aa r X i v : . [ m a t h . D S ] O c t MONOTONIC COCYCLES
ARTUR AVILA AND RAPHA¨EL KRIKORIAN
Abstract.
We develop a “local theory” of multidimensional quasiperiodicSL(2 , R ) cocycles which are not homotopic to a constant. It describes a C -open neighborhood of cocycles of rotations and applies irrespective of arith-metic conditions on the frequency, being much more robust than the localtheory of SL(2 , R ) cocycles homotopic to a constant. Our analysis is centeredaround the notion of monotonicity with respect to some dynamical variable.For such monotonic cocycles , we obtain a sharp rigidity result, minimality ofthe projective action, typical nonuniform hyperbolicity, and a surprising resultof smoothness of the Lyapunov exponent (while no better than H¨older can beobtained in the case of cocycles homotopic to a constant, and only under arith-metic restrictions). Our work is based on complexification ideas, extended “`ala Lyubich” to the smooth setting (through the use of asymptotically holo-morphic extensions). We also develop a counterpart of this theory centeredaround the notion of monotonicity with respect to a parameter variable, whichapplies to the analysis of SL(2 , R ) cocycles over more general dynamical sys-tems and generalizes key aspects of Kotani Theory. We conclude with a moredetailed discussion of one-dimensional monotonic cocycles, for which resultsabout rigidity and typical nonuniform hyperbolicity can be globalized using anew result about convergence of renormalization. Contents
1. Introduction 12. Monotonicity in parameter space 83. Monotonic cocycles 264. One-frequency cocycles: renormalization and rigidity 36Appendix A. Conformal barycenter 40Appendix B. Transitivity of the projective action 42References 431.
Introduction
Let f : X → X be a homeomorphism of a compact metric space, preserving aprobability measure µ . Given a map A ∈ C ( X, SL(2 , R )) the skew-product map on X × R given by ( x, w ) ( f ( x ) , A ( x ) · w ), denoted by ( f, A ), is called an SL(2 , R )cocycle over f .We will be particularly interested in quasiperiodic cocycles where X = R d / Z d and f is a translation, f ( x ) = x + α for some α ∈ R d and µ is Lebesgue measure.Since in this case f is a diffeomorphism of a manifold, it makes sense to considerquasiperiodic cocycles with various degrees of smoothness. Quasiperiodic cocycles ( f, A ) have been primarily studied in the case where A ishomotopic to a constant (in large part because this is the situation arising in theconsideration of quasiperiodic Schr¨odinger operators). One important aspect wasthe development of a local theory , starting with the KAM based work of Dinaburg-Sinai [DS]. This local theory concerns perturbations of the simplest cocycles ho-motopic to a constant, which are just the constant ones. The development of theKAM approach involves, as usual, non-resonance assumptions which here are codedin arithmetic conditions involving the frequency vector but also the fibered rotationnumber , and a key achievement, due to Eliasson [E], was the development of alocal theory covering all cocycles with Diophantine frequency vector. Except forthe one-dimensional case, where considerably more can be achieved by a range oftechniques (both non-KAM as in [BJ], [AJ], and non-standard KAM, [AFK]), thework of Eliasson remains basically the best description of the local theory in thecase of cocycles homotopic to a constant.One of our goals here is to develop a local theory of cocycles non-homotopic to aconstant, covering (in the ergodic case) perturbations of the simplest cocycles aris-ing in this case, which are the SO(2 , R )-valued ones. As it will turn out, the theorywe develop is considerably more robust then the usual one, in several respects. Forinstance, the frequency vector plays no role at all in our considerations, and weare able to treat quite low regularity ( C ) perturbations. Moreover, several of ourconclusions are in a sense also much stronger, and even surprising from the pointof view of the intuition developed in the case of cocycles homotopic to a constant.Specific questions addressed in this paper concern the regularity of the Lyapunovexponents, rigidity arising from zero Lyapunov exponents, and minimality of theassociated projective action.Our local theory centers around the crucial property of monotonicity with re-spect to some phase (dynamical) variable, a kind of twist condition that arisesnaturally (in the ergodic case) for SO(2 , R )-valued cocycles not homotopic to aconstant. There is a counterpart, which actually precedes logically the analysis ofmonotonic cocycles, which describes the consequences of monotonicity with respectto parameter variables, and works for cocycles over general dynamical systems. Ourresults, abstract and generalize key aspects of the theory of Schr¨odinger cocycles(particularly Kotani Theory), in particular to the case of non-analytic dependenceon parameters. Besides being fundamental to our analysis of monotonic cocycles,we would also like to point out that the larger flexibility afforded by this theory hasbeen recently applied back to address problems about the Schr¨odinger case [A2].Our third focus in this paper concerns the analysis of one-dimensional quasiperi-odic cocycles non-homotopic to a constant from the global point of view. As in[AK], the basic plan is to reduce global questions to local ones by renormalization.To this end, we prove a “convergence of renormalization” result that guaranteesthat, under a natural (from the point of view of parameter analysis) hypothesisthat renormalizations eventually become monotonic. As a consequence, we obtaina global L -rigidity result, and conclude that typical cocycles non-homotopic to aconstant are nonuniformly hyperbolic.We will next present more formally some key results of each of the three topicsmentioned above.1.1. Monotonic cocycles.
Below we will use the notation ( f n , A n ) for the n -iterate of the cocycle ( f, A ): thus if n ≥ A n ( x ) = A ( f n − ( x )) · · · A ( x ). Let us ONOTONIC COCYCLES 3 also recall basic definitions of the Lyapunov exponent(1.1) L ( f, A ) = lim n →∞ n Z ln k A n ( x ) k dx, and of a conjugacy between cocycles ( f, A ) and ( f, A ′ ), which is given by a map B : X → SL(2 , R ) satisfying(1.2) A ′ ( x ) = B ( f ( x )) A ( x ) B ( x ) − . Now, and for the remaining of this section, fix d ≥ X = R d / Z d . For α ∈ R d denote by f α : X → X the map f α ( x ) = x + α . Since such dynamics areregular, it makes sense to speak of regular cocycles and regular conjugacies.We say that A ∈ C ( X, SL(2 , R )) is monotonic if there exists some w ∈ R d suchthat for every x ∈ X and y ∈ R \ { } , (any determination of) the argument of A ( x + tw ) · y , t ∈ R , has positive derivative with respect to t .This condition clearly determines a C -open subset M of C ( X, SL(2 , R )). No-tice that the monotonicity condition only makes reference to A and is thus in-dependent of a frequency vector. For this reason, it is in particular not invari-ant by conjugacy. Given a frequency vector α , it is thus natural to define a set M α ⊂ C ( X, SL(2 , R )) consisting of all A for which there exists n ≥ f α , A ) n is C -conjugated to some monotonic cocycle. We call cocycles ( f α , A ) with A ∈ M α premonotonic . It is also clear that cocycles homotopic to a constant cannever be premonotonic. On the other hand, if ( f α , A ) is C conjugate to a cocycleof rotations (that is, to an SO(2 , R )-valued one), and f α is ergodic, then ( f α , A ) ispremonotonic as long as A is C and not homotopic to a constant.Obviously any cocycle of rotations, or merely measurably conjugate to such,must have a zero Lyapunov exponent. Our first result shows that for regular (pre-)monotonic cocycles, one can go the other way around: Theorem 1.1.
Let ( f α , A ) be C r , r = ∞ , ω and premonotonic. If L ( f α , A ) = 0 then ( f α , A ) is C r conjugate to an SO(2 , R ) -valued cocycle. By the previous discussion, Theorem 1.1 contains in it a rigidity result.
Corollary 1.2.
Let f α be ergodic. If ( f α , A ) be C r , r = ∞ , ω is non-homotopic toa constant and C conjugate to rotations then it is C r -conjugate to rotations. Next we look at the dependence of the Lyapunov exponent. We recall firstthat in the case of cocycles homotopic to a constant, simple examples show thatone should never expect more (as far as the modulus of continuity is concerned)than 1 / It is somewhat delicate (in the ergodic case) to construct examples of cocycles not homotopicto a constant which are not premonotonic. A non-negligible (positive measure on parameters) setof such examples can be obtained by forcing a certain behavior of the “critical points” that arisein the approach of Lai-Sang Young [Y] (of Benedicks-Carleson [BC] flavor). We will come back tothis issue elsewhere.
ARTUR AVILA AND RAPHA¨EL KRIKORIAN
Theorem 1.3.
Let ( f α , A s ) be a one-parameter analytic family of analytic pre-monotonic cocycles. Then s L ( f α , A s ) is analytic. Theorem 1.4.
Let ( f α ( s ) , A s ) be a one-parameter C ∞ family of C ∞ premonotoniccocycles. Then s L ( f α ( s ) , A s ) is C ∞ .Remark . Those results are all the more striking in view of the recent discov-ery by Wang-You [WY] of smooth one-frequency cocycles at which the Lyapunovexponent is discontinuous.Since the Lyapunov exponent is a regular function, its zero set can be expectedto be some kind of variety, hence Lyapunov exponents should be rare unless theequation L = 0 is very degenerate. In fact, since the Lyapunov exponent can notbecome negative, DL must be zero whenever L = 0. We are however able to shownon-degeneracy of D L , which implies: Theorem 1.5.
For fixed α , a typical C r , r = ω, ∞ , premonotonic cocycle over f α has positive Lyapunov exponent. Though we do obtain several other results, particularly addressing less regularsituations, we would like to conclude our discussion, at this introduction, with aresult of different flavor. Given a cocycle ( f, A ), we may define its projective action,which is just the projectivized skew-product on X × PR . The topological dynamicsof the projective action is a very interesting subject in itself (see for instance [Bj1],[Bj2], [BjJ], [J2]). Here we prove: Theorem 1.6. If f α is ergodic and ( f α , A ) is premonotonic with A ∈ C ǫ , thenthe projective action is minimal. Let us point out that dynamical notions of monotonicity also make sense fordynamical systems which are not strict translations, such as the skew-shifts, andsome of our results can be carried to a larger generality, see Remark 3.1.1.2.
Monotonicity in the parameter space.
