved the Zorich-Kontsevich conjecture.When the ergodic system ( f, µ ) is hyperbolic, M.Viana [37] proved for any s > C s − cocycles with positive Lyapunov exponents are dense in C s − topology.C.Bonatti and M.Viana [13] proved the cocycles of simple Lyapunov spectrumare dense in the space of fiber bunched H¨older continuous cocycles. For weakerhyperbolicity assumption on base dynamics, i.e. partial hyperbolicity, using thetechniques of partially hyperbolic systems, A.Avila, J.Santamaria, M.Viana [11]proved there is an open dense subset in the space of fiber bunched H¨older continuouscocycles with positive Lyapunov exponents.Of course there are many dynamical systems f without any hyperbolicity. A typ-ical one is quasiperiodic systems. A dynamics system ( f, X, µ ) is called quasiperi-odic if ( f, X ) is an irrational rotation on torus preserving the Lebesgue measure µ . The theory of Schr¨odinger and SL (2 , F )-cocycles over quasiperiodic systems areextensively studied, see [3], [9], [10], [8], [18], [4] for example. In particular, A.Avilaproved the stratified analyticity of Lyapunov exponents and obtained a global the-ory for one frequency quasiperiodic analytic Schr¨odinger cocycles (see [4]).Notice that for the results above in general we need some regularity restrictionfor the cocycles. For example some bunching condition (maybe non-uniformly) andH¨older continuity are usually necessary for discussing cocycles over hyperbolic basesystem, and analyticity is a suitable assumption for cocycles over quasi-periodicsystems. It is more difficult to study Lyapunov exponents for linear cocycles overgeneral base systems and regularity class. Using a semi-continuity argument andKotani theory in [29], [30], A.Avila and D.Damanik proved that if the system ( f, µ )is ergodic, then the set of cocycles with positive Lyapunov exponents is dense in C ( X, SL (2 , R )). A.Avila in [2] extended this result to all usual regularity classes of ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 3 SL (2 , R ) − cocycles using a local regularization formula proved by complexificationmethod.1.1. Main result for symplectic cocycles.
In this paper, we generalize theresult in [2] to symplectic, Hermitian-symplectic and pseudo unitary cocycles.
Definition 1.
The Symplectic group over F , denoted by Sp (2 d, F ) , is the group ofall matrices M ∈ GL (2 d, F ) satisfying M T JM = J, with J = (cid:18) I d − I d (cid:19) . The Hermitian-symplectic group
HSp (2 d ) is defined as: HSp (2 d ) = { M ∈ GL (2 d, C ) : M ∗ JM = J } . The pseudo unitary group U ( d, d ) ⊂ GL (2 d, C ) is defined as: U ( d, d ) := { A : A ∗ (cid:18) I d − I d (cid:19) A = (cid:18) I d − I d (cid:19) } . The special pseudo unitary group SU ( d, d ) is defined as: SU ( d, d ) := { A ∈ U ( d, d ) , det A = 1 } . The special Hermitian symplectic group
SHSp (2 d ) is defined as: SHSp (2 d ) := { A ∈ HSp (2 d ) , det A = 1 } . The Lie algebras of these groups are denoted respectively by sp (2 d, F ) , hsp (2 d ) , u ( d, d ) , su ( d, d ) , shsp (2 d ) . As in [2], we have the following definition for ample subspace of C ( X, G ), where G is a Lie group. Suppose g is the Lie algebra of G . Definition 2.
A topological space B continuously included in C ( X, G ) is ample ifthere exists a dense vector space b ⊂ C ( X, g ) , endowed with a finer (than uniform)topological vector space structure, such that for every A ∈ B , exp( b ) A ∈ B for all b ∈ b , and the map b exp( b ) A from b to B is continuous. Remark . If X is a compact smooth or analytic manifold, then the usual spacesof smooth or analytic maps X → G are ample in our sense. In this paper we prove the following theorem:
Theorem 1.
Suppose f is not periodic on supp ( µ ) , and let B ⊂ C ( X, Sp (2 d, R )) be ample. Then the set { A : L ( A ) > } is dense in B . Corollary 1.
The same result in Theorem 1 holds if we replace Sp (2 d, R ) by SHSp (2 d ) , SU ( d, d ) , HSp (2 d ) and U ( d, d ) . Stochastic Schr¨odinger operators and Jacobi matrices on the strip.
The most studied (Hermitian) symplectic cocycles are stochastic Schr¨odinger oper-ator and Jacobi matrices on the strip, coming from the study of solid physics. Forearlier studies of stochastic Jacobi matrices and Schr¨odinger operator on the stripand its relation to the Aubry dual of quasi-periodic Schr¨odinger operator, see [17],[27], [28], [35] for example.
DISHENG XU
Consider the following Jacobi matrices on the strip : h ω : l ( Z , C d ) → l ( Z , C d ) u ( h ω u )( n ) = u ( n + 1) + u ( n −
1) + v ω ( n ) · u ( n )(1.3)where the potential v ω ( n ) is a d × d Hermitian matrix, when d = 1, it is 1 − dimensionalSchr¨odinger operator (on the line).In this paper we always assume the potential is dynamically defined, i.e. v ( · ) ( n ) := v ( f n ( · )), v : X → Her ( d ) or Sym d R is a bounded measurable map, where ( f, X, µ )is defined as in the beginning of the paper, and Her ( d ), Sym d F are respectivelythe set of Hermitian matrices and symmetric d × d matrices over the field F .Then for energy E , the corresponding eigenequation is the following:(1.4) hu = Eu, with potential v ( f n ( x )) . Notice that for any u : Z → C d satisfies (1.4), we have(1.5) (cid:18) u ( n + 1 u ( n ) (cid:19) = (cid:18) E − v ( f n ( x )) − I d I d (cid:19) · (cid:18) u ( n ) u ( n − (cid:19) Then the associated linear cocycle ( f, A ( E − v ) ) : X × C n → X × C n is defined by(1.6) A ( E − v ) ( x ) = (cid:18) ( E · I d − v ( x )) − I d I d (cid:19) Notice that ( f, A ) is a (Hermitian) symplectic cocycle when E ∈ R .As in [2], we denote L ( A ( E − v ) ) = L ( E − v ). Using a similar method to the proofof Theorem 1, we prove the following result for (Hermitian) symplectic cocyclesrelated to the stochastic Jacobi matrices on the strip with form in (1.3) . Theorem 2.
Suppose f is not periodic on supp ( µ ) and let V ⊂ C ( X, Her ( d )) or C ( X, Sym d R ) be a dense vector space endowed with a finer topological vector spacestructure. Then for any E ∈ R , the set of v such that L ( E − v ) > is dense in V . Now we study Schr¨odinger operator on the strip. Suppose S ⊂ Z ν − is a finiteconnected set , we consider the following operator on l ( Z × S ).(1.7) ˜ u ( h ω ˜ u )( α ) = X | β − α | =1 , β ∈ Z × S ˜ u ( β ) + ˜ v ω ( α )˜ u ( α )where for p = ( x , . . . , x ν ) , q = ( y , . . . , y ν ) , | p − q | := P i | x i − y i | . ˜ v ω is a processergodic under the one-dimensional group of translations.As in [28], (1.7) can be viewed as an example of the stochastic Jacobi matrices onthe strip with form in (1.3). For example if S = { , .., d } ⊂ Z , then the associatedJacobi matrices on strip are as follows: for ˜ u satisfies (1.7), let u : Z → C d suchthat u ( n ) = ( u ( n ) , . . . , u d ( n )) , u i ( n ) = ˜ u ( i, n ) for more general Jacobi matrices with matrices entries, see [35], [33] for example. As in [28], S ⊂ Z ν − is a connected set means every two points of S can be joint by a sequenceof points in S , and any two consecutive points p n , p n +1 satisfy | p n − p n +1 | = 1 . ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 5 then u satisfies (1.3) with potential v ω ( n ) = ˜ v ω ( n,
1) 11 ˜ v ω ( n,
2) 11 ˜ v ω ( n,
3) 1. . . . . . . . .1 ˜ v ω ( n, k −
1) 11 ˜ v ω ( n, k ) For general S , v ω always has entries of ˜ v ω as diagonal elements and with non-randomoff-diagonal elements (be 0 or 1).For Schr¨odinger operator on the strip defined in (1.7), if S ) = d , and weconsider the embedding R d ֒ → Sym d R by identifying a vector with the diagonalelements of a symmetric matrix, then each measurable bounded map v ∈ ( X, R d ) ֒ → ( X, Sym d R ) induces a family of stochastic Schr¨odinger operator on strip, since theoff-diagonal elements of the potential matrix are non-random.For potential v : X → R d ֒ → Sym d R and energy E , We denote by A ( E − v ) the cocycle associated to the eigenequation hu = Eu (with potential v ( f n ( x ))) ofSchr¨odinger operator on the strip. Then we have the following similar result toTheorem 2. Corollary 2.
Suppose f is not periodic on supp ( µ ) and let V ⊂ C ( X, R d ) be adense vector space with a finer topological vector space structure. Then for any E ∈ R , the set of v such that L ( A ( E − v ) ) = L ( E − v ) > is dense in V . Main difficulty of the proof, novelty of the paper and some remarks.
The key step in the proof of our result is to prove Theorem 4 in Chapter 4. Basi-cally speaking, it proves that, if the one parameter family of cocycles R θ A satisfies L ( R θ A ) = 0 for positive Lebesgue measure of θ , then the function m + can deter-mine m − , where m + , m − are defined as in Kotani theory and [9], R θ is the higherdimensional rotation.Before that, we have to prove the monotonicity of fibered rotation function offamily R θ A , see section 2.4 and Lemma 4.13. To prove it, we consider a cone fieldon the Lagrangian Grassmannian induced by the invariant cone on Lie algebra ofHermitian type. This idea is inspired by the study of Maslov index, see [1] and [20]. The main difficulty in the proof of Theorem 4 is to generalize Kotani theory to(Hermitian) symplectic cocycles. Basically the generalization of Kotani theory toSchr¨odinger cocycle on the strip is done by S.Kotani and B. Simon in [28]. Ourapproach is for general symplectic cocycles and is closer to the work of monotoniccocycles in [9]. Considering the geometry of Hermitian symmetric spaces andΛ d ( C d ), Theorem 5 will be further proved by some tricky calculations.In addition, in Appendix, using techniques of monotonic symplectic cocycles (see[31]) we generalize some results of periodic Schr¨odinger operators and SL (2 , R )cocycles in [5],[2] to higher dimensions. After this paper was completed, I found my proof of monotonicity of fibered rotation functionin Chapter 3 partially coincided with the work of Hermann Schulz-Baldes in [34] and the work ofRoberta Fabbri, Russell Johnson and Carmen N´u˜nez in [19] . Thanks Qi Zhou for the references. In fact, we can define monotonicity for symplectic cocycles similarly. Using techniques in [9]and this paper, we can get similar results to [9] for symplectic cocycles, see [31].
DISHENG XU
These ideas are partially inspired by the studies of monotonic cocycles [9], higherdimensional cocycles in [33] and [8], the geometry of Hermitian symmetric space in[15] and [20].It is natural to ask whether the simplicity of Lyapunov spectrum holds for (Her-mitian) symplectic cocycles in a dense set of some regularity set over general basedynamics. The difficulty to prove it is that when only some of the Lyapunov ex-ponents coincides or vanishes we can not get too much dynamical information (forexample, deterministic , see [28] or section 5) on the cocycles . It is also difficultto get the density of positive Lyapunov exponents for cocycles taking values in SL ( n, F ) (except SL (2 , R )), or even to get the simplicity of Lyapunov spectrum.The basic difficulty lies in the following, Kotani theory, [2] and our paper heavilydepend on the fact that the associated groups act biholomorphically on some Her-mitian symmetric spaces (see details in section 2.1), which is not true for SL ( n, F )unless SL (2 , R ).1.4. The structure of the paper.
The outline of this paper is as follows: Chapter2-6 are dedicated to the proof for Theorem 1, in fact the proof can be easily adaptedto prove the rest of the results in this paper.Chapter 2 is a short introduction of the geometry of symplectic action on Siegelupper half plane and its boundary.Chapter 3 is dedicated to complexification of Lyapunov exponent and monotonic-ity of fibered rotation function, which implies an important equation in Lemma 16.Chapter 4 is the proof for Theorem 4, which is the most difficult part in thispaper.Chapter 5,6 are the rest of the proof of Theorem 1, based on the arguments(deterministic, semi-continuity, finitely-many-valued cocycles, local regularizationformula, etc.) for Schr¨odinger and SL (2 , R ) cocycles in [7],[2],[30],[36],[28].Chapter 7 shows how to adapt the proof of Theorem 1 to Hermitian symplecticgroup case, pseudo unitary group case and Schr¨odinger operator on the strip.The appendix is for some properties of generic periodic (Hermitian) symplecticcocycles which are used in the proof of Theorem 6. Acknowledgement.
I would like to express my thanks to my director of thesis,Professor Artur Avila, for his supervision and useful conversations. I would like tothank Xiaochuan Liu for reading the first version of this paper, Qi Zhou for someuseful suggestions for several versions. This research was partially conducted duringthe period when the author visited IMPA, supported by r´eseau franco-br´esilien enmath´ematiques.2.
Geometry of the symplectic group action
Hermitian symmetric space, Bergman Shilov boundary and somenotations.
Basically speaking, Kotani theory, techniques of monotonic cocyclesand Avila’s density result of Sch¨odinger and SL (2 , R )-cocycles heavily depend onthe fact that the group SL (2 , R ) (or SU (1 , C . In particular, SL (2 , R ) and SU (1 ,
1) preservecorresponding Poincar´e metric.The higher dimensional extension of Poincar´e upper half plane (or disc) are Her-mitian symmetric spaces of non compact type. A Hermitian symmetric space isa Hermitian manifold which at every point has an inversion symmetry preserving
ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 7 the Hermitian structure. Hermitian symmetric space appears in the theory of au-tomorphic forms and group representations. An example is the Siegel upper halfplane and its disc model, our proof heavily depends on the fact that symplecticgroup can act on it isometrically.Bergman discovered that, different from the one variable case, for a large classof domain, a holomorphic function of several variables is completely determined byits values on a proper closed subset of the topological boundary of the domain. Wecall the minimal one the Bergman-Shilov boundary or Shilov boundary.In this paper, Shilov boundaries of Hermitian symmetric spaces that we areinterested in are the set of unitary symmetric matrix U sym ( C d ) and unitary group U ( d ) which can be identified with real or complex Lagrangian Grassmannian.We consider the following notations which will be used later. Definition 3.
For a pair of complex d × d matrices M, N , we denote
M > N if ( M − N ) ∗ = M − N and M − N is positive definite. Definition 4.
For a square matrix M , we denote Im ( M ) := M − M ∗ i . For a complexnumber z , R ( z ) := z + z , I ( z ) := z − z . Remark . Later in Chapter 6 we will define R , I for an element in a complexLie algebra g C with its real form g , which coincides with the definition here when g = R , g C = C . Definition 5.
We denote by k · k HS the Hilbert-Schmidt norm of matrix. The symplectic action on the models of Siegel upper half plane.
Weconsider Siegel upper half plane and its disc model, which are the generalization ofPoincar´e upper half plane and Poincar´e disc.
Definition 6.
The Siegel upper half plane SH d is defined as the following: SH d := { X + iY ∈ Sym d C , X, Y ∈ Sym d R , Y > } Definition 7.
We define SD d as the set { Z ∈ Sym d C , I d − Z ¯ Z > } Notice that SD d is the set of complex d × d symmetric matrices with operator normless than . Now we consider the symplectic action on SH d and SD d . lemma . The symplectic group acts on the Siegel upper half plane transitively bythe generalized M¨obius transformations: M = (cid:18) A BC D (cid:19) ∈ Sp (2 d, R ) , Z ∈ SH d , M · Z := ( AZ + B )( CZ + D ) − The stabilizer of the point i · I d ∈ SH d is SO (2 d, R ) ∩ Sp (2 d, R ) .Proof. See [20]. (cid:3)
Consider the Cayley element(2.1) C := 1 √ (cid:18) I d − i · I d I d i · I d (cid:19) then for all 2 d × d complex matrix A , we denote ◦ A := CAC − . We have: DISHENG XU lemma . (1). The map A ◦ A is a Lie group isomorphism from Sp (2 d, R ) to U ( d, d ) ∩ Sp (2 d, C ) .(2). The group U ( d, d ) ∩ Sp (2 d, C ) acts on the set SD d transitively by the gen-eralized M¨obius transformations: suppose M = (cid:18) A BC D (cid:19) ∈ U ( d, d ) ∩ Sp (2 d, C ) then Z ∈ SD d , M · Z := ( AZ + B )( CZ + D ) − (3).The Cayley element induces a fractional transformation identifying SH d with SD d , i.e. for Z ∈ SH d , Φ C ( Z ) := ( Z − i · I d )( Z + i · I d ) − , we have the followingcommutative diagram: SH d A / / Φ C (cid:15) (cid:15) SH d Φ C (cid:15) (cid:15) SD d ◦ A / / SD d Proof.
See [20]. (cid:3)
Now we define the projective model for SH d and SD d . Consider the com-plex Grassmannian G d,d C , the set of all d − dimensional subspaces of C d , andlet M d,d ( C ) be the space of all full rank 2 d × d complex matrices and view thecolumns of these matrices as a basis of a subspace of C d .If we consider the action of GL ( d, C ) by right multiplication on M d,d ( C ), thenthe Grassmannian is G d,d = M d,d ( C ) /GL ( d, C )For each (cid:18) AB (cid:19) , we use (cid:20) AB (cid:21) to represent the class of (cid:18) AB (cid:19) . The projective model SP H d of SH d will be the set of all classes that admit a representative of the type (cid:18) ZI d (cid:19) with Z ∈ Sym d C , Im ( Z ) > SP H d is the left matrix multiplication by a representativeof the class: (cid:18) A BC D (cid:19) · (cid:20) ZI d (cid:21) = (cid:20) AZ + BCZ + D (cid:21) = (cid:20) ( AZ + B )( CZ + D ) − I d (cid:21) The map connecting SH d to SP H d is SH d → SP H d Z (cid:20) ZI d (cid:21) Similarly we can define the projective model
SP D d of the disc SD d as the setof classes in M d,d ( C ) that admit a representative of the type: (cid:18) ZI d (cid:19) with Z ∈ Sym d C , k Z k < SP D d and the identification between SP D d and SD d can be defined similarly.2.3. The boundaries of different models.
ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 9
Stratification of finite and infinite boundaries.
All the properties in this sub-section can be found in section 3 of [20].Consider the boundary of SD d in Sym d C . ∂SD d = { Z T = Z, k Z k = 1 } The M¨obius transform is well-defined on ∂SD d . Moreover, it has a stratification,the strata are, for 1 ≤ k ≤ d,∂ k SD d = { Z ∈ ∂SD d : rank( I − ZZ ) = d − k } In particular, ∂ d SD d = U sym ( C d ) = U d ∩ Sym d C , which is the Shilov boundary of SD d , and it is an orbit of U ( d, d ) ∩ Sp (2 d, C ) − action.We can also take the closure of the Siegel upper half plane in Sym d C , SH d = { Z ∈ Sym d C : Im( Z ) ≥ } and then map it to ∂SD d using the extensions of the map Φ C , Φ − C defined inLemma 2. Notice that Φ − C is not defined on the set { Z ∈ ∂SD d , ∈ the spectrum of Z } We call this set the infinite boundary and its complement in ∂SD d the finite bound-ary .The finite boundary contains a part of every stratum. We have the followingproperty: the image of the finite part of the stratum ∂SD d under the extension ofΦ − C is fin( ∂ k SH d ) = { Z ∈ Sym d C : Im( Z ) ≥ , rank(Im( Z )) = d − k } An atlas of Shilov boundary.
Consider fin( ∂ d SH d ) = Sym d R , then Φ C re-stricted to Sym d R gives a chart of { Z ∈ ∂ d SD d , / ∈ the spectrum of Z } Similarly, for an element g ∈ SL (2 , R ) , g = (cid:18) a bc d (cid:19) , composite with the Caylayelement, we get a chart of a dense subset of ∂ d SD d :Φ Cg : Sym d R → { Z ∈ ∂ d SD d , a − ica + ic / ∈ the spectrum of Z } Z (( a − ic ) Z + ( b − id ))(( a + ic ) Z + ( b + id )) − As a result, if we pick a sequence of g k such that a k − ic k a k + ic k take more than d differentvalues, then the family { Φ Cg k : Sym d R → ∂ d SD d } give an atlas for ∂ d SD d = U sym ( C d ).2.4. Invariant cone field and partial order.
Invariant cone field.
We construct an invariant cone field C on Shilov bound-ary. For fin( ∂ d SH d ) = Sym d R , we consider a cone field { h : h ∈ T Sym d R , h > } .Then using the tangent map of Φ Cg k defined in last subsection, we get a cone field C on T U sym ( C d ).It is easy to check that the cone field { h > } is invariant under symplecticaction. Therefore the cone field C is well-defined and invariant under U ( d, d ) ∩ Sp (2 d, C ) − action. Partial order defined on the universal covering space.
For U sym ( C d ), considerits universal covering space \ U sym ( C d ), denote the covering map by Π : \ U sym ( C d ) → U sym ( C d ). Then we can lift the cone field C to a cone field b C on \ U sym ( C d ), whichis invariant under the lift of the Sp (2 d, C ) ∩ U ( d, d ) − action.Using ˆ C , we define a partial order ” < ” on \ U sym ( C d ); we say ˆ Z < ˆ Z , if thereis an C path p : [0 , → \ U sym ( C d ) such that(2.2) p (0) = ˆ Z , p (1) = ˆ Z , p ′ ( t ) ∈ ˆ C ( p ( t ))For the determinant function restricted on U sym ( C d ), we pick a continuous liftˆdet : \ U sym ( C d ) → R , such that(2.3) π ◦ ˆdet = det ◦ Πwhere(2.4) π : R → S , π ( x ) = e ix To check ” < ” is actually a partial order and for later use, we have lemma . (1) Suppose ˆ Z ∈ \ U sym ( C d ) , for a path p : [0 , → \ U sym ( C d ) suchthat p (0) = ˆ Z, p ′ (0) ∈ b C ( Z ) , we have ddt | t =0 ˆdet( p ( t )) > The order ” < ” defined in (2.2) is a strict partial order, i.e. there is no ˆ Z, ˆ Z , ˆ Z ∈ \ U sym ( C d ) such that ˆ Z < ˆ Z and ˆ Z < ˆ Z < ˆ Z . (3) For any Z ∈ U sym ( C d ) , any continuous lift of the path θ e iθ Z is mono-tonic with respect to the order ” < ” . (4) Any lift of the Sp (2 d, C ) ∩ U ( d, d ) − action preserves the order ” < ” .Proof. Suppose 1 / ∈ the spectrum of Z = Π( ˆ Z ), by computation for D Π( p ′ (0)) := H ∈ C ( Z ), there is an h ∈ Sym d R , h > H = − i ( Z − h ( Z − p ( t ))) = det( Z + tH + o ( t ))= det( Z − it ( Z − h ( Z −
1) + o ( t ))= det( Z ) det(1 − it (1 − Z ∗ ) h ( Z −
1) + o ( t ))( Z is a unitary matrix.)= det( Z ) det(1 + it (1 − Z ∗ ) h (1 − Z ) + o ( t ))notice that (1 − Z ∗ ) h (1 − Z ) is positive definite, lift to the covering space we have ddt | t =0 ˆdet( p ( t )) > ∈ the spectrum of Z , we can get other expression of the tangentvectors in C ( Z ) by Φ Cg k , and the proof is similar. In summary we get the proof of(1).As a corollary, we have(2.5) if ˆ Z < ˆ Z then ˆdet( ˆ Z ) < ˆdet( ˆ Z )which implies (2).For (3). by taking the derivative, we need to prove iZ ∈ C ( Z ). We only prove itin the case 1 / ∈ the spectrum of Z , for other cases the proof is similar. ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 11
By computation, C ( Z ) = {− i ( Z − h ( Z − , h > , h ∈ Sym d R } . Take h = − Z (1 − Z ) − , then it can be checked that h ∈ Sym d R , h >
0, and − i ( Z − h ( Z −
1) = iZ , which implies iZ ∈ C ( Z ).(4). is the corollary of invariance of the cone field under the lift of the Sp (2 d, C ) ∩ U ( d, d ) − action. (cid:3) Bergman metric and the volume form on SD d . In this section, we definethe Bergman metric on SD d which is a generalization of Poincar´e metric on thePoincar´e disc. (see [32] for example) In particular, the symplectic group actionpreserves the Bergman metric. Definition 8.
Let D be a bounded domain of C n , dλ be the Lebesgue measure on C n , let L ( D ) be the Hilbert space of square integrable functions on D , and let L ,h ( D ) denote the subspace consisting of holomorphic functions in D , the L ,h ( D ) is closed in L ( D ) .For every z ∈ D , the evaluation ev z : f f ( z ) is a continuous linear func-tional on L ,h ( D ) . By the Riesz representation theorem, there is a function η z ( · ) ∈ L ,h ( D ) such that ev z ( f ) = Z D f ( ζ ) η z ( ζ ) dλ ( ζ ) The Bergman kernel K is defined by K ( z, ζ ) = η z ( ζ ) . Definition 9.
Let D ⊂ C n be a domain and let K ( z, w ) be the Bergman kernel on D , consider a Hermitian metric on the tangent bundle of T z C n by g ij ( z ) := ∂ ∂z i ∂ ¯ z j log K ( z, z ) for z ∈ D . Then the length of a tangent vector ξ ∈ T z C n is given by k ξ k B,z := vuut n X i,j =1 g ij ( z ) ξ ¯ ξ j This metric is called Bergman metric on D . We denoted by d the Bergman metric on SD d . We have the following lemmafor d . lemma . (1).For A ∈ Sp (2 d, R ) , Z , Z ∈ SD d , d ( ◦ AZ , ◦ AZ ) = d ( Z , Z ) . (2).For t ∈ (0 , , t · SD d := { tZ, Z ∈ SD d } is a bounded precompact set undermetric d . And we have d ( tZ , tZ ) ≤ td ( Z , Z ) Proof. (1) is the basic property of Bergman metric, i.e. Bergman metric is invariantunder bi-holomorphic map.(2) see Lemma 6 of [15]. (cid:3)
We give a explicit formula of Lebesgue density dλ on Sym d C . Let e ij denote thematrix such that all the entries are 0 except the one in i − th row and j − th columnis 1. Let E ii = e ii and E ij ( i = j ) = e ij + e ji , then E ij , i ≤ j form a basis of Sym d C .Then we can define the Lebesgue density on Sym d C , i.e.(2.6) | dE ∧ d ¯ E ∧ · · · ∧ dE ij ∧ d ¯ E ij ∧ · · · ∧ dE dd ∧ d ¯ E dd | , i ≤ j For Z ∈ SD d , let V ( Z ) dλ ( Z ) be a volume form on SD d induced by the Bergmanmetric on point Z . Without loss of generality, we can assume V (0) = 1. Then wehave: lemma . If σ i ( Z ) , ≤ i ≤ d are the singular values of Z , then V ( Z ) = Π ≤ i ≤ d (1 − σ i ( Z ) ) − ( d +1) Proof. see Lemma 6 in [15] for a computation of Riemann tensor for Bergmanmetrics for general Hermitian symmetric space. (cid:3) Fibered rotation function and complexification of Lyapunovexponents
Let us now fix A ∈ L ∞ ( X, Sp (2 d, R )). For σ ∈ R , t ≥ σ + it ∈ C + ∪ R , weconsider the following deformation of A : A σ + it := (cid:18) cos( σ + it ) · I d sin( σ + it ) · I d − sin( σ + it ) · I d cos( σ + it ) · I d (cid:19) · A Notice that ◦ A σ + it = (cid:18) e − t e t (cid:19) (cid:18) e iσ e − iσ (cid:19) ◦ A The main aim of this chapter is the following theorem, which gives the complex-ification of L d ( A ): Theorem 3.