We return now to the consider-ation of more general dynamics f : X → X . This time we will be interested inparametrized families of cocycles ( f, A θ ), and we will often assume some base reg-ularity of this dependence (just with respect to θ , since nothing beyond continuitycan be made sense with respect to the dynamical variable under our hypothesis).Moreover, we will require the dependence of A θ on θ displays monotonicity: as-suming that θ A θ is C , this means that for every x ∈ X , y ∈ R \ { } , (anydetermination of) the argument of θ A θ ( x ) · y has positive derivative.Let us consider two key examples where monotonicity arises.Recall that in the dynamical approach to ergodic Schr¨odinger operators, the basicobject considered is a one-parameter family of cocycles, depending on a parameter E , of the form A ( E ) ( x ) = (cid:18) E − v ( x ) −
11 0 (cid:19) . Though the family ( f, A ( E ) ) is notmonotonic in E , its second iterate is (which in fact is just as good for our purposes).This family has of course been considered intensively and much is known about it:Kotani Theory ([Ko], [S], and more dynamically [CJ]), in particular, gives much The set of premonotonic cocycles with zero Lyapunov exponent has infinite codimension whenthe number of frequencies d is at least 2, and finite codimension when d = 1. We will come backto this issue elsewhere. ONOTONIC COCYCLES 5 dynamical information about the set of parameters where the Lyapunov exponentvanishes.Another family type that has been considered is of the form θ R θ A , where A is arbitrary and R θ ∈ SO(2 , R ) is the rotation of angle 2 πθ . This family displaysobvious monotonicity. Some global aspects of this family were first exploited in[H] to yield examples of cocycles with positive Lyapunov exponents: the averageLyapunov exponent, with respect to θ , is zero if and only if A is a cocycle ofrotations. In fact, a later refinement [AB] shows that(1.3) Z R / Z L ( f, R θ A ) dθ = Z X ln k A k + k A k − dµ ( x ) , so the average Lyapunov exponent depends on A through a very simple formula.The dynamical aspects of Kotani Theory have also been extended to such families.Both examples we mentioned have in common, besides (some) monotonicity, avery nice global behavior of the holomorphic dependence on θ when θ is complex.Here we will show that a theory can be constructed without taking into accountglobal aspects: in fact even analyticity can be bypassed. But complexification isstill fundamental, and what allows us to consider the smooth case is the use of asymptotically holomorphic extensions, a tool first used in dynamics by Lyubich[Ly], in the context of unimodal maps.In doing this, our key motivation has been the understanding of monotoniccocycles. In fact, if ( f α , A ) is monotonic in the dynamical sense, then one canconstruct a monotonic family ( f α , A θ ) by setting A θ = A ( x + θw ), for some w ∈ R d .When we change the parameter, we are not really changing the dynamics (merelythe coordinates), but we do get something non-trivial out of it, by applying theparameter results we will obtain. For this purpose, we will obtain analogous ofTheorems 1.1 and 1.3. Those parametrized versions correspond respectively to anon-global version of (1.3) and to a well known result of Kotani Theory (see [CJ]).Instead of presenting formal versions of those here, we prefer to mention a differentKotani-type application. Let us say that ( f, A ) is L -conjugate to rotations if it ismeasurably conjugate and the conjugacy B satisfies R X k B ( x ) k dµ ( x ) < ∞ . Theorem 1.7.
Let A θ ∈ C ( X, SL(2 , R )) , be a one-parameter family which ismonotonic and C ǫ in θ . Then for almost every θ , if L ( f, A θ ) = 0 then ( f, A θ ) is L -conjugate to rotations. As we will see in the next section, L -conjugacy to rotations is a fundamentalhypothesis in renormalization theory of one-dimensional quasiperiodic cocycles, soin a sense this result enlarges the set of families along which parameter exclusionarguments can be made before applying renormalization. The ability to analyze inthis way arbitrary monotonic deformations turns out to be relevant even if one isultimately interested in the Schr¨odinger case, see [A2].1.3. One-dimensional quasiperiodic cocycles non-homotopic to a con-stant.
We consider again the quasiperiodic case, but now restrict attention tothe one-dimensional case. In this section X = R / Z and f α will always denote anergodic translation (thus α ∈ R \ Q ).Renormalization is a classical tool in the analysis of diffeomorphisms of the circle,where it can be used to reduce global questions to local ones [KS]. Application ofrenormalization ideas to the case of quasiperiodic cocycles has also proved to be ARTUR AVILA AND RAPHA¨EL KRIKORIAN fruitful though in this case the renormalization operator does not always lead tothe local situation due to the possible presence of positive Lyapunov exponents. Itshould thus be basically considered as a tool to explore cocycles with zero Lyapunovexponent; see [K1] for the case of SU (2)-valued cocycle. However, zero Lyapunovexponents are not a sufficient condition to achieve the global-local reduction. In [AK], it is shown that the existence of an L -conjugacy to rotations is enoughto guarantee precompactness of the renormalization operator, which is used to ex-tract limits which are cocycles of rotations (up to constant conjugacy). UsingKotani Theory and the local (KAM) description of cocycles homotopic to a con-stant, this yields a dicothomy (under suitable regularity requirements and arith-metic conditions) for typical energies: the associated Schr¨odinger cocycle has eithera positive Lyapunov exponent, or it is conjugate to a cocycle of rotations. Though cocycles of rotations which are not homotopic to a constant are pre-monotonic if the basis is ergodic , renormalization affects the base dynamics and inparticular may lead to non-ergodic limits.In order to obtain more precise results, we prove here that the limits of renor-malization are in fact of a very special kind, namely of the form x R θ + | deg | x ,where deg is the topological degree. In particular, if | deg | 6 = 0, one does reachmonotonicity. We conclude the following global rigidity result: Theorem 1.8.
Let ( f α , A ) , α ∈ R \ Q , be C r , r = ω, ∞ , and non-homotopicto a constant. If ( f α , A ) is L -conjugate to rotations, then it is C r -conjugate torotations. Combined with Theorems 1.7 and 1.5 we conclude:
Theorem 1.9.
For fixed α ∈ R \ Q , a typical C r , r = ω, ∞ , cocycle over f α whichis not homotopic to a constant has positive Lyapunov exponent.Remark . The first result about the existence of positive Lyapunov exponentsfor cocycles non-homotopic to a constant was obtained in [H], for cocycles of a veryspecific form (with respect to their global holomorphic extensions, say(1.4) x (cid:18) λ λ − (cid:19) R x ,λ > λ >>
1, positive Lyapunov exponents have large probability withrespect to the choice of the frequency vector. The method used by Young, quitedifferent from ours, of Benedicks-Carleson type [BC], is based on an inductivescheme which loses control of a positive measure set of parameters, and needs someinitial condition to start (creating the need for a largeness assumption).We would like to point out that our more precise results about convergence ofrenormalization have been recently applied [AFK] to the case of cocycles homotopicto a constant: together with developments in the local theory, it yields the basic[AK] dichotomy without arithmetic conditions. Particularly, the analysis of the spectrum of the critical almost Mathieu operator, where theLyapunov exponent is still zero, does not reduce to the local situation. Since this result indeed assumes at least a Diophantine condition on α , and the cocycle ishomotopic to a constant, the conclusion is in fact equivalent to the existence of a conjugacy to a constant cocycle of rotations. ONOTONIC COCYCLES 7
Structure of the paper.
In section 2, we analyze the consequences of mono-tonicity with respect to parameters. We start with some aspects of the dynamics ofcertain SL(2 , C ) cocycles, whose action by M¨obius transformations preserve the up-per half plane when going forward, but not necessarily backwards. We also discussthe crucial notion of variation of the fibered rotation number , which is necessaryeven to formulate several key results. After a few simple applications of the com-plexification idea in the analytic case, we describe an asymptotically holomorphicframework that allows us to exploit the basic monotonicity phenomenon, and wecarry out the basic computations of Kotani Theory in this setting.Basically, monotonicity is used to guarantee that when the parameter turns com-plex, the dynamics becomes uniformly hyperbolic , and everything depends nicelyon parameters. The focus is thus to recover some information when the imaginarypart approaches zero. In the analytic case, this is done by appealing to theoremsof complex analysis (such as Fatou’s Theorem on existence of non-tangential lim-its). In the asymptotically holomorphic setting, we would like to show that thediscrepancy from holomorphicity corresponds to a regular correction, say, of thenon-tangential limit. There are competing factors though: while the cocycle be-haves “more holomorphically” near real parameters, it also behaves “less uniformlyhyperbolic”. A key point is thus to give an estimate of the resulting regularity (inpractice giving up some derivatives in the process). After this, we are in good shapeto collect results such as Theorem 1.7.In section 3 we then move on to the analysis of monotonic cocycles. We proveeasily Theorems 1.1 and 1.3 using the results of section 2, and proceed to look atTheorem 1.4, whose proof does not really fall in the same context: since the dy-namics changes we have to revisit the estimates regarding the competition betweenasymptotic holomorphicity and uniform hyperbolicity, incorporating this additionalparameter. Using a somewhat different approach, we next discuss results in lowregularity, such as continuity of the Lyapunov exponent in the Lipschitz category.We come back to a more regular situation in the proof of Theorem 1.6, where weapply the Schwarz Reflection Principle, or rather, use that it can not be applied.We conclude with an estimate on the second derivative of the Lyapunov exponent,which implies Theorem 1.5.In section 4 we specify further to the one-dimensional case, but now with aglobal focus. We introduce formally the renormalization operator and explain how“convergence of renormalization” combined with the local theory indeed impliesTheorem 1.8. We conclude with a proof of convergence of renormalization.We include two appendices. The first discusses an estimate about the conformalbarycenter which is used when taking limits of L -conjugacies to rotations, whichis used to avoid unnecessary parameter exclusions. The second gives a proof oftransitivity of the projective action for quasiperiodic cocycles non-homotopic to aconstant. Acknowlegements:
R.K. would like to thank the hospitality of IMPA. Thisresearch was partially conducted during the period A.A. was a Clay Research Fel-low. We are grateful to Zhenghe Zhang for detailed comments about a preliminaryversion of this paper.
ARTUR AVILA AND RAPHA¨EL KRIKORIAN Monotonicity in parameter space
In this section, f : X → X is a fixed homeomorphism of a compact metric space X , preserving a fixed probability measure µ , assumed to have full support. Given A ∈ C ( X, SL(2 , C )), we use the dynamics f : X → X to define the iterated matrixproducts A n ( x ), n ∈ Z , by(2.1) A n ( x ) = A ( f n − ( x )) · A n − ( x ) , A − n ( x ) = A n ( f − n ( x )) − , n ≥ − , A ( x ) = id . The Lyapunov exponent is defined by(2.2) L ( A ) = lim n →∞ n Z X ln k A n ( x ) k dµ ( x ) . As discussed in the introduction, our aim here is to develop a “smooth” version ofKotani Theory, centered around the concept of monotonic dependence with respectto the parameter. Let I ⊂ R be an interval. We say that a continuous function f : I → R is ǫ -monotonic if for every x = x ′ we have(2.3) | f ( x ′ ) − f ( x ) || x ′ − x | ≥ ǫ. This definition naturally extends to functions defined on (or taking values on) R / Z (by considering lifts) and on the unit circle S ⊂ R ≡ C (by consideringthe identification with R / Z given by x e πix ). Naturally, we may distinguishbetween two types of ǫ -monotonicity, increasing or decreasing.We say that a continuous one-parameter family of matrices A θ ( · ) ∈ SL(2 , R ) is ǫ -monotonic if, for every w ∈ R ≡ C , the function θ A θ · w k A θ · w k is ǫ -monotonic.We will be interested in one-parameter families of SL(2 , R ) cocycles displayingmonotonicity with respect to the parameter variable. Thus, a continuous one-parameter family A θ ∈ C ( X, SL(2 , R )) is said to be ǫ -monotonic increasing (re-spectively, decreasing) if for every x ∈ X , the family θ A θ ( x ) is ǫ -monotonicincreasing (respectively, decreasing).For several results, we will need to assume further regularity with respect to theparameter. Let us say that the family A θ is C r in θ ∈ J if θ A θ ( x ) belongs tosome fixed compact subset of C r ( J, SL(2 , R )) (compact open topology) for each x .Among the results we will obtain in this section, we highlight the following ones: Theorem 2.1.