There is a function ζ defined on C + ∪ R satisfying the followingproperties: . ζ is a holomorphic on C + . . ζ ’s real part ρ is continuous on C + ∪ R , non-increasing on R . . − ζ ’s imaginary part = L d ( A σ + it ) , which is subharmonic on C + ∪ R . Remark . The function ρ defined here is the fibered rotation function (upto multiply π ) in [9] . It is a generalization of fibered rotation number for aSchr¨odinger or SL (2 , R ) − cocycle homotopic to identity. The strategy to prove Theorem 3 is similar to the discussion in section 2 of [9].But different from the 1 − dimensional case, the proof of monotonicity of the fiberedrotation function is not trivial. We have to use the partial order defined in the lastsection.3.1. Definition of ζ . Denoted by Υ the set { A ∈ Sp (2 d, R ) , ◦ A · SP D d ⊂ SP D d } where SP D d is the projective model of the disc SD d . For a matrix A ∈ Υ, we candefine the function τ A : SD d → GL ( d, C ) satisfying the following:(3.1) ◦ A (cid:18) Z (cid:19) = ◦ A · Z ! τ A ( Z )In fact the M¨obius transformation: ◦ A · Z is well-defined for A ∈ Υ , Z ∈ SD d :suppose ◦ A = (cid:18) ∗ ∗ C D (cid:19)
ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 13
Since ◦ A · SP D d ⊂ SP D d , then (cid:20) ∗ CZ + D (cid:21) ∈ SP D d , therefore CZ + D is invertible.As a result, τ A ( Z ) = CZ + D .For Υ, denoted by ˆΥ its universal cover considered as a topological semi-groupwith unity ˆid. Then there exists a unique continuous map ˆ τ :(3.2) ˆ τ : ˆΥ × SD d → C such that ˆ τ ( ˆid , Z ) = 0 and e i ˆ τ ( ˆ A,Z ) = det( τ A ( Z ))This map satisfies(3.3) ˆ τ ( ˆ A ˆ A , Z ) = ˆ τ ( ˆ A , ◦ A · Z + ˆ τ ( ˆ A , Z ))and the following lemma: lemma . For any ˆ A ∈ ˆΥ , and any Z, Z ′ ∈ SD d , I ˆ τ ( ˆ A, Z ) = −| ln det( τ A ( Z )) | (3.4) | R ˆ τ ( ˆ A, Z ) − R ˆ τ ( ˆ A, Z ′ ) | < dπ (3.5) Proof. (3.4) is the consequence of (3.2). For (3.5), suppose ◦ A = (cid:18) ∗ ∗ C D (cid:19) . No-tice that det( τ A ( Z )) = det( D ) det(1 + D − CZ ), and by Proposition 2.3 of [33], k D − C k ≤
1. Then by well-definedness of M¨obius transformation on SD d , thespectrum of matrix 1 + D − CZ, Z ∈ SD d is contained in a half plane, which im-plies (3.5). (cid:3) Now if γ : [0 , → Υ is continuous, and ˆ γ : [0 , → ˆΥ is a continuous lift, wedefine δ γ ˆ τ ( Z , Z ) = ˆ τ (ˆ γ (1) , Z ) − ˆ τ (ˆ γ (0) , Z ); notice that it is independent of thechoice of the lift.For x ∈ X , consider a path γ x ( s ) := A l z ,z ( s ) ( x ) , s ∈ [0 , l z ,z : [0 , → C + ∪ R is a continuous path such that l z ,z (0) = z , l z ,z (1) = z . Then we candefine δ z ,z ξ : X × SD d × SD d → C by δ z ,z ξ ( x, Z , Z ) = δ γ x ˆ τ ( Z , Z ). Noticethat δ z ,z ξ is independent of the choice of l z ,z .Using the dynamics f : X → X , we define δ z ,z ξ n : X × SD d × SD d → C δ z ,z ξ n ( x, Z , Z ) := 1 n n − X k =0 δ z ,z ξ ( f k ( x ) , ◦ A kz ( x ) · Z , ◦ A kz ( x ) · Z )where ◦ A k ( x ) := ◦ A ( f k − ( x )) · · · A ( x ).We denote δ z ξ short for δ ,z ξ . As in [9], we study the limit of δ z ξ n : lemma . The limit of I δ z ξ n ( x, Z , Z ) exists for µ − almost every x , all z ∈ C + ,and all Z , Z ∈ SD d . Moreover it is independent of the choice of Z .Proof. The proof of d = 1 can be found in Lemma 2.3 of [9]. For general d , for any Z ∈ SD d we identify Z with a vector v ( Z ) ∧ · · · ∧ v d ( Z ) ∈ Λ d ( C d ), where v i ( Z )are the column vectors of the matrix (cid:18) ZI d (cid:19) . Therefore we define L d ( A, x ) := lim n →∞ n ln || Λ d ( A )( x ) || (3.6) L d ( A, x, Z ) := lim n →∞ n ln || Λ d ( A )( x ) · Z || (3.7) By Oseledec theorem, the limit exists for µ − almost every x ∈ X and all Z ∈ SD d .We claim that for µ − almost every x ∈ X , for all z ∈ C + , Z , Z ∈ SD d ,(3.8) lim n →∞ I δ z ξ n ( x, Z , Z ) = L d ( A, x, Z ) − L d ( A z , x )The proof is basically the same as Lemma 2.3 of [9], we only need to check thefollowing: for z ∈ C + , Z ∈ SD d , (cid:20) ZI d (cid:21) transverses to all the Oseledec stable subspaceof ◦ A z ( x ).Consider A σ + it , t >
0. By Lemma 4, ◦ A σ + it uniformly contracts the Bergmanmetric of SD d , and there exists a measurable function m + ( σ + it, · ) : X → SD d which is bounded from ∂SD d and holomorphically depends on σ + it , such that(3.9) m + ( σ + it, f ( x )) = ◦ A σ + it ( x ) · m + ( σ + it, x )Moreover (cid:20) m + ( σ + it, x ) I d (cid:21) in the Grassmannian represents the unstable directionof the cocycle Λ d ( ◦ A σ + it ) (see remark of [33], or section 3 and section 6 of [8]).In particular, for all Z ∈ SD d , z ∈ C + , the distance d ( ◦ A nz ( x ) · Z, m + ( z, f n ( x )))goes to 0 exponentially fast, by Oseledec theorem, Z must transversal to all theOseledec stable subspace. (cid:3) lemma . For all z ∈ C + ∪ R , lim n →∞ R X R δ z ξ n ( x, Z , Z ) dµ ( x ) exists for all Z , Z ∈ SD d . Moreover, it is continuous on C + ∪ R and independent of the choiceof Z , Z .Proof. For z , z ∈ C + ∪ R , Z , Z ∈ SD d , let a n ( z , z , Z , Z ) := Z X R δ z ,z ξ n ( x, Z , Z ) dµ ( x )As in section 2 of [9], by (3.5) and (3.3), we have(3.10) | R δ z ,z ξ n ( x, Z , Z ) − R δ z ,z ξ n ( x, Z ′ , Z ′ ) | < dπn Then for any n, l >
0, by f − invariance of µ and last equation, | a n ( z , z , Z , Z ) − a l ( z , z , Z , Z ) | (3.11) ≤ | a n ( z , z , Z , Z ) − a nl ( z , z , Z , Z ) | + | a l ( z , z , Z , Z ) − a nl ( z , z , Z , Z ) |≤ l Z X n − X j =0 | R δ z ,z ξ n ( x, Z , Z ) − R δ z ,z ξ n ( x, A njz ( f − nj ( x )) · Z , A njz ( f − nj ( x )) · Z ) | dµ ( x ) ++ 1 n Z X l − X j =0 | R δ z ,z ξ l ( x, Z , Z ) − R δ z ,z ξ l ( x, A ljz ( f − lj ( x )) · Z , A ljz ( f − lj ( x )) · Z ) | dµ ( x ) ≤ dπn + 2 dπl ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 15
Since A ∈ L ∞ ( X, Sp (2 d, R )), for any Z , Z , a n ( · , · , Z , Z ) is continuous. By(3.11), a n ( · , · , Z , Z ) converges ( uniformly on any bounded subset ) to a continuousfunction on ( C + ∪ R ) . By (3.10) we know this limit function does not depend onthe choice of Z , Z . (cid:3) Remark . By the same proof, we can prove for any measurable sections Z , Z : X → SD d lim n →∞ R X R δ z ξ n ( x, Z ( x ) , Z ( x )) dµ ( x ) exists and does not depend on the choiceof Z , Z . Now we can define function ζ : ζ ( z ) := lim n →∞ Z X R δ z ξ n ( x, , dµ ( x ) − iL d ( A z )then we claim ζ satisfies all conditions of Theorem 3.3.2. subharmonicity and holomorphicity. We have the following lemma forthe subharmonicity of Lyapunov exponents. lemma . The map z L k ( A z ) is a subharmonic function for A ∈ L ∞ ( X, GL ( d, C )) .Proof. Notice that z L k ( A z ) is the limit of the decreasing sequence of sub-harmonic functions z n R X ln k Λ k ( A z ) n k HS dµ (see [2] Lemma 2.3 for exam-ple). (cid:3) By (3.8) and the subharmonicity of Lyapunov exponents we get ζ satisfies (3).of Theorem 3. By Lemma 8, ρ is a continuous function on C + ∪ R .Now we prove ζ is a holomorphic function on C + . For z ∈ C + , since m + ( z, x )depends on z holomorphically, Z X δ z ξ n ( x, , m + ( z, x )) dµ ( x )is a sequence of uniformly bounded holomorphic functions of z . By the proof ofLemma 7 and Remark 5,lim n →∞ Z X δ z ξ n ( x, , m + ( z, x )) dµ ( x ) = lim n →∞ Z X δ z ξ n ( x, , dµ ( x ) . Then by Montel theorem, lim n →∞ R X δ z ξ n ( x, , m + ( z, x )) dµ ( x ) depends on z holomorphically. By the definition of ζ we get ζ is a holomorphic function on C + .3.3. Fibered rotation function is non-increasing.
To prove Theorem 3, weonly need to prove ρ is non-increasing on R . At first, we give a proof for d = 1,which gives us the basic idea for the general case.For all z ∈ S , and any lift of A ∈ SL (2 , R ), we have the following equation:(3.12) ◦ A · z = e − i R (ˆ τ ( ˆ A,z )) z Notice that lim R X R δ θ ξ n ( x, z , z ) µ ( x ) does not depend on the choice of z , z ,we can assume z , z ∈ S . Then to prove ρ is non-increasing on R , by definition of ζ and (3.12) we onlyneed to prove for any x ∈ X, z ∈ S , n ∈ N and any continuous lift of the path ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · z, θ ∈ R , denoted as \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · z is monotonic with respect to θ . Here the lift ˆ γ for a curve γ : R → S is a continuousfunction on R such that π ◦ ˆ γ = γ , where π ( x ) = e ix .In fact, for θ > \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · z = \ e iθ ◦ A ( f n ( x )) e iθ ◦ A ( f n − ( x )) · · · e iθ ◦ A ( x ) · z> \ ◦ A ( f n ( x )) e iθ ◦ A ( f n − ( x )) · · · e iθ ◦ A ( x ) · z (the lift of the rotation is a translation) > \ ◦ A ( f n ( x )) ◦ A ( f n − ( x )) · · · e iθ ◦ A ( x ) · z (the lift of the ◦ A action preserves the order) > · · · > \ ◦ A ( f n ( x )) ◦ A ( f n − ( x )) · · · ◦ A ( x ) · z Then ρ is non-increasing when d = 1.For d >
1, we have the following lemma to replace (3.12), lemma . For all Z ∈ U sym ( C d ) , and any lift of A ∈ Sp (2 d, R ) , (3.13) det( ◦ A · Z ) = e − i R (ˆ τ ( ˆ A,Z )) det( Z ) Proof.
By (3.3), ˆ τ behaves well under the iteration, so by Cartan decomposition of Sp (2 d, R ), we only need to prove (3.13) for ◦ A = (cid:18) U ( U − ) T (cid:19) or (cid:18) ( S + S − ) ( S − S − ) ( S − S − ) ( S + S − ) (cid:19) where U is an arbitrary unitary matrix, S is an arbitrary real non-singular diagonal d × d matrix.For the first case,det( ◦ A · Z ) = det( U ZU T ) = det( U ) det( Z ) = e − i R (ˆ τ ( ˆ A,Z )) det( Z )For the second case,det( ◦ A · Z )= det(( S + S − ) Z + ( S − S − )) det(( S − S − ) Z + ( S + S − )) − = det(( S + S − ) + ( S − S − ) Z ) det(( S − S − ) Z + ( S + S − )) − · det( Z )( since Z ∈ U sym ( C d ) , Z − = Z )= e − i Arg(det(( S − S − ) Z +( S + S − ))) det( Z )( since S is a real matrix)= e − i R (ˆ τ ( ˆ A,Z )) det( Z ) (cid:3) ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 17
Come back to the proof of the non-increasing property of ρ . As in the case d = 1,by (3.13), we have to prove for all x ∈ X, Z ∈ U sym ( C d ), any continuous lift of thepath det( ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z ) , θ ∈ R is monotonic with respect to θ .In other words, we need to prove for any continuous lift of the path ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z , denoted as \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z , we have that ˆdet( \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z ) ismonotonic with respect to θ , where ˆdet is defined in (2.3) .By (3), (4) of Lemma 3, using the order defined in the subsection 2.4.2, as the1 − dimensional case, we have for θ > \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z > \ ◦ A ( f n ( x )) · · · ◦ A ( x ) · Z But by (1). of Lemma 3, ˆdet is monotonic with respect to the order ” < ”. Com-bining with last equation, we have that the function ˆdet( \ ◦ A θ ( f n ( x )) · · · ◦ A θ ( x ) · Z ) ismonotonic with respect to θ , which completes the proof of Theorem 3.4. A Kotani theoretic estimate
The main aim of this chapter is Theorem 4, which is a higher dimensional gen-eralization of Lemma 2.6 in [9]. We introduce the concept of m − − function firstly.4.1. The m − -function. By Lemma 4, ◦ A − σ − it , t > SD d . we can define m − ( σ − it, · ) ∈ SD d , t > σ − it holomorphically, such that(4.1) m − ( σ − it, f ( x )) = ◦ A σ − it ( x ) · m − ( σ − it, x )For later use, we consider the following property of m − : for t >
0, by (4.1) andthe definition of function τ ( · ) ( · ), there exists a function τ A σ − it ( x ) ( m − ( σ − it, x )) ∈ GL ( d, C ) such that(4.2) ◦ A σ − it (cid:18) m − ( σ − it, x ) I d (cid:19) = (cid:18) m − ( σ − it, f ( x )) I d (cid:19) τ A σ − it ( x ) ( m − ( σ − it, x ))Moreover we have: lemma . (4.3) ◦ A σ + it (cid:18) I d m − ( σ − it, x ) (cid:19) = (cid:18) I d m − ( σ − it, f ( x )) (cid:19) τ A σ − it ( x ) ( m − ( σ − it, x )) Proof.
We denote A for A σ + it , A − for A σ − it , m − for m − ( σ − it, x ), ˜ m − for m − ( σ − it, f ( x )), τ − for τ A σ − it ( x ) ( m − ( σ − it, x )). Recall that C is the Cayley element definedin (2.1), then by (4.2) we have ◦ A − (cid:18) m − I d (cid:19) = (cid:18) ˜ m − I d (cid:19) τ − CA − C − (cid:18) m − I d (cid:19) = (cid:18) ˜ m − I d (cid:19) τ − ( by definition of ◦ A ) A − C − (cid:18) m − I d (cid:19) = C − (cid:18) ˜ m − I d (cid:19) τ − Take complex conjugate for both sides of last equation, we have AC − (cid:18) m − I d (cid:19) = C − (cid:18) ˜ m − I d (cid:19) τ − AC − ( CC − (cid:18) m − I d (cid:19) ) = C − (cid:18) ˜ m − I d (cid:19) τ − CAC − ( CC − (cid:18) m − I d (cid:19) ) = CC − (cid:18) ˜ m − I d (cid:19) τ −◦ A ( CC − (cid:18) m − I d (cid:19) ) = CC − (cid:18) ˜ m − I d (cid:19) τ − Notice that CC − = (cid:18) I d I d (cid:19) , we have ◦ A (cid:18) I d m − (cid:19) = (cid:18) I d ˜ m − (cid:19) τ − (cid:3) Now we can state Theorem 4.
Theorem 4.
For almost every σ ∈ R such that L ( A σ ) = 0 , we have that:(1). lim sup t → + Z X − k m + ( σ + it, x ) k dµ ( x )+ Z X − k m − ( σ − it, x ) k dµ ( x ) < ∞ (2). lim inf t → + Z X k m + ( σ + it, x )) − m − ( σ − it, x ) k dµ ( x ) = 0To prove Theorem 4, we introduce the following concepts.4.2. q − function and Lyapunov exponents.Definition 10. Consider the derivative of the holomophic map Z ◦ A σ + it ( x ) · Z at point m + ( σ + it, x ) , denote q σ + it ( x ) as the Jacobian of the derivative with respectto the volume form induced by the Bergman metric. By the discussion before Lemma 5, we have the following expression of q − function: lemma . (4.4) q σ + it ( x ) = | dm + ( σ + it, f ( x )) dm + ( σ + it, x ) | V ( m + ( σ + it, f ( x )) V ( m + ( σ + it, x )) where | dm + ( σ + it,f ( x )) dm + ( σ + it,x ) | is the Jacobian of the map Z ◦ A σ + it ( x ) · Z at point m + ( σ + it, x ) with respect to the Lebesgue measure dλ defined in (2.6) . The following lemma gives a explicit formula of | dm + ( σ + it,f ( x )) dm + ( σ + it,x ) | . lemma . | dm + ( σ + it,f ( x )) dm + ( σ + it,x ) | = | det( τ A σ + it ( x )) | − d +1)ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 19 Proof.
We only need to prove the following: for an arbitrary Z ∈ Sym d C , (cid:18) A BC D (cid:19) ∈ Sp (2 d, C ) such that CZ + D is invertible, the map Sym d C → Sym d C X ( AX + B )( CX + D ) − at point Z has Jacobian (with respect to dλ ) | det( CZ + D ) | − d +1) .At first, by computation the tangent map of X → ( AX + B )( CX + D ) − atpoint Z is: Sym d C Sym d C H → ( A − ( AZ + B )( CZ + D ) − C ) · H · ( CZ + D ) − We need the following equation for symplectic group: lemma . For an arbitrary Z ∈ Sym d C , (cid:18) A BC D (cid:19) ∈ Sp (2 d, C ) such that CZ + D is invertible: we have that (4.5) ( A − ( AZ + B )( CZ + D ) − C ) = ( CZ + D ) − T Proof.
Since (cid:18)
A BC D (cid:19) ∈ Sp (2 d, C ), we have that(4.6) A T C, B T D are symmetric. A T D − C T B = 1Moreover, for all Z ∈ Sym d C such that CZ + D is invertible, we have that(4.7) ( AZ + B )( CZ + D ) − is symmetric.then(4.8) ( AZ + B )( CZ + D ) − = ( D T + ZC T ) − ( B T + ZA T )By (4.7), to prove Lemma 14, we have to prove:(4.9) ( A − ( D T + ZC T ) − ( B T + ZA T ) C )( CZ + D ) T = I d Multiply by D T + ZC T from the left to both sides, we need to prove(4.10) ( D T + ZC T ) A ( CZ + D ) T = ( B T + ZA T ) C ( CZ + D ) T + ( D T + ZC T )which is the consequence of (4.6). (cid:3) Come back to the proof of Lemma 13, by last lemma the tangent map of X → ( AX + B )( CX + D ) − at point Z is Sym d C Sym d C H → ( CZ + D ) − T · H · ( CZ + D ) − So Lemma 13 is the consequence of the following lemma: lemma . Suppose g ∈ GL ( d, C ) , the linear map Sym d C → Sym d C H g T Hg has jacobian | det g | d +1) with respect to the density dλ on Sym d C . Proof.
The Jacobian behaves well under the multiplication on GL ( d, C ). By thepolar decomposition of GL ( d, C ), we only need to prove the lemma in the case g is diagonal or g is contained in the unitary group. When g is diagonal, the lemmacan be verified by computation directly. Notice the Jacobian of the map gives ahomomorphism from GL ( d, C ) to ( R + , × ). So it maps the unitary group to theunique compact subgroup of ( R + , × ): the identity. (cid:3)(cid:3) By our construction of m + , for t > (cid:20) m + I d (cid:21) represents the unstable direction ofthe cocycle Λ d ( ◦ A ). As a result, we have(4.11) L d ( A σ + it ) = Z X ln | det τ A σ + it ( x ) ( m + ( σ + it, x )) | dµ ( x )Combining (4.11) and Lemma 12, 13, we get(4.12) L d ( A σ + it ) = 12( d + 1) Z X − ln q σ + it ( x ) dµ ( x )4.3. Boundary behavior of Lyapunov exponents.
As in Kotani theory and [9],using the results of Theorem 3, we get the following lemma for boundary behaviorof Lyapunov exponents. lemma . For almost every σ ∈ R such that L ( A σ ) = 0 , (4.13) lim t → + L d ( A σ + it ) t − ∂L d ( A σ + it ) ∂t = 0 Proof.
We follow the proof of Theorem 2.5 in [9]. By upper semi-continuity of L d ,for every σ ∈ R such that L ( A σ ) = 0, we have(4.14) lim t → + L d ( A σ + it ) = 0Then lim t → + L d ( A σ + it ) t = lim t → + L d ( A σ + it ) − L d ( A σ + i + ) t = lim t → + R t ∂L d ( A σ it ) ∂t dtt To prove Lemma 16, we only need to prove the following limit exists for almostevery σ ∈ R .(4.15) lim t → + ∂L d ( A σ + it ) ∂t By Cauchy-Riemann equations,(4.16) ∂L d ( A σ + it ) ∂t = − ∂ρ∂σ ( σ + it )By Theorem 3, since the map ρ is harmonic on C + , continuous on C + ∪ R , non-increasing on R , one can say that for Lebesgue almost every σ ∈ R , (see Theorem2.5 of [9])(4.17) lim t → ∂ρ∂σ ( σ + it ) = ddσ ρ ( σ ) ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 21
Since ρ is non-increasing, the derivative of ρ on R exists almost every where, whichimplies the limit in (4.15) exists for almost every σ . (cid:3) Proof of Theorem 4.
Now we come back to the proof of theorem. ByLemma 16, for almost every σ ∈ R such that L ( A σ ) = 0, we have (4.13) holds,lim t → + ∂L d ( A σ it ) ∂t exists and is finite.We claim that for these σ , equations (1).(2).of Theorem 4 hold. From now to theend of the proof of Theorem 4, we denote for simplicity m ± for m ± ( σ ± it, x ), ˜ m ± for m ± ( σ ± it, f ( x )), τ for τ A σ it ( x ) ( m + ( σ + it, x )), τ − for τ A σ − it ( x ) ( m − ( σ − it, x )), A for A σ + it ( x ), A − for A σ − it ( x ), L d for L d ( A σ + it ), q for q σ + it ( x ) .4.4.1. proof of (1). of Theorem 4. Notice that ◦ A σ + it = (cid:18) e − t e t (cid:19) ◦ A σ , we havean expression of q by the singular values of ˜ m . lemma . (4.18) q − = e − t ( d + d ) · Π di =1 ( e t (1 − σ i ( ˜ m + ) )1 − e t σ i ( ˜ m + ) ) d +1 Proof.
By Lemma 12 and the definition of q , q − = V ( m + ) V ( ˜ m + ) · | dm + d ˜ m + | = V ( e t ˜ m + ) V ( ˜ m + ) · V ( m + ) V ( e t ˜ m + ) | dm + de t ˜ m + | · e t ( d + d ) ( since SD d has d + d real dimension)= V ( e t ˜ m + ) V ( ˜ m + ) · e t ( d + d ) ( since m e t ˜ m + is an isometry for Bergman metric )= e − t ( d + d ) · Π di =1 ( e t (1 − σ i ( ˜ m ) )1 − e t σ i ( ˜ m ) ) d +1 ( by Lemma 5) (cid:3) Using that for r > , ≤ s < e − r we have(4.19) ln( e r (1 − s )1 − e r s ) ≥ r − s by last lemma, we getln q − ≥ − t ( d + d ) + d X i =1 ( d + 1) · t − σ i ( ˜ m + ) = 2( d + 1) t d X i =1 σ i ( ˜ m + ) − σ i ( ˜ m + ) By (4.12), since L d = d +1) R X ln q − dµ , we have(4.20) L d ≥ t Z X d X i =1 σ i ( ˜ m + ) − σ i ( ˜ m + ) dµ An analogous argument yields(4.21) L d ≥ t Z X d X i =1 σ i ( ˜ m − ) − σ i ( ˜ m − ) dµ Then we conclude that(4.22) L d t ≥ Z X d X i =1 ( 1 + σ i ( ˜ m + ) − σ i ( ˜ m + ) + 1 + σ i ( ˜ m − ) − σ i ( ˜ m − ) ) dµ By our assumption of σ , we get the proof of (1).4.4.2. map Λ and basis B ( · , · ) . To prove (2)., we consider the following map:
Definition 11.
Let
M at d,d ( C ) be the space of all d × d complex matrices, we candefine the map: Λ :
M at d,d ( C ) → Λ d ( C d ) X x ∧ · · · · · · ∧ x d where { x i , ≤ i ≤ d } are the column vectors of X . The following lemma lists some properties of Λ we will use later. Recall thatfor A ∈ GL (2 d, C ), Λ k ( A ) is the natural action induced by A on Λ k ( C d ). Forarbitrary two 2 d × d matrices X, Y , denote D Λ( X )( Y ) := lim t → Λ( X + tY ) − Λ( X ) t (4.23) lemma . For A ∈ GL (2 d, C ) , B ∈ GL ( d, C ) , X, Y ∈ M at d,d ( C ) , supppose that X = ( x , . . . , x d ) , Y = ( y , . . . , y d ) , where { x i , ≤ i ≤ d } , { y i , ≤ i ≤ d } are thecolumn vectors of X, Y respectively, then we have the following equations: Λ d ( A ) · Λ( X ) = Λ( A · X )(4.24) D Λ( X )( Y ) = d X i =1 x ∧ · · · ∧ x i − ∧ y i ∧ x i +1 ∧ · · · ∧ x d (4.25) D Λ( AX )( AY ) = Λ d ( A ) · D Λ( X )( Y )(4.26) Λ( X · B ) = det( B )Λ( X )(4.27) Proof.