Let A θ ∈ C ( X, SL(2 , R )) be monotonic and C r +1+ ǫ , ≤ r < ∞ , C ∞ , or C ω in θ . If L ( A θ ) = 0 for every θ in some open interval J then there exists B θ ∈ C ( X, SL(2 , R )) , θ ∈ J depending C r , C ∞ or C ω on θ and conjugating A θ toa cocycle of rotations. Theorem 2.2.
Let A θ,s ∈ C ( R / Z , SL(2 , R )) , θ ∈ R / Z , s a one-dimensional realparameter, be monotonic in θ and C r +1+ ǫ , ≤ r < ∞ , C ∞ or C ω , in ( θ, s ) . Then (2.4) s Z R / Z L ( A θ,s ) dθ is C r , C ∞ , or C ω . This theorem implies for instance that A R R / Z L ( R θ A ) dθ is an analytic func-tion of A ∈ C ( R / Z , SL(2 , R )). Indeed, it can be shown (see [AB]) that(2.5) Z R / Z L ( R θ A ) dθ = Z X ln k A ( x ) k + k A ( x ) k − dµ ( x ) . ONOTONIC COCYCLES 9
This generalization beyond families with specific form such as R θ A will be crucialwhen we start to mix phase and parameter in the analysis of quasiperiodic cocyclesdisplaying monotonicity with respect to some phase variable.We will also obtain several other results whose statements depend on the conceptof “variation of the fibered rotation number”, which we will first need to introduce.2.1. Complexification.
Much of the information we will get from matrices inSL(2 , C ) will come from their action on the Riemann Sphere C through M¨obiustransformations:(2.6) (cid:18) a bc d (cid:19) · z = az + bcz + d , For A ∈ SL(2 , C ), let ˚ A = QAQ − where(2.7) Q = −
11 + i (cid:18) − i i (cid:19) . The map A ˚ A maps bijectively SL (2 , R ) to SU (1 , (cid:18) u ¯ vv ¯ u (cid:19) , u, v ∈ C such that | u | − | v | = 1. Variation of the fibered rotation number.
In the analysis of Schr¨odingercocycles, the notion of fibered rotation number plays a fundamental role (oftenthrough the analysis of its close cousin, the integrated density of states). In oursetting, it turns out that it is not always possible to define “the” fibered rotationnumber of a cocycle. However, we will be able to define the notion of variation ofthe fibered rotation number along a path.Define Υ as the space of SL(2 , C ) matrices A such that ˚ A · D ⊂ D (or equivalently A · H ⊂ H ).Let A ∈ Υ. Define τ A : D → C \ { } by(2.8) ˚ A (cid:18) z (cid:19) = τ A ( z ) · (cid:18) ˚ A · z (cid:19) . Since D is simply connected there exists a map ˆ τ A : D → C such that e πi ˆ τ A ( z ) = τ A ( z ); any other lift is obtained by the addition of an integer. If we denote byˆΥ the universal cover of Υ considered as a topological semi-group with unity ˆid,there exists a unique continuous map ˆ τ : ˆΥ × D → C such that ˆ τ ( ˆid , z ) = 0 and e πi ˆ τ ( ˆ A,z ) = τ A ( z ). This map satisfies(2.9) ˆ τ ( ˆ A ˆ A , z ) = ˆ τ ( ˆ A , ˚ A · z ) + ˆ τ ( ˆ A , z ) . We note that for any ˆ A ∈ ˆΥ and any z, z ′ ∈ D ℑ ˆ τ ( ˆ A, z ) = − π | ln τ A ( z ) ||ℜ ˆ τ ( ˆ A, z ) − ℜ ˆ τ ( ˆ A, z ′ ) | < / . The equality is trivial, while the inequality follows from the fact that for any A , τ A ( D ) is contained in an open half plane (it is enough to observe that if ˚ A = If one identifies the complex one-dimensional projective space CP with C , by associating tothe line through (0 , = ( x, y ) ∈ C the complex number x − iyx + iy , the action of A ∈ SL(2 , C ) on CP is given precisely by z ˚ A · z . In this identification, the real one-dimensional projective space RP corresponds to the unit circle ∂ D , and SL(2 , R ) matrices preserve the unit disk D . (cid:18) u vv u (cid:19) then τ A ( z ) = vz + u , so τ A ( D ) does not intersect the line through iu , since | u | − | v | = 1).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 this is independent of thechoice of the lift ˆ γ .Let us note a nice composition rule: given γ and γ ′ , let γ ′ γ ( t ) = γ ′ ( t ) γ ( t ). Then(2.10) δ γ ′ γ ˆ τ ( z , z ) = δ γ ′ ˆ τ (˚ γ (0) z , ˚ γ (1) z ) + δ γ ˆ τ ( z , z ) . Consider now a continuous path γ ∈ C ([0 , , C ( X, Υ)) = C ([0 , × X, Υ).Define δ γ ξ : X × D × D → C by δ γ ξ ( x, z , z ) = δ γ x ˆ τ ( z , z ), where γ x ( t ) = γ ( t, x ).Using the dynamics f : X → X , we define paths γ n ∈ C ([0 , , C ( X, Υ)) byputting γ n ( t, x ) = γ ( t, f n − ( x )) · · · γ ( t, x ). Define δ γ ξ n ∈ C ( X × D × D , C ) by δ γ ξ n = n δ γ n ξ . We have an expression for δ γ ξ n as a Birkhoff average (for thedynamical system ( x, z , z ) ( f ( x ) , ˚ γ (0 , x ) · z , ˚ γ (1 , x ) · z ) acting on X × D × D ):(2.11) δ γ ξ n ( x, z , z ) = 1 n n − X k =0 δ γ ξ ( f k ( x ) , ˚ γ k (0 , x ) · z , ˚ γ k (1 , x ) · z ) . This is obtained by the composition formula (2.10).We claim that lim n →∞ δ γ ξ n ( x, z , z ) exists for µ -almost every x , and is inde-pendent of z , z ∈ D . We will show this by proving convergence for the real andimaginary parts.Since for every z , z ∈ D we have |ℜ ( δ γ ξ n ( x, z , z ) − δ γ ξ n ( x, z ′ , z ′ )) | < n , itfollows from Birkhoff Ergodic Theorem that(2.12) lim n →∞ ℜ δ γ ξ n ( x, z , z )exists and is independent of z , z ∈ D for µ -almost every x . We call the µ -averageof (2.12) the variation of the fibered rotation number along γ and we denote itby δ γ ρ . It is obviously invariant by homotopy, and it is a continuous function of γ ∈ C ([0 , , C ( X, Υ)).
Remark . If µ is ergodic, (2.12) is µ -almost everywhere constant. If f is uniquelyergodic, (2.12) exists for every x and is constant. If the iterates of f are uniformlyequicontinuous (for instance, if f is a translation of the torus, but perhaps non-ergodic), (2.12) exists for every x , and is a continuous function of x ∈ X .The convergence of the imaginary part of δ γ ξ n ( x, z , z ) is somewhat more deli-cate, and it is certainly less robust. For A ∈ C ( X, SL(2 , C )) the Lyapunov expo-nent is defined by(2.13) L ( A, x ) = lim n →∞ n ln k A n ( x ) k (which exists µ -almost everywhere by subadditivity). When A ∈ C ( X, Υ), a simpleapplication of the Oseledets Theorem shows that for any z ∈ D (2.14) L ( A, x ) = lim n →∞ n ln (cid:13)(cid:13)(cid:13)(cid:13) ˚ A n ( x ) · (cid:18) z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) since (cid:18) z (cid:19) can not belong to the stable direction (this is due to the fact that in thatcase A k ( x ) ∈ Υ for every k ≥ ONOTONIC COCYCLES 11
Lemma 2.3.
For µ -almost every x ∈ X and for every z , z ∈ D , (2.15) lim n →∞ ℑ δ γ ξ n ( x, z , z ) = 12 π ( L ( γ (0) , x ) − L ( γ (1) , x )) . Proof.
Notice that if A ∈ C ( X, SL(2 , C ))(2.16) ln (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y k = n − ˚ A ( f k ( x )) · (cid:18) z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ln | τ A n ( x ) ( z ) | + ln (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Q k = n − ˚ A ( f k ( x )) · z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , and the second term in the right hand side is bounded. (cid:3) We let δ γ ζ be the µ -average of lim δ γ ξ n ( x, , δ γ ρ and its imaginary part is π ( L ( γ (0)) − L ( γ (1)), where L ( A ) is defined as inthe introdution:(2.17) L ( A ) = lim n →∞ n Z ln k A n k dµ. Remark . Notice that δ γ ζ behaves well under concatenation: if γ , γ ′ and ˜ γ aresuch that γ (1) = γ ′ (0) and ˜ γ is homotopic to the concatenation of γ and γ ′ then δ ˜ γ ζ = δ γ ′ ζ + δ γ ζ .2.2.1. Invariant section.
Assume that for every x ∈ X , we have A ( x ) · D ⊂ D . Inthis case, by the Schwarz Lemma, A ( · ) uniformly contracts the Poincar´e metric ofthe disk and there exists m ∈ C ( X, D ) satisfying(2.18) m ( f ( x )) = ˚ A ( x ) · m ( x ) , and we have for every ( x, z ) ∈ X × D ,(2.19) lim n →∞ ˚ A n ( f − n ( x )) · z = m ( x ) . Thus(2.20) δ γ ζ = Z δ γ ξ ( x, m γ (0) ( x ) , m γ (1) ( x )) dµ ( x ) , where m γ ( t ) stands for the invariant section corresponding to A = γ ( t ).Notice that there is another formula for the Lyapunov exponent in terms of m .Let q ( x ) be the norm of the derivative of the holomorphic function z ˚ A ( x ) · z at z = m ( x ), with respect to some conformal Riemannian metric k · k z on D . In thiscase(2.21) L = 12 Z − ln q ( x ) dµ ( x ) . The most convenient metric to consider is the Poincar´e metric (1 − | z | ) − | dz | ,since it enables us to apply the Schwarz Lemma. Notice that whenever(2.22) ˚ A (cid:18) z (cid:19) = τ (cid:18) ˜ z (cid:19) one has(2.23) (cid:12)(cid:12)(cid:12)(cid:12) d ˜ zdz (cid:12)(cid:12)(cid:12)(cid:12) = 1 | τ | , q = (cid:12)(cid:12)(cid:12)(cid:12) d ˜ zdz (cid:12)(cid:12)(cid:12)(cid:12) − | z | − | ˜ z | , and (2.21) can be also obtained as a consequence of (2.18) (which implies that L = R ln | τ A ( x ) ( m ( x )) | dµ ( x )). The Schwarz Lemma yields, for instance, the following estimate of the Lyapunovexponent. If ˚ A ( x ) · D ⊂ D e − ǫ for every x ∈ X then the composition m ˜ m =˚ A ( x ) · m e ǫ ˜ m sends D to itself and hence weakly contracts Poincar´e metric:(2.24) | d ( e ǫ ˜ m ) | − | e ǫ ˜ m | ≤ | dm | − | m | from which one gets(2.25) | d ˜ m | − | ˜ m | ≤ e − ǫ − e ǫ | ˜ m | − | ˜ m | | dm | − | m | . Whence(2.26) q ( x ) − ≥ e ǫ − | m ( f ( x )) | − e ǫ | m ( f ( x )) | ≥ e ǫ so that L ≥ ǫ .2.2.2. Fibered rotation function.
Let us now consider a continuous family A θ ∈ C ( X, Υ), where θ belongs to some connected Banach manifold M . Since δ γ ζ depends only on the homotopy class of γ : [0 , → C ( X, Υ), and it behaves wellunder concatenation, see Remark 2.2, we can define a map ζ : ˜ M → C ( ˜ M beingthe universal cover of M ) such that for every path ˜ γ : [0 , → ˜ M , we have δ γ ζ = ζ (˜ γ (1)) − ζ (˜ γ (0)), where γ ( t ) = A π (˜ γ ( t )) and π : ˜ M → M is the canonical projection.Moreover, we can take ζ so that for π (˜ θ ) = θ we have − π ℑ ζ (˜ θ ) = L ( A θ ). We shallthen denote ρ (˜ θ ) = ℜ ζ (˜ θ ) or with an abuse of notation ρ ( θ ) or ρ A θ . Though ρ isonly defined up to a real constant, it makes sense to speak about the derivative of ρ , and if M = R or M = R / Z , we may ask whether ρ is monotonic or not.2.3. Simple applications.
We now turn to one-parameter continuous families θ A θ ( · ) ∈ C ( X, SL(2 , R )). To keep definite and to avoid superfluous notations,we will consider only the case when the parameter space is R / Z . To fix ideas, wewill always assume in the proofs below that A θ ( · ) is monotonic decreasing. Notethat due to our identification of PR with ∂ D , the projectivization map S → PR is orientation reversing. Thus if θ A θ ( · ) is monotonic decreasing then for each x ∈ X , z ∈ ∂ D , θ ˚ A θ ( x ) · z is monotonic increasing.Let us show how to deduce the analytic case of Theorem 2.2. Let Ω δ = { z ∈ C / Z , |ℑ ( z ) | < δ } , Ω ± δ = { z ∈ C / Z , < ±ℑ ( z ) < δ } , and let ˜Ω δ , ˜Ω ± δ , be theiruniversal covers.If θ A θ is assumed to be monotonic and analytic, the Cauchy Riemann equa-tions imply that there exists δ > A θ to θ ∈ Ω + δ satisfies(2.27) ˚ A θ · D ⊂ D e − ǫ ℑ ( θ )+ κ ( ℑ ( θ )) , < ℑ θ ≤ δ, where κ ( t ) < ǫt for 0 < t ≤ δ and lim t → κ ( t ) /t = 0. As we saw in section 2.2.1,this implies that L ( A θ ) ≥ ǫ ℑ θ − κ ( ℑ θ ).Indeed, for fixed z ∈ ∂ D , x ∈ X , the Cauchy-Riemann equations applied to theanalytic function F x,z : Ω + δ → D defined by F x,z ( σ + it ) = ˚ A σ + it ( x ) · z shows that( ∂ σ F x,z ( θ ) , ∂ t F x,z ( θ )) is a directed orthogonal basis with vectors of equal lengthwhen θ ∈ R / Z . Since ∂ σ F x,z ( θ ) is tangent to the circle ∂ D , and of length at least2 ǫ (by monotonicity), we see that ∂ t F x,z ( θ ) is radial (pointing to the origin because ONOTONIC COCYCLES 13 of the sign assumption) and of length at least 2 ǫ for any θ ∈ R / Z . This proves theabove estimate (2.27).By the previous discussion, we can choose δ > < t ≤ δ we have˚ A σ + it,s ( x ) · D ⊂ D and we are thus in a situation where the discussion of section 2.2applies since A σ + it,s ( · ) ∈ C ( X, Υ): there exists a function ζ : ˜Ω + δ → C such that onthe closure of ˜Ω + δ the map θ ρ ( θ ) = ℜ ζ ( θ ) is continuous and ℑ ζ ( θ ) = − π L ( A θ );also, since θ A θ is holomorphic on Ω + δ , the same is true for ζ . We will use thenotation m + ( z, x ) for the D -valued invariant section of the cocycle A z .Now if ( θ, s ) A θ,s is analytic and if s is in some fixed neighborhood of s , wecan choose δ > < t ≤ δ we have ˚ A σ + it,s ( x ) · D ⊂ D . Let then(2.28) U ( t, s ) = Z R / Z L ( A σ + it,s ) dσ, so that for t = 0 we have(2.29) U ( t, s ) = Z X × R / Z ln | τ A σ + it,s ( x ) ( m + s ( σ + it, x )) | dµ ( x ) dσ. Then for 0 < t < δ the map s U ( t, s ) is analytic. Moreover, the map( σ + it, z ) τ A σ + it,s ( x ) ( z ) (defined on Ω + δ × D is holomorphic and non zero andso R X ln | τ A σ + it,s ( x ) ( m + s ( σ + it, x )) | dµ ( x ) is harmonic w.r.t σ + it ∈ Ω + δ ; its integralw.r.t σ , given by U ( t, s ), is thus an affine function of 0 < t < δ since it is harmonicin σ + it and does not depend on σ .On the other hand, since σ + it A σ + it,s ( x ) is holomorphic, the functions σ + it ln k ( A σ + it,s ) n ( x ) k are subharmonic and the same is true for σ + it L ( A σ + it,s ). Notice that U ( t, s ) = U ( − t, s ) since σ + it L ( A σ + it,s ) is real sym-metric, so by subharmonicity, U ( t, s ) is an affine function of | t | for 0 ≤ | t | < δ ( t U ( t, s ) is convex and thus continuous in t ). Thus, for 0 < t < δ we have(2.30) Z R / Z L ( A θ,s ) dθ = 2 U ( t , s ) − U ( t, s ) , is analytic on s , as desired. Remark . With a little bit more work, one can get the formula(2.31) Z R / Z L ( A θ,s ) dθ = U ( t, s ) − πt deg , for 0 < t < δ , where deg is the variation of the fibered rotation number as θ runsonce around R / Z . Indeed, for fixed s , let ρ ( σ + it, s ) : ˜Ω + δ → R be a continuousdetermination of ρ A σ + it,s , so that deg = ρ ( σ + it + 1 , s ) − ρ ( σ + it, s ). Then thefunction(2.32) Z σ +1 σ − i π L ( A y + it,s ) + ρ ( y + it, s ) dy is holomorphic in σ + it ∈ ˜Ω + δ and its real part is an affine function of σ of slopedeg. Thus the function U ( t, s ) defined above is an affine function of 0 < t < δ (Cauchy-Riemann) with slope 2 π deg, and (2.31) follows. It is easy to see that deg is just the µ -average of deg( x ), where deg( x ) is the topological degreeof θ A θ,s ( x ) as a map R / Z → SL(2 , R ) (in particular, if X is connected, deg is an integer). Let us now describe another application, the analogous of a basic derivativebound in Kotani Theory. To set it up, let us state an obvious consequence ofmonotonicity (following directly from the definitions).
Lemma 2.4.
Let A θ ( · ) ∈ C ( X, SL(2 , R )) , be a one-parameter family monotonicdecreasing in θ . Then the fibered rotation number is non-increasing as a functionof θ . Theorem 2.5.
Let A θ ( · ) ∈ C ( X, SL(2 , R )) , θ ∈ R / Z , be analytic and monotonicdecreasing in θ . For Lebesgue a.e θ ∈ R / Z , if L ( A θ ) = 0 then (2.33) − ddθ ρ ( θ ) ≥ ǫ π > , where ǫ is the monotonicity constant of θ A θ ( · ) .Proof. Using the analyticity in θ of A θ , we can conclude that θ ζ ( θ ) can bedefined as a holomorphic function ˜Ω + δ → H . We know that the real part of ζ ( θ )(the “fibered rotation number”) is continuous up to the closure. For σ ∈ R ,(2.34) ℑ ( ζ ( σ + it )) = ℑ ( ζ ( σ + i + )) + Z t + ∂ s ℑ ( ζ ( σ + is )) ds, and using the Cauchy-Riemann equations(2.35) ℑ ( ζ ( σ + it ) = ℑ ( ζ ( σ + i + )) + Z t + ∂ σ ℜ ζ ( σ + is ) ds, Since the map ℜ ζ ( · ) is harmonic on Ω + δ , continuous on the closure of ˜Ω + δ and itsrestriction to ℑ θ = 0 is non-increasing, one can say that for Lebesgue a.e σ ∈ R / Z (2.36) lim s → ∂ σ ℜ ζ ( σ + is ) = ddσ ρ ( σ ) . Since the Lyapunov exponent is upper semicontinuous (it is by subadditivity theinfimum of the continuous functions θ (1 /n ) R X ln k ( A θ ) n ( x ) k dµ ( x )), if we knowadditionally that L ( A σ ) = 0 it becomes continuous at the point σ and we have(2.37) − lim 12 πt L ( A σ + it ) = ddσ ρ ( σ ) , almost surely, and the result follows (since L ( A σ + it ) ≥ ǫt − κ ( t )). (cid:3) General framework.