By computation directly. (cid:3)
From now on we identify Λ d ( C d ) with C as the following: Identification If ̟ ∈ Λ d ( C d ) = c ( ̟ ) · e ∧ · · · ∧ e d , then we identify ̟ with c ( ̟ ). Here e i are standard basis of C d .Now we define a collection of basis of Λ d ( C d ) for later use. Definition 12.
Suppose
X, Y ∈ M at d,d ( C ) are with rank d , and the column vec-tors { x i , ≤ i ≤ d } , { y i , ≤ i ≤ d } of X, Y are linearly independent, then thefollowing subset in Λ d ( C d ) forms a basis of Λ d ( C d ) : { x i ∧ · · · ∧ x i | I | ∧ y j ∧ · · · ∧ y j | J | : I, J ⊂ { , . . . , d } , | I | + | J | = d, i < i < . . . , j < j < . . . } denoted by B ( X, Y ) . ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 23
For any element ω ∈ Λ d ( C d ) , the coefficient of x ∧ · · · ∧ x d for the expansionof ω with respect to the basis B ( X, Y ) is (4.28) ω ∧ ( y ∧ · · · ∧ y d ) x ∧ · · · ∧ x d ∧ y ∧ · · · ∧ y d Here we use the identification above.
We will use the following lemma later. lemma . Suppose
X, Z ∈ M at d,d ( C ) are with rank d , and the column vectors { x i , ≤ i ≤ d } , { z i , ≤ i ≤ d } of X, Z are linearly independent, then the coefficientof Λ( X ) for the expansion of D Λ( X )( Y ) with respect to the basis B ( X, Z ) is thetrace of the matrix (cid:0) X, Z (cid:1) − · Y . (the trace of a d × d matrix is the sum of thediagonal entries)Proof. Suppose (cid:0)
X, Z (cid:1) − · Y = (cid:18) a ij b ij (cid:19) ≤ i ≤ d, ≤ j ≤ d , then(4.29) Y = (cid:16) . . . P dk =1 ( a ki x k + b ki z k ) . . . (cid:17) ≤ i ≤ d By (4.25) we get that D Λ( X )( Y ) = d X i =1 x ∧ · · · ∧ x i − ∧ y i ∧ x i +1 ∧ · · · ∧ x d = d X i =1 x ∧ · · · ∧ x i − ∧ ( d X k =1 ( a ki x k + b ki z k )) ∧ · · · ∧ x d = d X i =1 a ii x ∧ · · · ∧ x d + other term in B ( X, Z )= (trace of (cid:0)
X, Z (cid:1) − · Y )Λ( X ) +other terms in B ( X, Z )which gives the proof. (cid:3) proof of (2). of Theorem 4.
Come back to the proof of (2). At first,(4.30) ◦ A (cid:18) m + I d (cid:19) = (cid:18) ˜ m + I d (cid:19) τ Take the inverse,(4.31) ◦ A − (cid:18) ˜ m + I d (cid:19) = (cid:18) m + I d (cid:19) τ − Let the operator Λ acting on both sides of (4.31), we get:Λ( ◦ A − (cid:18) ˜ m + I d (cid:19) ) = Λ( (cid:18) m + I d (cid:19) τ − )(4.32)Then differentiate with respect to t , ∂∂t Λ( ◦ A − (cid:18) ˜ m + I d (cid:19) ) = ∂∂t ( 1det τ Λ( (cid:18) m + I d (cid:19) )( by (4.27))(4.33) Using Lemma 18 to compute the derivative, we getleft of (4.33) = D Λ( ◦ A − (cid:18) ˜ m + I d (cid:19) )( ∂∂t ( ◦ A − (cid:18) ˜ m + I d (cid:19) ))= D Λ( ◦ A − (cid:18) ˜ m + I d (cid:19) )( − ◦ A − ( ∂∂t ◦ A ) ◦ A − (cid:18) ˜ m + I d (cid:19) + ◦ A − (cid:18) I d (cid:19) ∂ ˜ m + ∂t )= − Λ d ( ◦ A − ) · D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) − ˜ m + I d (cid:19) − (cid:18) I d (cid:19) ∂ ˜ m + ∂t )where we use (4.26) and ∂∂t ◦ A = (cid:18) − I d I d (cid:19) ◦ A in the last equality.right of (4.33) = − τ ) ∂ det τ∂t Λ( (cid:18) m + I d (cid:19) )+ 1det τ D Λ( (cid:18) m + I d (cid:19) )( (cid:18) I d (cid:19) ∂m + ∂t )Notice that Λ d ( ◦ A ) · Λ( (cid:18) m + I d (cid:19) ) = Λ( ◦ A (cid:18) m + I d (cid:19) )= Λ( (cid:18) ˜ m + I d (cid:19) τ )= det τ · Λ( (cid:18) ˜ m + I d (cid:19) )Applying − Λ d ( ◦ A ) to both sides of (4.33), by previous discussion we have the key equation D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) − ˜ m + I d (cid:19) ) − D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) I d (cid:19) ∂ ˜ m + ∂t ) =1det τ ∂ det τ∂t Λ( (cid:18) ˜ m + I d (cid:19) ) − τ Λ d ( ◦ A ) · D Λ( (cid:18) m + I d (cid:19) )( (cid:18) I d (cid:19) ∂m + ∂t )4.4.4. the key equation. To analyse each term of the key equation, for (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ∈ M at d,d ( C ), we consider the basis B ( (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ). This is actually a basis since k ˜ m + k , k ˜ m − k <
1, det (cid:18) ˜ m + I d I d ˜ m − (cid:19) = 0.For the key equation, the following lemmas give the coefficients of Λ( (cid:18) ˜ m + I d (cid:19) ) forthe expansion of each term with respect to the basis B ( (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ). ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 25 lemma . The coeffcient of Λ( (cid:18) ˜ m + I d (cid:19) ) for the expansion of D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) − ˜ m + I d (cid:19) ) with respect to the basis B ( (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ) is the trace of ( I d − ˜ m − ˜ m + ) − ( I d +˜ m − ˜ m + ) .Proof. By Lemma 19, to prove lemma 20, we only need to compute(4.34) (cid:18) ˜ m + I d I d ˜ m − (cid:19) − · (cid:18) − ˜ m + I d (cid:19) In fact (cid:18) ˜ m + I d I d ˜ m − (cid:19) − = (cid:18) ( ˜ m − ˜ m + − I d ) − ˜ m − ( I d − ˜ m − ˜ m + ) − I d + ˜ m + ( I d − ˜ m − ˜ m + ) − ˜ m − − ˜ m + ( I d − ˜ m − ˜ m + ) − (cid:19) Then we get(4.35) (cid:18) ˜ m + I d I d ˜ m − (cid:19) − · (cid:18) − ˜ m + I d (cid:19) = (cid:18) ( I d − ˜ m − ˜ m + ) − ( I d + ˜ m − ˜ m + ) ∗ (cid:19) By Lemma 19, we get the proof of Lemma 20. (cid:3) lemma . The coeffcient of Λ( (cid:18) ˜ m + I d (cid:19) ) for the expansion of D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) I d (cid:19) ∂ ˜ m + ∂t ) with respect to the basis B ( (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ) is det (cid:18) ˜ m + I d I d ˜ m − (cid:19) − · D Λ( (cid:18) ˜ m + I d (cid:19) )( (cid:18) I d (cid:19) ∂ ˜ m + ∂t ) ∧ Λ( (cid:18) I d ˜ m − (cid:19) ) Proof.
Using (4.28). (cid:3) lemma . The coeffcient of Λ( (cid:18) ˜ m + I d (cid:19) ) for the expansion of τ Λ d ( ◦ A ) · D Λ( (cid:18) m + I d (cid:19) )( (cid:18) I d (cid:19) ∂m + ∂t ) with respect to the basis B ( (cid:18) ˜ m + I d (cid:19) , (cid:18) I d ˜ m − (cid:19) ) is det (cid:18) m + I d I d m − (cid:19) − · D Λ( (cid:18) m + I d (cid:19) )( (cid:18) I d (cid:19) ∂m + ∂t ) ∧ Λ( (cid:18) I d m − (cid:19) ) Proof.
Let X = (cid:18) m + I d (cid:19) , W = (cid:18) I d (cid:19) ∂m + ∂t , W = (cid:18) I d m − (cid:19) By (4.28) we getthe coefficient= 1det τ (Λ d ( ◦ A ) · D Λ( X )( W )) ∧ Λ( (cid:18) I d ˜ m − (cid:19) ) · det (cid:18) ˜ m + I d I d ˜ m − (cid:19) − = 1det τ · det τ − (Λ d ( ◦ A ) · D Λ( X )( W )) ∧ (Λ d ( ◦ A ) · Λ( W )) · det (cid:18) ˜ m + I d I d ˜ m − (cid:19) − (use Lemma 11)= 1det τ · det τ − Λ d ( ◦ A ) · ( D Λ( X )( W ) ∧ Λ( W )) · det (cid:18) ˜ m + I d I d ˜ m − (cid:19) −
16 DISHENG XU
To prove Lemma 22, we only need to prove the following equation:(4.36) det( ◦ A ) det (cid:18) m + I d I d m − (cid:19) = det τ det τ − det (cid:18) ˜ m + I d I d ˜ m − (cid:19) which is just a corollary of (4.30) and Lemma 11. (cid:3) Now come back to the key equation. By Lemma 20,21,22, taking the coefficientof Λ( (cid:18) ˜ m + I d (cid:19) ) in the key equation and integrating with respect to the measure µ ,we have(4.37) Z X tr(( I d − ˜ m − ˜ m + ) − ( I d + ˜ m − ˜ m + )) dµ = Z X τ ∂ det τ∂t dµ Consider the real part, which gives(4.38) Z X R (tr(( I d − ˜ m − ˜ m + ) − ( I d + ˜ m − ˜ m + ))) dµ = ∂L d ∂t A trace inequality and the rest of the proof.
By (4.22) and Lemma 16, wehave that lim inf t → + Z X d X i =1 ( 1 + σ i ( ˜ m + ) − σ i ( ˜ m + ) + 1 + σ i ( ˜ m − ) − σ i ( ˜ m − ) ) − R (tr(( I d − ˜ m − ˜ m + ) − ( I d + ˜ m − ˜ m + ))) dµ ≤ lim t → + L d ( A σ + it ) t − ∂L d ( A σ + it ) ∂t = 0Compare with (2). of Theorem 4, to finish the proof, we only need to prove thefollowing inequality: lemma . d X i =1 ( 1 + σ i ( ˜ m + ) − σ i ( ˜ m + ) + 1 + σ i ( ˜ m − ) − σ i ( ˜ m − ) ) − R ( tr (( I d − ˜ m − ˜ m + ) − ( I d + ˜ m − ˜ m + ))) ≥ k ˜ m + − ˜ m − k HS Proof.
Notice that for k x k < , x − x = 2(1 − x ) − − − P ∞ k =0 x k , and˜ m − = ( ˜ m − ) ∗ We have that:left of Lemma 23= d X i =1 − σ i ( ˜ m + ) + 11 − σ i ( ˜ m − ) − R tr(( I d − ˜ m − ˜ m + ) − = ∞ X k =0 ( d X i =1 σ i ( ˜ m + ) k + σ i ( ˜ m − ) k ) − R tr((( ˜ m − ) ∗ ˜ m + ) k )= ∞ X k =0 tr((( ˜ m + ) ∗ ˜ m + ) k ) + tr((( ˜ m − ) ∗ ˜ m − ) k ) − R tr((( ˜ m − ) ∗ ˜ m + ) k ) ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 27
Then the proof of Lemma 23 is the consequence of the following matrix inequalities:for arbitrary d × d complex matrices X, Y, k > X ∗ X ) k + ( Y ∗ Y ) k ) ≥ R tr(( X ∗ Y ) k )tr( X ∗ X ) + tr( Y ∗ Y ) − R tr( X ∗ Y ) = k X − Y k HS (cid:3) Density of positive Lyapunov exponents for continuous symplecticcocycle
Herglotz function.
We recall that m is called a Herglotz (matrix valued)function if m is an analytic matrix valued function defined on C + and Im ( m ( z ))is a positive definite Hermitian matrix for all z ∈ C + , we list some basic propertieswe will use (see [23]). lemma . The function m ( · ) has a finite normal limit m ( σ + i + ) = lim t → + m ( σ + it ) for a.e. σ ∈ R . Moreover if two Herglotz function m , m have the same limiton a positive measure set on R , then m = m . Notice that Φ − C · m + ( · , x ) , Φ − C · m − ( −· , x ) are Herglotz functions.5.2. M − function. Consider the following definition of M − function, which is in-troduced in [7]. Definition 13.
For A ∈ L ∞ ( X, Sp (2 d, R )) , we denote (5.1) M ( A ) := the Lebesgue measure of { θ ⊂ [0 , π ] , L ( A θ ) = 0 } We hope to prove for generic A , M ( A ) = 0. At first, we prove it for a family ofsymplectic cocycles taking finitely many values.5.3. Symplectic cocycles taking finitely many values.