After the motivation above, we are ready to introducea more general framework for the complexification argument.It may look like analyticity is crucial in order to exploit the complexificationapproach. This is not the case: in the non-analytic case, we can still complexify theproblem using asymptotically holomorphic extensions (this idea is inspired fromthe work of Lyubich on smooth unimodal maps [Ly]).Let ∆ δ be the space of all continuous families A σ + it ( · ) ∈ C ( X, SL(2 , C )), σ + it ∈ Ω δ , which are C and real-symmetric in σ + it , satisfying A σ + it ∈ int Υ, σ + it ∈ Ω + δ ,(2.38) ∂ z A z = 0 , if ℑ ( z ) = 0 , A continuous harmonic function f on the disk is the Poisson integral of its restriction ρ to theboundary of the disk. If ρ is of bounded variation, the tangential derivative ∂ σ f is the Poissonintegral of the measure dρ . Fatou theorem asserts that for a.e point on the boundary, the radiallimit of ∂ σ f is dρ/dσ . The situation on the strip is easily reduced to the one on the disk. ONOTONIC COCYCLES 15 and such that σ A σ ( · ) is monotonic in σ . Condition (2.38) is an asymptoticholomorphicity assumption, some stronger forms of which we will later introduce.Let us fix A ∈ ∆ δ . Then we have functions m + ( σ + it, x ) ∈ D , τ + ( σ + it, x ) ∈ C \ { } , σ + it ∈ Ω + δ , x ∈ X , characterized by(2.39) ˚ A σ + it ( x ) · (cid:18) m + ( σ + it, x )1 (cid:19) = τ + ( σ + it, x ) (cid:18) m + ( σ + it, f ( x ))1 (cid:19) . Notice that A σ + it ( x ) − ∈ int Υ for σ + it ∈ Ω − δ . Thus we have also functions m − ( σ + it, x ), τ − ( σ + it, x ), σ + it ∈ Ω − δ , x ∈ X , characterized by(2.40) ˚ A σ + it ( x ) · (cid:18) m − ( σ + it, x )1 (cid:19) = τ − ( σ + it, x ) (cid:18) m − ( σ + it, f ( x ))1 (cid:19) . Since A σ + it is real-symmetric in σ + it , letting(2.41) m + ( σ + it, x ) = 1 m + ( σ − it, x ) , τ + ( σ + it, x ) = 1 τ + ( σ − it, x ) , σ + it ∈ Ω − δ and(2.42) m − ( σ + it, x ) = 1 m − ( σ − it, x ) , τ − ( σ + it, x ) = 1 τ − ( σ − it, x ) , σ + it ∈ Ω + δ , we have that (2.39) and (2.40) are valid for σ + it ∈ Ω δ \ R / Z .The following key computation generalizes estimates of Kotani [K2] (see also [S])and Deift-Simon [DeS]. Lemma 2.6.
Let A ∈ ∆ δ and let σ ∈ R / Z . Then (1) If (2.43) lim inf t → L ( σ + it ) t < ∞ then (2.44)lim inf t → Z X − | m + ( σ + it, x ) | dµ ( x ) + Z X − | m − ( σ − it, x ) | dµ ( x ) < ∞ . (2) If (2.45) lim sup t → L ( σ + it ) t < ∞ then (2.46)lim sup t → Z X − | m + ( σ + it, x ) | dµ ( x ) + Z X − | m − ( σ − it, x ) | dµ ( x ) < ∞ and (2.47) lim inf t → Z X | m + ( σ + it, x ) − m − ( σ − it, x ) | dµ ( x ) = 0 . Proof.
Let us assume that(2.48) ( ∂ t ˚ A σ + it ( x )) ˚ A σ + it ( x ) − = u ( x ) (cid:18) − (cid:19) + C ( σ + it, x ) , where u ( x ) < t → sup x ∈ X k C ( σ + it, x ) k = 0. The general case can bereduced to this one by conjugacy. Indeed, if B : X → SL(2 , R ) is continuous,then it is indifferent to prove the results for the original A or for its conjugate B ( f ( x )) A σ + it ( x ) B ( x ) − , so it is enough to select B such that(2.49) ˚ B ( f ( x ))( ∂ t ˚ A σ ( x )) ˚ A σ ( x ) − ˚ B ( f ( x )) − is a matrix of the form (cid:18) u ( x ) 00 − u ( x ) (cid:19) , u ( x ) <
0, for each x . Such a B can befound due to the monotonicity hypothesis as we will now show.Since ¯ ∂ z A z = 0 at ℑ z = 0, we have ( ∂ t ˚ A ) ˚ A − = i ( ∂ σ ˚ A ) ˚ A − at σ + it = σ , andsince ˚ A σ is in SU(1 ,
1) for any σ ∈ R / Z , we can write(2.50) ( ∂ t ˚ A ) ˚ A − = (cid:18) a − i ¯ ν − iν − a (cid:19) , a ∈ R , ν ∈ C at σ + it = σ . Moreover, since ˚ A σ + it · D ⊂ D e − ǫt + o ( t ) and ˚ A σ · D = D , we have(2.51) (cid:12)(cid:12)(cid:12)(cid:12) (1 + ta ) m − iνt + o ( t ) − iνmt + (1 − at ) + o ( t ) (cid:12)(cid:12)(cid:12)(cid:12) < − ǫt + o ( t )for any m ∈ ∂ D and any small t >
0; this implies that a − | ν | sin φ ≤ − ǫ/ φ and consequently a < ∂ t ˚ A ) ˚ A − ) = | ν | − a ≤ − ǫ /
4. It is thenclear that ( ∂ t ˚ A ) ˚ A − is conjugated by a matrix in SU(1 ,
1) to a matrix of the form (cid:18) u − u (cid:19) with u <
0. (the sign of u cannot be changed to be positive because theconjugacy is in SU(1 , m + for m + ( σ + it, x ), m − for m − ( σ + it, x ), ˜ m + for m + ( σ + it, f ( x )), ˜ m − for m − ( σ + it, f ( x )), τ + for τ + ( σ + it, x ), τ − for τ − ( σ + it, x ), A for A σ + it ( x ), L for L ( σ + it ) and u for u ( x ).We now estimate the Lyapunov exponent using (2.21), (2.23) and the fact that A σ + it = (cid:18) e tu e − tu (cid:19) A σ + o ( t ) by evaluating the contraction coefficient q in thePoincar´e metric of D . A straightforward computation yields(2.52) q − = e − tu + o ( t ) − | ˜ m + | − e − tu | ˜ m + | = e tu + o ( t ) e − tu (1 − | ˜ m + | )1 − e − tu | ˜ m + | . Using that for r > ≤ s < e − r we have(2.53) ln (cid:18) e r (1 − s )1 − e r s (cid:19) ≥ r − s , we get(2.54) ln q − ≥ tu + o ( t ) − tu − | ˜ m + | = − tu | ˜ m + | − | ˜ m + | + o ( t ) . Since L = R X ln q − dµ , we get(2.55) L ≥ − t Z X u | ˜ m + | − | ˜ m + | dµ + o ( t ) . An analogous argument yields(2.56) L ≥ − t Z X u | ˜ m − | − − | ˜ m − | − dµ + o ( t ) . ONOTONIC COCYCLES 17
We conclude that(2.57) lim inf t → Lt + 12 Z X u (cid:18) | ˜ m + | − | ˜ m + | + 1 + | ˜ m − | − − | ˜ m − | − (cid:19) dµ ≥ . Since u <
0, this gives the first item and the first part of the second item.Differentiating(2.58) ˚ A − · (cid:18) ˜ m + (cid:19) = 1 τ + (cid:18) m + (cid:19) with respect to t , and applying − ˚ A to both sides, we get(2.59) ( ∂ t ˚ A ) ˚ A − · (cid:18) ˜ m + (cid:19) − ∂ t ˜ m + (cid:18) (cid:19) = ∂ t τ + ( τ + ) ˚ A · (cid:18) m + (cid:19) − τ + ∂ t m + ˚ A · (cid:18) (cid:19) . Using that(2.60) (cid:18) (cid:19) = 1 m + − m − (cid:18)(cid:18) m + (cid:19) − (cid:18) m − (cid:19)(cid:19) = 1˜ m + − ˜ m − (cid:18)(cid:18) ˜ m + (cid:19) − (cid:18) ˜ m − (cid:19)(cid:19) , we get( ∂ t ˚ A ) ˚ A − · (cid:18) ˜ m + (cid:19) − ∂ t ˜ m + ˜ m + − ˜ m − (cid:18)(cid:18) ˜ m + (cid:19) − (cid:18) ˜ m − (cid:19)(cid:19) (2.61) = ∂ t τ + τ + (cid:18) ˜ m + (cid:19) − ∂ t m + m + − m − (cid:18)(cid:18) ˜ m + (cid:19) − τ − τ + (cid:18) ˜ m − (cid:19)(cid:19) . On the other hand, we can compute using (2.48)(2.62)( ∂ t ˚ A ) ˚ A − · (cid:18) ˜ m + (cid:19) = u (cid:18) ˜ m + + ˜ m − ˜ m + − ˜ m − (cid:18) ˜ m + (cid:19) − m + ˜ m + − ˜ m − (cid:18) ˜ m − (cid:19)(cid:19) + c + (cid:18) ˜ m + (cid:19) + c − (cid:18) ˜ m − (cid:19) , where c + ≡ c + ( σ + it, x ), c − ≡ c − ( σ + it, x ) satisfy(2.63) | c ± ( σ + it, x ) | ≤ K k C ( σ + it, x ) k (cid:18) − | ˜ m + | + 11 − | ˜ m − | − (cid:19) . for some constant K > (cid:18) ˜ m + (cid:19) and integrat-ing with respect to µ we get(2.64) Z X u ˜ m + + ˜ m − ˜ m + − ˜ m − dµ + Z X c + dµ = Z X ∂ t τ + τ + dµ. We can now consider the real part, which gives(2.65) Z X u | ˜ m + | | ˜ m − | − − (cid:12)(cid:12)(cid:12) ˜ m + ˜ m − − (cid:12)(cid:12)(cid:12) dµ + Z X ℜ c + dµ = ∂ t L. Using (2.63) we conclude(2.66) lim t → − Z X u | ˜ m + | | ˜ m − | − − (cid:12)(cid:12)(cid:12) ˜ m + ˜ m − − (cid:12)(cid:12)(cid:12) dµ + ∂ t L = 0 . Write(2.67) I = 12 1 + | ˜ m + | − | ˜ m + | + 12 1 + | ˜ m − | − − | ˜ m − | − + | ˜ m + | | ˜ m − | − − (cid:12)(cid:12)(cid:12) ˜ m + ˜ m − − (cid:12)(cid:12)(cid:12) . Using (2.57) and (2.66) we get(2.68) lim inf t → (cid:18) Lt − ∂ t L + Z X uIdµ (cid:19) ≥ . Notice that(2.69) I ≥ (cid:12)(cid:12)(cid:12)(cid:12) ˜ m + − m − (cid:12)(cid:12)(cid:12)(cid:12) ≥ , and we conclude(2.70) lim inf t → (cid:18) Lt − ∂ t L (cid:19) ≥ lim inf t → − Z X uIdµ ≥ . Since lim sup t → Lt < ∞ , we must have(2.71) lim inf t → (cid:18) Lt − ∂ t L (cid:19) = − lim sup t → (cid:18) t∂ t Lt (cid:19) ≤ , so lim inf t → − R X uIdµ = 0, and since − u is positive and bounded away from 0we have lim inf t → R X Idµ ( x ) = 0, which gives the second part of the second itemby (2.69). (cid:3) The following estimates will allow us to work in the higher order asymptoticallyholomorphic setting:
Lemma 2.7.
Let A z,s ∈ ∆ δ be a one-parameter family. Assume that s A z,s ( x ) is C r , ≤ r < ∞ and (2.72) k ∂ ks A z,s ( x ) k = O (1) , ≤ k ≤ r. Then (2.73) | ∂ ks m + s ( z, x ) | = O ( |ℑ ( z ) | − k +1 ) , ≤ k ≤ r. Moreover, if additionally s ∂ z A z,s ( x ) is C r − and we have the estimate (2.74) k ∂ ks ∂ z A z,s ( x ) k = o ( |ℑ ( z ) | η − k − ) , ≤ k ≤ r − , for some η ∈ R then (2.75) | ∂ ks ∂ z m + s ( z, x ) | = o ( |ℑ ( z ) | η − k − ) , ≤ k ≤ r − . Remark . Implicit in the statement of Lemma 2.7 is the existence of the deriva-tives taken in the left hand sides of (2.73) and (2.75). This is just a consequenceof usual normal hyperbolicity theory [HPS], but naturally the estimates depend onthe strength of the uniform hyperbolicity, and hence may degenerate as ℑ z → Proof.
Let F sz ( x, w ) = ˚ A z,s ( x ) · w , m sz ( x ) = m + s ( z, x ). Our estimates will come fromthe study of the hyperbolicity of F with respect to the variable w , as measured inthe Poincar´e metric. The way we exploit this hyperbolicity is contained in thefollowing. ONOTONIC COCYCLES 19
Proposition 2.8.
There exists
K > such that if ( s, z, x ) u sz ( x ) is any boundedfunction then (2.76) | u sz ( x ) | ≤ K |ℑ ( z ) | − sup y ∈ X | u sz ( f ( y )) − ( ∂ w F sz )( y, m sz ( y )) u sz ( y ) | . Proof.
For s and z fixed, let x satisfy(2.77) | u sz ( f ( x )) | − | m sz ( f ( x )) | = M = sup y ∈ X | u sz ( y ) | − | m sz ( y ) | (we assume that the suppremum is achieved to keep the argument transparent, thegeneral case is obtained by approximation). Then for every y ,(2.78) | u sz ( y ) | ≤ | u sz ( y ) | − | m sz ( y ) | ≤ M, so it is enough to estimate(2.79) M ≤ K |ℑ ( z ) | − | u sz ( f ( x )) − ( ∂ w F sz )( x, m sz ( x )) u sz ( x ) | . We have u sz ( f ( x )) − ( ∂ w F sz )( x, m sz ( x )) u sz ( x ) = (1 − | m sz ( f ( x )) | )(2.80) · (cid:18) u sz ( f ( x ))1 − | m sz ( f ( x )) | − ( ∂ w F sz )( m sz ( x )) 1 − | m sz ( x ) | − | m sz ( f ( x )) | u sz ( x )1 − | m sz ( x ) | (cid:19) . Noticing that by the Schwarz Lemma(2.81) (cid:12)(cid:12)(cid:12)(cid:12) ( ∂ w F sz )( x, m sz ( x )) 1 − | m sz ( x ) | − | m sz ( f ( x )) | (cid:12)(cid:12)(cid:12)(cid:12) < , we get | u sz ( f ( x )) − ( ∂ w F sz )( x, m sz ( x )) u sz ( x ) | ≥ M (1 − | m sz ( f ( x )) | )(2.82) · (cid:18) − | ( ∂ w F sz )( x, m sz ( x )) | − | m sz ( x ) | − | m sz ( f ( x )) | (cid:19) . The Schwarz Lemma hyperbolicity bound (cf. (2.26))(2.83) | ∂ w F sz ( x, m sz ( x )) | − | m sz ( x ) | − | m sz ( f ( x )) | ≤ e − ǫ ℑ ( z ) − e ǫ ℑ ( z ) | m sz ( f ( x )) | − | m sz ( f ( x )) | , for some constant ǫ >
0, gives(1 − | m sz ( f ( x )) | ) (cid:18) − | ( ∂ w F sz )( x, m sz ( x )) | − | m sz ( x ) | − | m sz ( f ( x )) | (cid:19) (2.84) ≥ − e − ǫ ℑ z + ( e ǫ ℑ z − | m sz ( f ( x )) | ≥ − e − ǫ ℑ ( z ) , which together with (2.82) implies (2.79). (cid:3) Let us complete the proof of lemma 2.7. Differentiating (taking ∂ ks )(2.85) m sz ( f ( x )) = F sz ( x, m sz ( x )) (as we are allowed to do, see Remark 2.4), we get ∂ ks m sz ( f ( x )) = ∂ w F sz ( x, m sz ( x )) · ∂ ks m sz ( x )(2.86) + X l ≥ , ≤ i ≤ ... ≤ i l
Remark . The estimates above are still valid if the parameter space is allowedto be multidimensional, or, more generally, a Banach manifold, but the notation ismore cumbersome.
Remark . As a particular case of the previous estimates (zero-dimensional pa-rameter space), if A ∈ ∆ + δ satisfies(2.96) k ∂ z A z ( x ) k = o ( |ℑ ( z ) | η − )then(2.97) | ∂ z m + ( z, x ) | = o ( |ℑ ( z ) | η − ) . Asymptotically holomorphic extensions.
In order to apply the estimatesobtained in the previous section to one-parameter families of SL(2 , R ) cocycles, weneed to consider appropriate asymptotic holomorphic extensions.For η ∈ [1 , ∞ ), a C η function defined in some neighborhood of R / Z satisfying(2.98) d k dt k ∂F ( σ + it ) = 0 , σ + it ∈ R , k = 0 , ..., [ η − η -asymptotically holomorphic.Let AH η be the set of η -asymptotically holomorphic functions defined on thewhole C / Z . it is easy to see that one can define (linear) sections Φ η of the restrictionoperator AH η → C η ( R / Z , C ). For instance, one can let(2.99) Φ η ( f )( σ + it ) = Z K ( x ) f ( σ + tx ) dx, where K : R → C is a C ∞ function with compact support satisfying(2.100) Z x k K ( x ) dx = i k , k = 0 , ..., [ η + 1] . We can also define η -asymptotically SL(2 , C )-valued functions by requiring eachcoefficient to be asymptotically holomorphic. In order to obtain asymptoticallyholomorphic extensions of a matrix valued function A = (cid:18) a bc d (cid:19) ∈ C η ( R / Z , SL(2 , R )),it is enough to consider(2.101) Φ η ( A ) = (Φ η ( a )Φ η ( d ) − Φ η ( b )Φ η ( c )) − / (cid:18) Φ η ( a ) Φ η ( b )Φ η ( c ) Φ η ( d ) (cid:19) , which is a well defined function Ω δ → SL(2 , R ), where δ only depends on the C -norm of A . Lemma 2.9.
Let A θ ( · ) ∈ C ( X, SL(2 , R )) , θ ∈ R / Z , be monotonic and C η in θ . Then there exists δ > and an extension A ∈ ∆ δ which is η -asymotitucallyholomorphic in θ .Proof. Consider the η -asymptoticaly holomorphic extension given above defined onsome Ω δ . We just need to check that A ( σ + it, x ) ∈ int Υ for t > (cid:3) The following result will illustrate the use of higher order asymptotically holo-morphic extensions:
Lemma 2.10. If A z ∈ ∆ δ is ǫ -asymptotically holomorphic, then L ( A σ + it ) is acontinuous function of ≤ t < δ for almost every σ ∈ R / Z . The proof will involve the following decomposition technique which will play arole in several other arguments. Given a function u : Ω + δ → C / Z which is continuouswith derivatives in L and satisfies(2.102) | ∂u ( z ) | = O ( |ℑ ( z ) | ǫ − ) , for some ǫ >
0, let us write a canonical decomposition u = u h + u c where u h :Ω + δ → C / Z is holomorphic and u c : C / Z → C is a real-symmetric continuousfunction given by the Cauchy transform(2.103) u c ( z ) = lim t →∞ − π Z [ − t,t ] × [ − δ,δ ] φ ( w ) z − w dxdy, where φ ( z ) = ∂u ( z ) if 0 < ℑ ( z ) < δ and φ ( z ) = ∂u ( z ) for 0 < −ℑ ( z ) < δ .Notice that if(2.104) | ∂u ( z ) | = O ( |ℑ ( z ) | k + ǫ ) , then u c ( z ) is complex differentiable at each z ∈ R / Z , and u c : R / Z → R is C k +1 . Proof of Lemma 2.10.
By Lemma 2.7, k ∂ z m + ( z, x ) k = O ( |ℑ z | ǫ − ) in Ω + . Thus wecan write a decomposition m + = m + h + m + c where z m + c ( z, x ) is continuous upto R / Z (uniformly in x ), and m + h is of course uniformly bounded for z near R / Z .Thus m + h admits non-tangential limits as ℑ z → m + also does. ByFubini’s Theorem, for almost every σ , the right hand side of the expression(2.105) L ( A σ + it ) = Z X ln | τ A σ + it ( x ) ( m + ( σ + it, x )) | dµ ( x ) , originally defined for 0 < t < δ , makes sense up to t = 0 and defines a continuousfunction of 0 ≤ t < δ . Thus we just need to show that we can identify the righthand side (when it makes sense) with L ( A θ ) for t = 0. Indeed, if θ is such thatthe non-tangential limits m + ( θ, x ) exist, then they provide an invariant section forthe cocycle A θ . Assuming ergodicity, by the Oseledets Theorem the left hand sidemust be ± L ( A θ ), and (since it is non-negative (by continuity), it must be L ( A θ ).The general case reduces to this one by ergodic decomposition. (cid:3) Derivative bound.
Another simple application of the asymptotically holo-morphic technique is a generalization of Theorem 2.5.
Theorem 2.11.
Let A θ ∈ C ( X, SL(2 , R )) , θ ∈ R / Z , be C ǫ and monotonicdecreasing in θ . For almost every θ ∈ R / Z , if L ( A θ ) = 0 then (2.106) − ddθ ρ ( θ ) ≥ ǫ π > , where ǫ is the monotonicity constant of θ A θ .Proof. For δ > A z ∈ ∆ δ , some fixed 2 + ǫ -asymptoticallyholomorphic extension of A θ , thus in particular(2.107) | ∂ z A z ( x ) | = O ( |ℑ ( z ) | ǫ ) . ONOTONIC COCYCLES 23
Notice that estimate (2.27) obtained in section 2.3 in the analytic case is still truesince the asymptotically holomorphic extension is complex differentiable on R / Z .Let us show that our hypothesis imply that for almost every σ ∈ R ,(2.108) ∂ σ ρ ( σ ) = lim t → L ( σ + it ) − lim t → L ( σ + it ) t , since the result then follows as in Theorem 2.5.We have(2.109) | ∂ z m + ( z, x ) | = O ( |ℑ ( z ) | ǫ ) , which implies from equation (2.20)(2.110) | ∂ z ζ ( z ) | = O ( |ℑ ( z ) | ǫ )as well, when 0 < ℑ z < δ . Like in Theorem 2.5 we get using the fact that ζ asymptotically satisfies Cauchy-Riemann equations(2.111) ℑ ( ζ ( σ + it )) = ℑ ( ζ ( σ + i + )) + Z t + ∂ σ ℜ ζ ( σ + is ) ds + o ( | t | η ) . Notice that by Lemma 2.10, ℑ ζ ( σ ) = ℑ ζ ( σ + i + ) for almost every σ ∈ R / Z .From equation (2.110), decomposing ζ = ζ h + ζ c , ζ c ( z ) is complex differentiable at z ∈ R / Z and ζ c : R / Z → R is C . Since ℑ ζ > σ ρ ( σ ) = lim t → ℜ ζ ( σ + it )is monotonic, this is enough to conclude that (2.108) holds for almost every σ . (cid:3) For further use, let us remark that an argument analogous to the proof of The-orem 2.11 also gives:
Proposition 2.12.