We introduce the fol-lowing definition of deterministic , which is similar to the definition for Sch¨odingeroperator in [36] and [30].
Definition 14.
For A ∈ L ∞ ( X, Sp (2 d, R )) , we say A is deterministic if A ( f n ( x )) , n ≥ is a.e., a measurable function of { A ( f n ( x )) , n < } . As [30], we have the following theorem for the M − function for symplectic cocy-cles. Theorem 5.
Suppose A ∈ L ∞ ( X, Sp (2 d, R )) such that(1). A ( x ) , x ∈ X only takes finitely many values.(2). A ( f n ( x )) , n ∈ Z , is not periodic for almost every x ∈ X .(3).If A ( x ) = A ( y ) , x, y ∈ X , then ◦ A ( x ) − (0) = ◦ A ( y ) − (0) .We have M ( A ) = 0 .Proof. We know that for almost every x ∈ X, A ( f n ( x )) , n ≥ m − ( x ). In fact, for z such that I ( z ) < , ◦ A z ( x ) − uniformly contracts theBergman metric on SD d , so like the property of m − function in Kotani theory, wehave that(5.2) m − ( z, x ) = lim n →∞ ◦ A z ( x ) − · · · ◦ A z ( f n ( x )) − · lemma . If a cocycle A ∈ L ∞ ( X, Sp (2 d, R )) satisfies (1), (3) of Theorem 5, thenthe function m − ( z, · ) , z ∈ C + determines { A ( f n ( · )) , n ≥ } in the sense that if x, y ∈ X such that A ( f n ( x )) , A ( f n ( y )) , n ≥ are bounded, and m − ( · , x ) = m − ( · , y ) ,then A ( f n ( x )) = A ( f n ( y )) , n ≥ .Proof. Let z tends to ∞ along the line { R ( z ) = 0 , I ( z ) < } in (5.2), we get(5.3) lim R ( z )=0 , I ( z ) →−∞ m − ( z, x ) = ◦ A ( x ) − (0)By (3) of Theorem 5, we know that m − ( · , x ) can determine ◦ A ( x ), by(5.4) ◦ A z ( x ) · m − ( z, x ) = m − ( z, f ( x ))it implies m − ( · , x ) can determine m − ( · , f ( x )), using the same method again, wecan determine ◦ A ( f ( x )). Repeat this process, we determine all A ( f n ( x )) , n ≥ (cid:3) Come back to the proof of Theorem 5. Suppose M ( A ) >
0, we claim that underthe assumptions (1),(3), A must be deterministic. Then by Kotani’s argument in[30], A must be periodic, which contradicts the assumption (2).In fact, the set { A ( f n ( x )) , n < } determines m + ( · , x ). If M ( A ) >
0, by (2). ofTheorem 4, m + ( · , x ) determines m − ( · , x ) on a full mesure subset of { θ : L ( A θ ) = 0 } .By Lemma 24, since Φ − C · m + ( · , x ) , Φ − C · m − ( −· , x ) are Herglotz functions, m + ( · , x ) determines m − ( −· , x ) on all of C + . By Lemma 25, { A ( f n ( x )) , n ≥ } is determined by { A ( f n ( x )) , n < } . That means A is deterministic. (cid:3) Continuous symplectic cocycles.Theorem 6.
Suppose f is not periodic on supp ( µ ) , then the set of A such that L ( A ) > is dense in C ( X, Sp (2 d, R )) .Proof. At first we consider the following lemma: lemma . Suppose f : ( X, µ ) → ( X, µ ) ( f = id ) is ergodic, then there is a residualsubset of cocycles A in C ( X, Sp (2 d, R )) such that M ( A ) = 0 .Proof. We follow the proof in [7]. At first we consider the following lemma: lemma . There exists a dense subset Z of L ∞ ( X, Sp (2 d, R )) satisfying all con-ditions of Theorem 5.Proof. By Lemma 2 of [7], the cocycles in L ∞ ( X, Sp (2 d, R )) satisfying the firsttwo conditions of Theorem 5 are dense in L ∞ ( X, Sp (2 d, R )). But for each cocycle A satisfying the first two condition of Theorem 5, we can find a new cocycle A ′ satisfying all conditions in Theorem 5 and arbitrarily close to A . (cid:3) lemma . For every r > , the map ( L ( X, Sp (2 d, R )) ∩ B r ( L ∞ ( X, Sp (2 d, R ))) , k · k ) → R ,A M ( A ) is upper semi-continuous.Proof. The proof is the same as the SL (2 , R ) case, since we have the formula in[33] to replace the Herman-Avila-Bochi formula in [6] for SL (2 , R ) case. And byTheorem 3, L d ( A z ) is harmonic for z ∈ C + and subharmonic on C + ∪ R , we canmove the proof for SL (2 , R ) case in [7] to here. (cid:3) ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 29 lemma . For A ∈ C ( X, Sp (2 d, R )) , ǫ > , δ > , there is an A ′ ∈ C ( X, Sp (2 d, R )) such that k A − A ′ k ∞ < ǫ, M ( A ) < δ .Proof. The proof is almost the same as Lemma 3 of [7], we only need to use the set Z in Lemma 27 and Theorem 5 to replace the set Z and Kotani result in Lemma3 of [7]. (cid:3) Come back to the proof of Lemma 26, for δ >
0, we define M δ = { A ∈ C ( X, Sp (2 d, R ) : M ( A ) < δ } By Lemma 28, M δ is open, and by Lemma 29, M δ is dense. It follows that { A ∈ C ( X, Sp (2 d, R ) : M ( A ) = 0 } = ∩ δ> M δ is residual. (cid:3) Come back to the proof of Theorem 6, Let P ⊂ X be the set of periodic orbitsof f . If µ ( P ) <
1, then using Lemma 26 we get the proof.Assume µ ( P ) = 1, we follow the argument of Lemma 3.1 in [2]. Let P k ⊂ X be the set of periodic orbits of period k ≥
1. Since f is not periodic on supp ( µ ), P n = ∪ k ≤ n P k = supp ( µ ) for every n ≥
1. Thus there are arbitrarily large n suchthat µ ( P n \P n − ) > n , and take x ∈ supp ( µ ) |P n \P n − . We can approximate any A ∈ C ( X, Sp (2 d, R )) by some A ′ which is constant in a compact neighborhood K of { f k ( x ) } n − k =0 . The details of the following argument can be found in the Appendix A,here we only give an outline. We will prove that there is a constant C independent of n, f such that for generic { A ′ ( f k ( x )) } , there exist θ ∈ ( − Cn , Cn ) with L ( A ′ θ ( x )) > A ′ is locally constant near the orbit of x ∈ supp ( µ ), we have L ( A ′ θ ) >
0. )Otherwise there is an open interval I contained 0 and | I | > O ( n ), such that forall θ ∈ I , all the eigenvalues of A ′ ( n ) θ ( x )) are norm 1, where | I | is the length of I .But in the Appendix A we will prove for any interval I ′ such that(5.5) ∀ θ ∈ I ′ , all the eigenvalues of A ′ ( n ) θ ( x )) are simple and norm 1we have | I ′ | ≤ O ( 1 n )As a result, there are two intervals I , I ⊂ I satisfying (5.5) and sharing commonboundary point θ such that A ′ ( n ) θ ( x ) has repeated eigenvalues with norm 1. Butthis can not happen for generic choice of { A ′ ( f k ( x )) } . (cid:3) The proof of Theorem 1
By Theorem 6, as in the SL (2 , R ) − case, to prove Theorem 1, we need a localregularization formula similar to Theorem 7 in [2].At first we need the following lemma: lemma . Suppose A ∈ C ( X, Sp (2 d, R )) , Ω ⊂ C is a domain. Suppose an analytic Sp (2 d, C ) -valued map B is defined on Ω such that for all z ∈ Ω , x ∈ X , ◦ B ( z ) · SD d ⊂ SD d . Then the Lyapunov exponent L d ( B ( z ) A ) harmonically depends on z ∈ Ω . Proof.
The proof is basically the same as the discussion in Chapter 3 for holomor-phicity of ζ − function. See the remark at page 7 of [33], and section 3 and 6 of[8]. (cid:3) As in [2], let k · k ∗ denote the sup norm in the space C ( X, sp (2 d, R )) and C ( X, sp (2 d, C )). And for r >
0, let B ∗ ( r ) , B C ∗ ( r ) be the corresponding r − ball.For A ∈ C ( X, Sp (2 d, R )), a, b ∈ C ( X, sp (2 d, R )), we define the following function(6.1) Φ ǫ ( A, a, b ) := Z − − t | t + 2 it + 1 | L d ( e ǫ ( tb +(1 − t ) a ) A ) dt The following local regularization formula is the main result of this chapter.
Theorem 7.
There exists η > such that if b ∈ C ( X, sp (2 d, R )) is η − close to (cid:18) I d − I d (cid:19) , then for every ǫ > , and every A ∈ C ( X, Sp (2 d, R )) , (6.2) ◦ e ǫ ( zb +(1 − z ) a ) · SD d ⊂ SD d when(1). z ∈ {| z | = 1 } ∩ I ( z ) > or z = ( √ − i, a ∈ B C ∗ ( η ) ,(2). z ∈ {| z | < } ∩ I ( z ) > , a ∈ B ∗ ( η ) .Moreover (6.3) a Φ ǫ ( A, a, b ) is a continuous function of a ∈ B ∗ ( η ) and depends continuously (as an analyticfunction) on A .Proof. In fact we only need to prove (6.2), (6.3) is the consequence of (6.2) andLemma 30, see Theorem 7 of [2].To prove (6.2), we claim there exists a positive number η > Z ∈ ∂SD d , { Z T = Z, k Z k = 1 } , for ǫ > Z ǫ := ◦ e ǫ ( zb +(1 − z ) a ) · Z is contained in SD d for z and a in either case (1) or (2). Thisimplies there exists ǫ > ǫ < ǫ , ◦ e ǫ ( zb +(1 − z ) a ) · SD d ⊂ SD d . Byiteration, ◦ e ǫ ( zb +(1 − z ) a ) takes SD d into SD d for every ǫ > ǫ is small, k X k , k Y k ≤ e ǫ ( X + Y ) = e O ( ǫ k X k·k Y k ) e ǫX e ǫY which means there exist a vector W in the Lie algebra with norm less than O ( ǫ k X k·k Y k ), such that e ǫ ( X + Y ) = e W e ǫX e ǫY .In addition, we need some notations for a real Lie algebra g and its complex-ification g C = g ⊕ i g . For an element c ∈ g C , a, b ∈ g such that c = a + ib , wedenote(6.5) R ( c ) = a, I ( c ) = b From now to the end of this chapter, we always consider g is the Lie algebra of U ( d, d ) ∩ Sp (2 d, C ) or R . Then g C is sp (2 d, C ) or C . ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 31
Now we denote R ( a, b, z ) = R ( z ◦ b + (1 − z ) ◦ a ) = R ( z ) ◦ b + R ((1 − z ) ◦ a ) and I ( a, b, z ) = I ( z ◦ b + (1 − z ) ◦ a ) = I ( z ) ◦ b + I ((1 − z ) ◦ a ).Let η be small, then for z, a in either case (1) or (2) we have the followingequations: Z ǫ = ◦ e ǫ ( zb +(1 − z ) a ) · Z = e ǫ ( R + iI ) · Z (6.6) e ǫR · Z ∈ ∂SD d (6.7) I ( a, b, z ) = I ( z )( (cid:18) i − i (cid:19) + O ( η ))(6.8) k R ( a, b, z ) k ∗ ≤ k I ( a, b, z ) k ∗ ≤ I ( z )(6.10)Here (6.8) comes from the fact that k I ((1 − z ) ◦ a ) k ∗ ≤ O ( η I ( z )) holds for eithercase (1) or (2).Denote Z ′ = e ǫR Z . Then by (6.7) we know k Z ′ k = 1, and we have: Z ǫ = e ǫ ( R + iI ) · Z = e O ( ǫ k R k ∗ k I k ∗ ) e ǫiI e ǫR · Z by (6.4)(6.9)(6.10)= e O ( ǫ I ( z )) e ǫiI · Z ′ by (6.9)(6.10)= e O ( ǫ I ( z )) e ǫ I ( z )( − + O ( η )) · Z ′ by (6.8)= e O ( ǫ I ( z )) e O ( ǫ η I ( z ) ) e O ( ǫη I ( z )) · ( e − ǫ I ( z ) Z ′ ) by (6.4)= e O ( ǫ ( ǫ + η ) I ( z )) · ( e − ǫ I ( z ) Z ′ ) since η is small.If ǫ is small enough, since the action on the equation above is a M¨obius transfor-mation, then by computation we have k Z ǫ k = k e O ( ǫ ( ǫ + η ) I ( z )) · ( e − ǫ I ( z ) Z ′ ) k≤ e − ǫ I ( z ) for ǫ small, which implies Z ǫ ∈ SD d . Then we get the proof of Theorem 7. (cid:3) To finish the proof of Theorem 1 we need the following short lemma. lemma . Let A ∈ C ( X, Sp (2 d, R )) , a, b ∈ C ( X, sp (2 d, R )) and ǫ > , if L d ( e ǫa ) > , then Φ ǫ ( A, a, b ) > .Proof. As in the proof of Lemma 3.2 and Lemma 2.3 in [2], the map γ : t L d ( e ǫ ( tb +(1 − t ) a ) A )is a subharmonic function.By subharmonicity, if γ ( t ) = 0 for almost every t ∈ ( − , γ ( t ) = 0 for all t ∈ ( − , γ (0) >
0, Φ ǫ ( A, a, b ) must be positive. (cid:3)