Let A ∈ ∆ δ satisfy (2.112) k ∂ z A z k = O ( |ℑ ( z ) | ǫ ) . Then, for every σ ∈ R / Z , if (2.113) lim sup σ → σ | ρ A σ − ρ A σ || σ − σ | < ∞ then (2.114) lim sup t → | L ( A σ + it ) − L ( A σ ) || t | < ∞ . Proof of Theorem 2.2.
It is enough to consider the case of finite differen-tiability, since we have already proved the analytic case in section 2.3.Let A z,s ∈ ∆ δ be an asymptotically holomorphic extension of A θ,s satisfying(2.115) k ∂ ks A z,s ( x ) k = O (1) , ≤ k ≤ r, (2.116) k ∂ ks ∂ z A z,s ( x ) k = O ( |ℑ ( z ) | r − k + ǫ ) , ≤ k ≤ r. Define U ( t, s ) by formula (2.28). It still satifies (2.29) for t = 0.We must show that s U (0 , s ) is C k . By Lemma 2.10, t U ( t, s ) is continuousup to t = 0, for each s . For t = 0, the functions s U ( t, s ) are in C k , so we justneed to show that as t →
0, those functions converge uniformly in C k if 0 ≤ k ≤ r .We have the estimate(2.117) | ∂ ks ∂ z ζ s ( z ) | = O ( |ℑ ( z ) | r − k − ǫ ) , ≤ k ≤ r − . Fix 0 < δ ′ < δ ′′ < δ . Use Stoke’s Theorem to integrate ζ s ( z ) dz on [0 , × [ δ ′ , δ ′′ ],and take the imaginary part to get12 π U ( δ ′ , s ) = 12 π U ( δ ′′ , s ) + Z X ×{ δ ′ < ℑ z<δ ′′ } ∂ z ζ s ( z ) dz ∧ dz (2.118) + Z X Z δ ′′ δ ′ ℜ ζ s (1 + it ) − ℜ ζ s ( it ) dtdµ ( x )The first term on the right hand side is a fixed C k function of s . The second is alsoin C k and converges in C k as δ ′ → ℜ ζ s (1 + it ) − ℜ ζ s ( it )is independent of s and t : it is the integral over x of the topological degree of A z,s as z runs once around Ω + δ . Thus the third term is a linear function δ ′′ − δ ′ (and independent of s ). This shows that U ( δ ′ , s ) converges uniformly in C k , asdesired. (cid:3) L -estimates. We continue with some crucial L -estimates, which play animportant role in renormalization theory, obtaining a somewhat more precise formof Theorem 1.7, Theorem 2.14.First, let us recall the basic connection between estimates for D -valued invariantsections and the existence of conjugacy to rotations. Lemma 2.13.
Let A : X → SL(2 , R ) be measurable. The following are equivalent: (1) There exists a measurable B : X → SL(2 , R ) such that R X k B ( x ) k dµ ( x ) < ∞ and B ( f ( x )) A ( x ) B ( x ) − ∈ SO(2 , R ) for almost every x , (2) There exists a measurable m : X → D such that R X −| m ( x ) | dµ ( x ) < ∞ and ˚ A ( x ) · m ( x ) = m ( f ( x )) for almost every x .Proof. Let B ( x ) be such that ˚ B ( x ) = −| m ( x ) | ) / (cid:18) − m ( x ) − m ( x ) 1 (cid:19) . (cid:3) If A ∈ C ( R / Z , SL(2 , R )) satisfies the equivalent conditions of the previouslemma, we will say that the corresponding cocycle is L -conjugate to a cocycleof rotations.Our aim in this section is to prove the following. Theorem 2.14.
Let A θ ∈ C ( X, SL(2 , R )) , θ ∈ R / Z , be C ǫ and monotonic in θ . For every θ ∈ R / Z , if (2.119) lim sup θ ′ → θ | ρ A θ ′ − ρ A θ || θ ′ − θ | < ∞ (in particular if θ ρ A θ is Lipschitz) and L ( A θ ) = 0 then A θ is L -conjugate to acocycle of rotations. We will need a simple compactness result:
Proposition 2.15.
Let A k ∈ C ( X, Υ) be a sequence converging to A . Assumethere exists measurable functions m k : X → D satisfying ˚ A k ( x ) · m k ( x ) = m k ( f ( x )) ,such that lim inf R X −| m k ( x ) | dµ ( x ) < ∞ . Then there exists a measurable m : X → D such that ˚ A ( x ) · m ( x ) = m ( f ( x )) and R X −| m ( x ) | dµ ( x ) < ∞ . More generally, we can also let the dynamics vary, thus considering a sequence of µ -preservinghomeomorphisms f k : X → X converging to f . ONOTONIC COCYCLES 25
The proof uses the notion of conformal barycenter [DE], and we leave it for theAppendix A.
Corollary 2.16.
Let A ∈ ∆ δ . If σ ∈ R / Z satisfies (2.120) lim inf t → L ( σ + it ) t < ∞ then A σ is L -conjugate to a cocycle of rotations.Proof. The corollary follows from the previous Proposition 2.15 and Lemma 2.6. (cid:3)
Proof of Theorem 2.14.
It is enough to apply Proposition 2.12 and the corollaryabove. (cid:3)
Proof of Theorem 1.7.
Since θ ρ A θ is monotonic (2.119) holds for almost every θ . We can thus apply Theorem 2.14. (cid:3) Remark . If one is only concerned with a result valid for almost every θ (as in thecase of Theorem 1.7), one can bypass the use of the conformal barycenter argument.Indeed, the most usual argument in such situations is to apply the Lemma of Fatouto guarantee convergence of m + ( σ + it, x ) as t →
0+ for almost every x , and thenapply Fubini’s Lemma to obtain a set of σ of full Lebesgue measure for whichlim t → m + ( σ + it, x ) exists for almost every x .2.9. Proof of Theorem 2.1.
We will consider only the finitely differentiable case,the other cases following basically the same argument. We shall assume that J = R / Z for simplicity. Consider an asymptotically holomorphic extension of A θ satisfying(2.121) k ∂ z A z k = O ( |ℑ ( z ) | r + ǫ ) . Then we have(2.122) k ∂ z m + ( z, x ) k = O ( |ℑ ( z ) | r − ǫ ) , ℑ ( z ) > k ∂ z m − ( z, x ) k = O ( |ℑ ( z ) | r − ǫ ) , ℑ ( z ) < . Let(2.124) φ ( z, x ) = ∂ z m ± ( z, x ) , ±ℑ ( z ) > , and let u : C / Z × X → C be given by(2.125) u ( z, x ) = lim t →∞ − π Z [ − t,t ] × [ − δ,δ ] φ ( w, x ) z − w dxdy. A compactness argument shows that u ( z, x ) is continuous on both variables. More-over, R / Z ∋ y u ( y, x ) is C r (uniformly in x ). Let(2.126) m ( z, x ) = m ± ( z, x ) , z ∈ Ω ± δ . Then lim t → m ( σ + it, x ) exists for almost every σ and almost every x by Lemma2.6. Thus for almost every x ∈ X , z m ( z, x ) − u ( z, x ) extends to a holomorphicfunction defined on Ω δ . A compactness argument shows that this holds indeedfor all x ∈ X , and that the function Ω δ × R / Z ∋ ( z, x ) m ( z, x ) − u ( z, x ) iscontinuous (as in the classical De Concini-Johnson argument [CJ]). It also followsthat R / Z ∋ y m ( y, x ) is C r (uniformly on x ). To conclude, it is enough to showthat m ( y, x ) takes values on D . Let us assume first that f : X → X is minimal. If y is such that m ( y, x ) ∈ ∂ D for some x ∈ X , then for every x ∈ X we also have m ( y, x ) ∈ ∂ D (by invariance).However, since L ( A y ) = 0 for every y , ρ A y is C (by Schwarz Reflection and r ≥ y we have(2.127) lim sup t → Z R / Z − | m + ( y + it, x ) | dµ ( x ) < ∞ , so by continuity m ( y, x ) ∈ D for almost every x .Let us now consider the general case. Notice that if µ ′ is any ergodic invariantprobability measure, the Lyapunov exponent L ′ ( A y ) with respect to µ ′ is still 0 forevery y ∈ R / Z . Indeed,(2.128) L ′ ( A z ) = ± Z ln | τ A z ( m ( z, x )) | dµ ′ ( x ) ≥ , z ∈ Ω ± δ , and since m is continuous, we have R ln | τ A y ( m ( y, x )) | dµ ′ ( x ) = 0. On the otherhand, since m is an invariant section, if L ′ ( A y ) = 0 then R ln | τ A y ( m ( y, x )) | dµ ′ ( x )coincides with L ′ ( A y ) up to sign, as desired.Since any minimal set supports an ergodic invariant measure, we can apply theprevious argument to conclude that m ( y, x ) ∈ D whenever x belongs to a minimalset. To conclude, notice that for each y , the set of all x such that m ( y, x ) ∈ ∂ D iscompact and invariant, so if it is non-empty it must contain a minimal set. (cid:3) Monotonic cocycles
We now turn to the study of quasiperiodic cocycles presenting monotonicity inphase space. In this section, d ≥ f : R d / Z d → R d / Z d is a translation f ( x ) = x + α , where α is assumed to be fixed except whenotherwise noted. The underlying probability measure will be Lebesgue measure.Given w ∈ R d , we shall say that A ∈ C ( R d / Z d , SL(2 , R )) is w - monotonic if A wθ ( x ) = A w ( x + θw ) is monotonic decreasing.We say that A is monotonic if it is w -monotonic for some w . If A is C then the set of w such that A is w -monotonic is an open convex cone, hence ifnon-empty it contains primitive vectors of Z d , and up to linear automorphism of R d / Z d it contains (1 , , ..., x , ..., x d ), x A ( x , x , ..., x d ) is a map R / Z → SL(2 , R ) with negativetopological degree. Thus monotonic cocycles are never homotopic to a constant.Indeed the phenomena we will uncover for monotonic cocycles collide often with theintuition developed for Schr¨odinger cocycles, which are homotopic to a constant.A straightforward application of our previous results yields: Theorem 3.1.