Now we can prove Theorem 1. We follow the argument in [2]. For all δ > A ∈ B ⊂ C ( X, Sp (2 d, R )), where B is ample, we need to prove there is a v ∈ b ⊂ C ( X, sp (2 d, R )) such that k v k b < δ, L d ( e v A ) > Choose a positive number η satisfying the conditions in Theorem 7, and take b ∈ b such that b is η − close to (cid:18) I d − I d (cid:19) , let ǫ > ǫ k b k b < δ . By Theorem6, there is an element a ∈ B ∗ ( η ) ⊂ C ( X, sp (2 d, R )) such that L d ( e ǫa A ) >
0. ByLemma 31, we have Φ ǫ ( A, a, b ) > b is dense in C ( X, sp (2 d, R )), and by Theorem 7 we know the map in (6.3)is continuous, there is an element a ′ ∈ B ∗ ( η ) ∩ b such that Φ ǫ ( A, a ′ , b ) > γ ′ : s Φ ǫ ( A, sa ′ , b ) is an analytic function of s ∈ [ − , γ ′ (1) >
0, we can choose0 < s < min { , δ ǫ k a ′ k b } such that γ ′ ( s ) >
0. Then there exists t ∈ ( − ,
1) such that L d ( e ǫ ( tb +(1 − t ) sa ′ ) A ) > v = ǫ ( tb +(1 − t ) sa ′ ), then v ∈ b , k v k b < ǫ ( k b k b + s k a ′ k b ) < δ and L d ( e v A ) > proof of the rest of results Proof of Corollary 1:
SHSp (2 d ) and SU ( d, d ) cocycles. The proof ofCorollary 1 for
SHSp (2 d ) and SU ( d, d ) cocycles is similar to Theorem 1. Considerthe following correspondences between the concepts used in symplectic cocycle andspecial Hermitian symplectic cocycle. Almost all the following concept also worksfor any biholomorphic transformation group for (non compact) Hermitian symmet-ric space. Sp (2 d, R ) ↔ SHSp (2 d ) U ( d, d ) ∩ Sp (2 d, C ) ↔ SU ( d, d ) SH d ↔ { Z = X + iY,X, Y ∈ Her ( d ) , Y > } SD d ↔ { Z, I d − Z ∗ Z > } Caylay element 1 √ (cid:18) I d − i · I d I d i · I d (cid:19) ↔ √ (cid:18) I d − i · I d I d i · I d (cid:19) A ◦ A := CAC − ↔ A ◦ A := CAC − ,Sp (2 d, R ) ∼ = U ( d, d ) ∩ Sp (2 d, C ) SHSp (2 d ) ∼ = SU ( d, d )M¨obius transformation ↔ M¨obius transformationΦ C : SH d → SD d ↔ Φ C : { Z = X + iY, X, Y ∈ Her ( d ) ,Y > } → { Z, I d − Z ∗ Z > } ∂SD d ↔ ∂ { Z, I d − Z ∗ Z > } = {k Z k = 1 } ∂ k SD d ↔ {k Z k = 1 , rank(1 − Z ∗ Z ) = d − k } ∂ d SD d = U sym ( C d ) ↔ U ( d )fin ∂ d SH d = Sym d R ↔ her ( d )Φ C gives chart : Sym d R ↔ Φ C : her ( d ) → { det( Z − = 0 } ∩ ∂ d SD d → { Z, k Z k = 1 , det( Z − = 0 } ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 33 atlas Φ C gk ↔ Φ C gk be defined similarly.Bergman metric, volume form ↔ Bergman metric, volume formon {k Z k < } V ( Z ) ↔ V ( Z ) = Π ≤ i ≤ d (1 − σ i ( Z ) ) − d Consider A σ + it , ◦ A σ + it defined as in the symplectic case, we hope to prove the sameresult as Theorem 3 firstly. Obviously we can define τ A , ˆ τ as in the symplectic case,and we also have similar results to Lemma 6, 8.By [15], Lemma 4 holds for all (non compact) bounded Hermitian symmetricspace, then we can define m + , m − as in the symplectic case. So Lemma 7 alsoworks for special pseudo unitary group. So we can define function ζ which satisfiescondition 1 and 3 in Theorem 3.To prove statement in Theorem 3 for SHSp (2 d ), we need to prove the fiberedrotation function, the real part of ζ is non-increasing on R .Define the cone field { h > } on her ( d ), and then use the tangent map of Φ C gk to map it to T U ( d ). As in the symplectic case, it gives well-defined cone fields C , ˆ C on U ( d ) and [ U ( d ). Using ˆ C we give a partial order on [ U ( d ) as the \ U sym ( C d ) case.By Cartan decomposition of SHSp (2 d ) and identitydet(1 + XY ) = det(1 + Y X )we can get the same equation as in Lemma 10. Then we can prove the non-increasingproperty of the fibered rotation function as Theorem 3.To prove the same result as Theorem 4. We can use the following equation toreplace Lemma 11.Suppose t > m − = m − ( σ − it, x ) , ˜ m − = m − ( σ − it, f ( x )) satisfying ˜ m − = ◦ A σ − it ( x ) · m − . There exists τ − ∈ GL ( d, C ) such that(7.1) ◦ A (cid:18) I d m ∗ (cid:19) = (cid:18) I d ˜ m ∗ (cid:19) τ − As in the symplectic case, we can define the q − function, we have the followingproperties for q − function for SHSp (2 d ) and SU ( d, d ) to replace Lemma 13 and(4.12). L d ( A σ + it ) = 14 d Z X − ln q σ + it ( x ) dµ ( x )(7.2) | dm + ( σ + it, f ( x )) dm + ( σ + it, x ) | = | det( τ A σ + it ( x )) | − d (7.3)Now we can prove Theorem 4 as the following. By non-increasing property offibered rotation function, Lemma 16 holds. And by the same proof as Lemma 17,we have(7.4) q − = e − td · Π di =1 ( e t (1 − σ i ( ˜ m + ) )1 − e t σ i ( ˜ m + ) ) d then we have the same inequality for L d t as (4.22). For the estimate of ∂L d ∂t forHermitian symplectic case, we use (cid:18) I d m −∗ (cid:19) to replace (cid:18) I d m − (cid:19) , τ − defined in (7.1) to substitute τ − defined in Lemma 11, use (7.1) to replace Lemma 11, we can getthe following equation(7.5) Z X R (tr(( I d − ˜ m − ∗ ˜ m + ) − ( I d + ˜ m − ∗ ˜ m + ))) dµ = ∂L d ∂t then the rest of the proof of Corollary 1 for SHSp (2 d ) , SU ( d, d ) cocycles is thesame as the proof of Theorem 1. The proof of Corollary 1:
HSp (2 d ) and U ( d, d ) cocycles. To prove corol-lary 1 for
HSp (2 d ) and U ( d, d ) cocycles, we consider the following lemma whichsimilar to Lemma 27: lemma . There exists a dense subset Z of L ∞ ( X, HSp (2 d )) satisfying all condi-tions of Theorem 5. As a result, for all A ∈ Z , M ( A ) = 0 .Proof. Consider a
HSp (2 d ) − cocycle A taking finitely many values, Notice thatthere are cocycles B and C also taking finitely many values and satisfying(7.6) B ( x ) ∈ SHSp (2 d ) , C ( x ) = c ( x ) · I d , A ( x ) = B ( x ) C ( x )And the statement of Lemma 27 also holds for the space of SHSp (2 d ) − cocycle. Sowe can find a SHSp (2 d ) − cocycle B ′ , L ∞ − close to B satisfying all conditions ofTheorem 5.As a result, the HSp (2 d ) − cocycle A ′ := B ′ C is L ∞ − close to A and satisfyingall conditions of Theorem 5. In particular M ( A ′ ) = 0. (cid:3) The rest of the proof is the same as the part after Lemma 27 for the proof ofTheorem 1. Remark . In general, for a
HSp (2 d ) − cocycle A ( x ) , there is no cocycles B, C inthe same regularity class as A and satisfying (7.6) . So we can not use the result of SHSp (2 d ) and SU ( d, d ) cocycles directly to get the proof for HSp (2 d ) and U ( d, d ) cocycles. Moreover, since Lemma 10 does not holds for general scalar matrix A , wecan not prove corollary 1 for HSp (2 d ) and U ( d, d ) cocycles by mimicking the proofof Theorem 1 step by step. Proof of Theorem 2 and Corollary 2.
Firstly we consider the followingclassical result for stochastic Jacobi matrices on the strip in (1.3) proved by B.Simonand S.Kotani (see [28]).
Definition 15.
For potential v we define the following function, M ( v ) := the Lebesgue measure of { E ∈ R , L ( A E − v ) = 0 } . And we say v is deterministic if for n ≥ , v ( f n ( x )) is a measurable function of { v ( f k ( x ) , k < } . lemma . Suppose M ( v ) > , then v is deterministic. see appendix A for a discussion corresponds to the final part of the proof of Theorem 6 forspecial Hermitian symplectic cocycles. see Appendix A for a discussion corresponds to the final part of Theorem 6 for Hermitiansymplectic cocycles. ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 35
Combining Lemma 33 with semi-continuity argument in subsection 5.4 (the proofis the same as in [7], using harmonicity of L d on upper half plane), we can proveresult similar to Theorem 6 for stochastic Schr¨odinger operators and Jacobi matriceson the strips since non-periodic potentials which taking finitely many values aredense in L ∞ ( X, Her ( d )) , L ∞ ( X, Sym d R ) and L ∞ ( X, R d ). Then using local regularization formula in [2] (the proof is similar to Theorem6.2), we can get the proof of Theorem 2 and Corollary 2.
Appendix A. Generic periodic (Hermitian) symplectic cocycles
In this section we finish the proof of Theorem 6 (also for Hermitian symplecticcocycles) by considering the generic periodic cocycles. Without loss of generality, weconsider the dynamics ( f, X, µ ) as the following: f ( x ) = x + 1 for x ∈ X := Z /n Z ,and for any Z ⊂ X , µ ( Z ) = n Z ).It is easy to see there is a set O ⊂ C ( X, HSp (2 d )) = HSp (2 d ) n such that O , O ∩ Sp (2 d, R ) , O ∩
SHSp (2 d ) are residual sets in HSp (2 d ) , Sp (2 d, R ) , SHSp (2 d )respectively and satisfy the following property: for all θ ∈ R / π Z , the geometric andalgebraic multiplicities of eigenvalues of A ( n ) θ ( x ) are 1 and at most 2 respectively.And there are only finite θ ∈ R / π Z such that A ( n ) θ ( x ) has repeated eigenvalues.We only need to prove the following lemma: lemma . There is a constant C independent of n such that for A ∈ O , thereexists θ ∈ ( − Cn , Cn ) such that L ( A θ ) > .Proof. Without loss of generality, we only need to consider those A with deter-minant equal to 1. We fix such an A ∈ O . Suppose there is an open interval I containing 0 and | I | > O ( n ), such that(A.1) ∀ θ ∈ I, σ ( A ( n ) θ ( x )) ⊂ {| z | = 1 } where | I | is the length of I . We prove the following lemma: lemma . For any interval I ′ such that (A.2) ∀ θ ∈ I ′ , all eigenvalues of A ( n ) θ ( x ) are simple and norm we have | I ′ | ≤ O ( 1 n ) Proof.
Suppose I ′ = ( a, b ), for θ ∈ I ′ , we conjugate it ◦ A ( n ) θ to the matrix with formdiag( e iρ ( θ ) , . . . , e iρ d ( θ ) , e − iρ ′ ( θ ) , . . . , e − ρ ′ d ( θ ) )where ρ, ρ ′ depend continuously on θ , P di =1 ρ i = P di =1 ρ ′ i and ρ i + ρ ′ j / ∈ π Z . in that case where µ − almost all points are periodic, see Appendix B. Therefore ρ ′ i ( b − ) + ρ i ( b − ) − ρ i ( a +) − ρ ′ i ( a +) < π . As a result, by the definitionand property of fibered rotation function ρ , we have ρ ( a ) − ρ ( b ) = 1 n d X i =1 ρ ′ i ( b − ) − n d X i =1 ρ ′ i ( a +)(A.3) = 12 n d X i =1 ρ ′ i ( b − ) + ρ i ( b − ) − ρ ′ i ( a +) − ρ i ( a +) ≤ dπn But as in the proof of Lemma 16 and (4.22), for almost all θ ∈ I ′ , we have(A.4) − dρdθ = lim t → + ∂L d ( A θ + it ) ∂t = lim t → L d ( A θ + it ) t ≥ d Combined with (A.3), we get the bound of | I ′ | . (cid:3) As a result, there are two intervals I , I ⊂ I satisfying (A.1) and sharing com-mon boundary point θ such that det( λI d − A ( n ) θ ( x )) has a double root e iρ ( θ ) .Denote the corresponding general eigenspace by V θ .To state the next lemma and for later use we introduce some basic concepts ofHermitian symplectic geometry on C d (see for example [26]). Definition 16.
The two-form < · , · > is linear in the second argument and conju-gate linear in the first argument, is a hermitian symplectic form if < φ, ψ > = − < ψ, φ > Let e i be the standard basis of C d , then we can define a classical Hermitiansymplectic form < · , · > such that < e i , e d + i > = 1 for 1 ≤ i ≤ d and < e i , e j > = 0if | i − j | 6 = d . The Hermitian symplectic groups are all transformation preservingthis form. A basis satisfying the same relation is called a canonical basis . Asubspace V of C d is called a Hermitian symplectic subspace if < · , · > | V isnon-degenerate. If an even dimensional Hermitian symplectic subspace V admitsa canonical basis then we say V is canonical. A subspace N is called isotropic if N ⊂ N ⊥ := { v, < v, w > = 0 , ∀ w ∈ N } . For a 2 l − dimensional Hermitian symplecticsubspace V , if there is an l − dimensional isotropic subspace N ⊂ V , then we call N is a Lagrange plane of V . It can be proved that a Hermitian symplectic subspace V contains a Lagrange plane if and only if it is canonical (see [26]).Now we state the lemma: A general eigenspace of eigenvalue λ of linear transform A is defined by V λ := { v : ∃ n, ( A − λ ) n · v = 0 } ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 37 lemma . By a suitable choice of canonical basis, we can assume A ( n ) θ , A ( n ) θ | V θ to be e iρ ( θ ) β ( θ ) ∗ ( d − × ( d − ∗ ( d − × ( d − e iρ ( θ ) ∗ ( d − × ( d − ∗ ( d − × ( d − (A.5) e iρ ( θ ) β ( θ )0 ( d − × ( d − ( d − × ( d − e iρ ( θ ) ( d − × ( d − ( d − × ( d − (A.6) respectively, where β ( θ ) = 0 .Proof. Notice that different general eigenspaces are mutually orthogonal (in theHermitian symplectic sense). So V θ is a Hermitian symplectic subspace of C d .Moreover consider the only one eigenvector v ∈ V θ − { } and w ∈ V θ − C · v ,then(A.7) A · v = e iρ ( θ ) v, A · w − e iρ ( θ ) w ∈ C · v − { } Therefore computing < Av, Aw > − < v, w > we get(A.8) < v, v > = 0Since V θ contains a Lagrange plane, it is a canonical Hermitian symplectic subspaceof C d . Then we can pick a vector v ′ ∈ V θ − C · v such that < v ′ , v ′ > = 0 and byfurther normalization we can assume < v, v ′ > = 1.It is easy to prove that V ⊥ θ is also a canonical Hermitian symplectic subspace.Then we can extend v, v ′ to a canonical basis of C d . Then A ( n ) θ , A ( n ) θ | V θ have theform in (A.5) with respect to this canonical basis. (cid:3) Choose a contour Γ enclosing e iρ ( θ ) but no other points of σ ( A ( n ) θ ), then for θ close to θ , Γ encloses exactly two points of σ ( A ( n ) θ ). Therefore there is a2 − dimensional invariant space V θ for eigenvalues of A ( n ) θ contained in the boundedregion with boundary Γ.Consider the spectral projection πi R Γ 1 z − A ( n ) θ dz of A ( n ) θ onto V θ . Since σ ( A ( n ) θ ) ⊂{| z | = 1 } , we have (by suitable choice of branch of z
7→ √ z ):tr( πi R Γ zz − A ( n ) θ dz ) q tr(Λ ( πi R Γ zz − A ( n ) θ dz )) ≤ πi R Γ zz − A ( n ) θ dz ) r tr(Λ ( πi R Γ zz − A ( n ) θ dz )) = 2(A.10)Therefore(A.11) ddθ | θ = θ tr( πi R Γ zz − A ( n ) θ dz ) q tr(Λ ( πi R Γ zz − A ( n ) θ dz )) = 0 Then we get 2 ddθ | θ = θ tr( 12 πi Z Γ zz − A ( n ) θ dz ) · tr(Λ ( 12 πi Z Γ zz − A ( n ) θ dz ))(A.12) = tr( 12 πi Z Γ zz − A ( n ) θ dz ) · ddθ | θ = θ tr(Λ ( 12 πi Z Γ zz − A ( n ) θ dz ))By (A.6), we have tr(Λ ( 12 πi Z Γ zz − A ( n ) θ dz )) = e iρ ( θ ) (A.13) tr( 12 πi Z Γ zz − A ( n ) θ dz ) = 2 e iρ ( θ ) (A.14)Since ddθ | θ = θ A ( n ) θ A ( n ) θ − ∈ shsp (2 d ), we can assume(A.15) ddθ | θ = θ A ( n ) θ A ( n ) θ − = (cid:18) X YZ − X ∗ (cid:19) where Y = Y ∗ , Z = Z ∗ and tr( X − X ∗ ) = 0. Let X, Y, Z be ( x ij ) di,j =1 , ( y ij ) di,j =1 , ( z ij ) di,j =1 respectively.By computation we get,(A.16) ddθ | θ = θ tr( 12 πi Z Γ zz − A ( n ) θ dz ) = tr( ddθ | θ = θ A θ · πi Z Γ z ( z − A ( n ) θ ) dz )Notice that πi R Γ z ( z − A ( n ) θ ) dz is the spectral projection of A ( n ) θ onto V θ , we have ddθ | θ = θ tr( 12 πi Z Γ zz − A ( n ) θ dz )(A.17) = tr( ddθ | θ = θ A ( n ) θ A ( n ) θ − · A ( n ) θ | V θ )= e iρ ( θ ) ( x − x ) + β ( θ ) z Denote A ( n ) θ | V θ by B θ = ( b i,j ( θ )) ≤ i,j ≤ d , then ddθ | θ = θ tr(Λ ( 12 πi Z Γ zz − A ( n ) θ dz ))(A.18) = ddθ | θ = θ tr(Λ ( A ( n ) θ | V θ ))= ddθ | θ = θ tr(Λ ( B θ ))= X i By (A.5),(A.6),(A.15) we have (cid:18) b , ( θ ) b ,d +1 ( θ ) b d +1 , ( θ ) b d +1 ,d +1 ( θ ) (cid:19) = (cid:18) e iρ ( θ ) β ( θ ) e iρ ( θ ) (cid:19)(cid:18) b ′ , ( θ ) b ′ ,d +1 ( θ ) b ′ d +1 , ( θ ) b ′ d +1 ,d +1 ( θ ) (cid:19) = (cid:18) e iρ ( θ ) x β ( θ ) x + e iρ ( θ ) y e iρ ( θ ) z β ( θ ) z − e iρ ( θ ) x (cid:19) Then combining with (A.18) we get(A.19) ddθ | θ = θ tr(Λ ( 12 πi Z Γ zz − A ( n ) θ dz )) = e iρ ( θ ) ( x − x )Combining (A.19),(A.17),(A.13),(A.14),(A.12) we get(A.20) z = 0Now we define the monotonic special Hermitian symplectic cocycles: consider acone C on shsp (2 d, R ) defined by(A.21) C := { W ∈ shsp (2 d, R ) : J · W is negative definite } where J is defined in Definition 1. Obviously for any W , W ∈ C , W + W ∈ C . Definition 17. Let θ D θ ∈ C ( X, SHSp (2 d, R )) be a one parameter family ofcontinuous symplectic cocycles and C in θ . We say it is monotonic at θ if (A.22) ddθ | θ = θ D θ ( x ) D − θ ( x ) ∈ C for any x ∈ X . We have: lemma . If the one parameter family of symplectic cocycle D θ is monotonic at θ , then for any n > , D ( n ) θ is also monotonic at θ .Proof. First of all we prove the invariance of the cone C under inner automorphism. lemma . For any g ∈ SHSp (2 d, R ) , g C g − = C Proof. Notice that for any g ∈ SHSp (2 d, R ), Jg = ( g − ) ∗ J . Then for any W ∈ C , JgW g − = ( g − ) ∗ JW g − (A.23)is negative definite (since g is invertible and JW is negative definite). Thereforewe have for any g , g C g − ⊂ C . It is easy to prove that the equality holds. (cid:3) Suppose D θ is monotonic at θ , then for any x ∈ X , using Lemma 38, we have ddθ | θ = θ D ( n ) θ ( x ) D ( n ) θ − ( x )(A.24) = n X i =0 D θ ( f n − ( x )) · · · D θ ( f i ( x )) · ( ddθ | θ = θ D θ ( f i ( x )) D θ ( f i ( x )) − ) · ( D θ ( f n − ( x )) · · · D θ ( f i ( x ))) − ∈ C which implies D ( n ) θ is also monotonic at θ . (cid:3) Come back to the discussion of periodic cocycle A θ . It is easy to check thatthe one parameter family cocycles θ A θ is monotonic on R , then by Lemma37, θ A ( n ) θ is monotonic at θ . As a result, J · ddθ | θ = θ A ( n ) θ A ( n ) θ − is negativedefinite, using (A.15) we get Z is negative definite, which contradicts with (A.20).In summary, I , I cannot have common boundary point. (cid:3) Appendix B. Generic periodic Schr¨odinger operator and Jacobimatrices on the strip For periodic Sch¨odinger operator and Jacobi matrices on the strip, consider thecorresponding (Hermitian) symplectic cocycle: A ( E − v ) : Z /n Z → Sp (2 d, R ) or SHSp (2 d, R )where A ( v ) ( x ) := (cid:18) v ( x ) − I d I d (cid:19) , v ( x ) ∈ R d ֒ → Sym d R , Sym d R or her ( d ) is thecorresponding potential. As in Appendix A, we hope to prove the following lemma: lemma . There is a constant C independent of n such that for generic potential v in ( R d ) n , ( Sym d R ) n or her ( d ) n , there exists E ∈ ( − Cn , Cn ) such that L ( A ( E − v ) ) > .Proof. At first we prove the following fact: lemma . For any Jacobi matrices on the strip, the corresponding one parametercocycle E ( A ( E − v ) ) (2) is monotonic on R .Proof. By computation J ddE | E = E ( A ( E − v ) ( f ( x )) A ( E − v ) ( x )) · ( A ( E − v ) ( x )) − ( A ( E − v ) ( f ( x )) − = (cid:18) − I d E − v ( f ( x )) E − v ( f ( x )) − I d − ( E − v ( f ( x ))) (cid:19) = − (cid:18) I d − ( E − v ( f ( x ))) I d (cid:19) · (cid:18) I d − ( E − v ( f ( x ))) I d (cid:19) which is negative definite, by definition of monotonicity in Appendix A, we get theproof. (cid:3) Denote B E ( x ) := ( A ( E − v ) ) (2) . Since E B E ( x ) is monotonic, mimic the proofof Theorem 3, we can define the m − function (taking values in { Z, k Z k < } ),fibered rotation function (which is non increasing) and complexify the Lyapunovexponent E + it ( L d + iρ )( B E + it ). In fact these properties are already knownfor Jacobi matrices on the strip with form in (1.3), see [28], [34] for example or [31]for general monotonic symplectic cocycles.The proof of Lemma 39 is basically the same as the discussion in Appendix A.At first we prove the following Lemma similar to Lemma 35. lemma . For any interval I ′ such that (B.1) ∀ E ∈ I ′ , all eigenvalues of B ( n ) E ( x ) are simple and norm we have | I ′ | ≤ O ( 1 n ) ENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SYMPLECTIC COCYCLES 41 Proof. As in the proof of Lemma 35, we only need to prove for almost all E ∈ I ′ ,we have(B.2) lim sup t → + L d ( B E + it ) t ≥ c ( d )where c ( d ) is a constant independent of n . Letˆ B E + it ( x ) := C ( f ( x )) B E + it ( x ) C ( x ) − . where C ( x ) := (cid:18) I d E − v ( f − ( x )) I d (cid:19) . Then all the dynamical properties of cocy-cles B E + it and ˆ B E + it are the same. Moreover by computation,(B.3) ∂ ˆ B E + it ( x ) ∂t = (cid:18) i · I d − i · I d (cid:19) · ˆ B E + it ( x )Therefore we have(B.4) ◦ ˆ B E + it ( x ) = ( (cid:18) e − t e t (cid:19) + o ( t )) · ◦ ˆ B E ( x )where ◦ ˆ B ( x ) := C ˆ B E ( x ) C − , C is the Cayley element defined in (2.1).Without loss of generality, we fix an E such that lim t → + Φ − C · ˆ m ( E + it, x )exists and be finite, where ˆ m ( E + it, · ) is the associated m − function of cocycle ◦ ˆ B E + it . Since Φ − C · ˆ m is the classical m − function taking values in { Z, Im ( Z ) > } for Jacobi matrices on the strip (see [28] for example), by Lemma 24 we know thatalmost every E ∈ R satisfies our condition.As Definition 10, we define by ˆ q E + it ( x ) the Jacobian (with respect to the volumeform induced by the Bergman metric) of the map Z ◦ ˆ B E + it ( x ) · Z at point Z = ˆ m ( E + it, x ). Then similar to the case of symplectic cocycles, we have(B.5) L d ( ◦ ˆ B E + it ) = 14 d Z X ln ˆ q − E + it dµ. Moreover as the proof of Lemma 17, we haveˆ q − E + it ( x ) = V ( e t ˆ m ( E + it, f ( x )) + o ( t )) V ( ˆ m ( E + it, f ( x ))) · e td + o ( t ) = e td + o ( t ) · Π di =1 − σ i ( ˆ m ( E + it, f ( x )))1 − ( e t σ i ( ˆ m ( E + it, f ( x ))) + o ( t ))(B.6)By our choice of E , lim t → + σ i ( ˆ m ( E + it, f ( x ))) exists and not greater than 1.Therefore using the inequality (4.19), we have(B.7) 1 − σ i ( ˆ m ( E + it, f ( x )))1 − ( e t σ i ( ˆ m ( E + it, f ( x ))) + o ( t )) ≥ 1, when t is smallCombining with (B.6)(B.5) we get,(B.8) L d ( ◦ ˆ B E + it ) ≥ dt + o ( t ), when t is smallfor almost all E ∈ I ′ , which implies our lemma. (cid:3) As a result, to prove Lemma 39, we only need to prove for generic potential v ,two intervals I , I satisfying the condition of Lemma 41 cannot share a commonboundary point. But in the corresponding part of Appendix A, we only use thecondition that the one parameter family θ A θ is monotonic, then by Lemma 40,the proof is the same here. 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