Let A ∈ C r ( R d / Z d , SL(2 , R )) , r = ω, ∞ be monotonic. If L ( A ) = 0 then A is C r -conjugate to a cocycle of rotations.Proof. If A is w -monotonic, then the family A wθ ( x ) is obviously C r in θ and satisfies L ( A wθ ) = L ( A ) for every θ . If L ( A ) = 0, the proof of Theorem 2.1 gives m wθ ∈ C ( R d / Z d , D ), C r in θ , such that ˚ A wθ ( x ) · m wθ ( x ) = m wθ ( x + α ).We claim that m wθ ( x ) = m w ( x + θw ), where m w = m w . Recall that m wθ ( x )is obtained, almost everywhere, as a non-tangential limit lim ǫ → m wθ + ǫi ( x ) where m wθ + ǫi is the unique D -valued invariant section of an asymptotically holomorphic We apologize for this collision of definition with the notion of ǫ -monotonic cocycle. ONOTONIC COCYCLES 27 extension of A wθ . This asymptotically holomorphic extension can be chosen here tosatisfy A wθ + ǫi ( x ) = A wǫi ( x + θw ): this is in fact automatic if we follow the proceduredescribed in section 2.5 for the construction of the asymptotically holomorphicextension. In this case, we get m wθ + ǫi ( x ) = m wǫi ( x + θw ), and hence this equality issatisfied almost surely by the non-tangential limit. Since it is continuous, the claimfollows.Let us now consider another monotonicity vector w ′ . We claim that m w ′ = m w .Let us first assume ergodicity. In this case, if m w = m w ′ at some x ∈ R d / Z d then this must happen at every x , and in fact the hyperbolic distance between m w ′ ( x ) and m w ( x ) must be some constant c , by ergodicity of f and the fact thatthe projective action preserves the Poincar´e distance of the disk. Then we candefine B : R d / Z d → PSL(2 , R ) by ˚ B ( x ) · m w ( x ) = 0 and ˚ B ( x ) · m w ′ ( x ) ∈ ti ,where t > ti is c . It followsthat ˚ B ( f ( x )) ˚ A ( x ) ˚ B ( x ) − takes 0 to 0 and ti to ti , so B ( f ( x )) A ( x ) B ( x ) − is theidentity in PSL(2 , R ). This is clearly impossible since A is non-homotopic to aconstant. In the non-ergodic case, the open set U where m w ′ ( x ) = m w ( x ) is notnecessarily everything, but it is foliated by periodic subtori where the dynamicsis ergodic. The previous argument shows that there exists a continuous B : U → PSL(2 , R ) such that B ( f ( x )) A ( x ) B ( x ) − = id. By monotonicity, for each x ∈ U there exists C = C ( x ) > θ >
0, for any line l ∈ PR ,and for every k ∈ Z , there exists γ = γ ( x, k, θ, l ) with C − θ < γ < Cθ and B ( f k +1 ( x )) A ( f k ( x + θw )) B ( f k ( x )) − · l = R γ · l . This shows that for every ǫ > x ∈ U , if θ > l ∈ PR the sequence( B ( f n ( x )) A n ( x + θw ) B ( x ) − ) n ∈ Z is ǫ -dense in PR . But this is impossible sincefor small θ > B ( f n ( x )) A n ( x + θw ) B ( x ) − = B ( f n ( x )) B ( f n ( x + θw )) − B ( x + θw ) B ( x ) − is close to the identity for every n ∈ Z .We conclude that there exists a single m ∈ C ( R d / Z d , D ) which coincides withall m w ’s. Thus for each w in an open cone, θ m ( x + θw ) is C r . By Journ´e’sTheorem [Jo], it is C r as a function of x . (cid:3) Theorem 3.2.
Let A s ∈ C ( R d / Z d , SL(2 , R )) be a one-parameter family whichis C r in x and s , r = ω, ∞ . If A s is monotonic then s L ( A s ) is C r in aneighborhood of s .Proof. Let s and w be such that A s is w -monotonic. We may assume that w is aprimitive vector of Z d . Consider the two-parameter family A θ,s , θ ∈ R / Z given by A θ,s ( x ) = A s ( x + wθ ). By Theorem 2.2, the θ -average of L ( A θ,s ) is C r in s near s . But L ( A θ,s ) = L ( A s ) for every s , which gives the result. (cid:3) Remark . Much of our analysis generalizes to some other dynamical systems,including the usual skew-shift ( x, y ) ( x + α, y + x ). Consider a skew-product f : X × R d / Z d → X × R d / Z d , f ( x, y ) = ( φ ( x ) , y + ψ ( x )), and define a cocycle over f to be monotonic if there exists w ∈ R d such that y A ( x, y ) is w -monotonicfor every x ∈ X . Our arguments imply that if A is monotonic and C r , r = ω, ∞ ,with respect to the second coordinate then L ( A ) = 0 implies that A ( x, y ) is C conjugated to rotations (the conjugacy being C r in y ). We can also show that if A s ( x, y ) is a family of monotonic cocycles which is C r with respect to ( s, y ) then s L ( A s ) is C r . Varying the frequency.
Let us briefly allow the dynamics to vary, in orderto obtain a result about the regularity of the Lyapunov exponent with respect tosuch more general perturbations.We first need a replacement for Lemma 2.7. Given a one-parameter family A z,s ∈ ∆ δ which is monotonic, and a C one-parameter family of frequencies α ( s ) ∈ R d ,define invariant sections m + s ( z, x ) ∈ D so that ˚ A z,s ( x ) · m + s ( z, x ) = m + s ( z, x + α ( s )). Lemma 3.3.
Assume that α ( s ) and ( s, x ) A z,s ( x ) are C r , ≤ r < ∞ and (3.1) k ∂ is ∂ jx A z,s ( x ) k = O (1) , ≤ k = i + j ≤ r. Then (3.2) | ∂ is ∂ jx m + s ( z, x ) | = O ( |ℑ ( z ) | − k − i +1) ) , ≤ k = i + j ≤ r. Moreover, if additionally s ∂A z,s ( x ) is C r − and we have the estimate (3.3) k ∂ is ∂ jx ∂ z A z,s ( x ) k = O ( |ℑ ( z ) | η − k − ) , ≤ k = i + j ≤ r − , for some η ∈ R then (3.4) | ∂ is ∂ jx ∂ z m + s ( z, x ) | = O ( |ℑ ( z ) | η − k − i − ) , ≤ k = i + j ≤ r − . Proof.
One can basically repeat the proof of Lemma 2.7, since the added complica-tions are not very serious. Write m sz ( x ) for m + s ( z, x ) and F sz ( x, w ) for ˚ A z,s ( x ) · w .Consider first the first estimate. Notice that when i = 0 things reduce to Lemma2.7. Let us assume by induction (first on k and then on i ) that we have the desiredbounds when i ′ + j ′ < k and also for i ′ + j ′ = k , 0 ≤ i ′ < i . Consider the derivative ∂ is ∂ jx of m sz ( x + α ( s )) = F sz ( x, m sz ( x )). The left hand side has a main term of theform ( ∂ is ∂ jx m sz )( x + α ( s )) and a lower order term of the form(3.5) i X l
0, 1 ≤ n ≤ l , P ln =1 i n = i ′ ≤ i , P ln =1 j n = j ′ ≤ j and i ′ + j ′ < k , and C are now constant multiples of ∂ i − i ′ s ∂ j − j ′ x ∂ lw F . The lower orderterm can be estimated by induction to be O ( |ℑ z | − (2 k + i − ). Rearranging we get(3.7) | ( ∂ is ∂ jx ) m sz ( f + α ( s )) − ∂ w F sz ( x, m sz ( x )) · ∂ is ∂ jx m sz ( x ) | = O ( |ℑ z | − (2 k + i − ) , and the first estimate follows from Proposition 2.8.Consider now the second estimate. Let us assume by induction that we have thedesired bounds when i ′ + j ′ < k and also for i ′ + j ′ = k , 0 ≤ i ′ < i . We consider the ONOTONIC COCYCLES 29 derivative ∂ is ∂ jx ∂ z of m sz ( x + α ( s )) = F sz ( x, m sz ( x )). The left hand side has a mainterm of the form ( ∂ is ∂ jx ∂ z m sz )( x + α ( s )) and a lower order term(3.8) i X l
0, 1 ≤ n ≤ l , P ln =1 i n = i ′ ≤ i , P ln =1 j n = j ′ ≤ j , and C arenow constant multiples of ∂ i − i ′ s ∂ j − j ′ x ∂ z ∂ lw F . Using the first estimate, we see thatit is O ( |ℑ z | η − k − i ) except when k = 0, where it is O ( |ℑ z | η − ). The second has theform(3.10) X D · ∂ i s ∂ j x ∂ z m sz ( x ) l Y n =1 ∂ i n s ∂ j n x m sz ( x ) , where the sum runs over all 0 ≤ l ≤ k , 0 ≤ i ≤ i , 0 ≤ j ≤ j , i + j < k , and( i , j ) ≤ ... ≤ ( i l , j l ) (lexicographic order) with i n , j n ≥
0, 1 ≤ n ≤ l , P ln =0 i n = i ′ ≤ i , P ln =0 j n = j ′ ≤ j , and D are constant multiples of ∂ i − i ′ s ∂ j − j ′ x ∂ lw F . Thisterm can be estimated using the first estimate and the induction hypothesis to be O ( |ℑ z | η − k − i − ). Rearranging we get(3.11) | ( ∂ is ∂ jx ∂ z ) m sz ( f + α ( s )) − ∂ w F sz ( x, m sz ( x )) · ∂ is ∂ jx ∂ z m sz ( x ) | = O ( |ℑ z | η − k − i − ) , and the second estimate follows from Proposition 2.8. (cid:3) Theorem 3.4.
Let A θ,s ∈ C ( R d / Z d , SL(2 , R )) be monotonic on θ and C ∞ on θ, s , and let s α ( s ) be C ∞ . Then the average with respect to θ of the Lyapunovexponent of A θ,s over x x + α ( s ) is a C ∞ function of s .Proof. Fixing k ≥
0, choose η large and let A z,s ∈ ∆ δ be an η -asymptoticallyholomorphic extension of A θ,s . As in the proof of Theorem 2.2, we define for every s and 0 < t < δ (3.12) U ( t, s ) = Z R / Z L ( A σ + it,z ) dσ. Then for 0 < t < δ the functions s U ( t, s ) are C k and as t → C k . For each fixed s , the limit is seen to be the θ -average of theLyapunov exponent of A θ,s : since s is fixed, we can just apply Lemma 2.10. (cid:3) Remark . Even if everything is analytic, we do not, in general, get analyticdependence when varying the frequency by this argument. Analytic dependenceshould not be expected, since when the frequency is complexified the domain of an-alyticity does not remain invariant by the dynamics. A special case where analyticdependence holds is when A θ,s = R θ A s , since the dynamics does not influences the θ -average of the Lyapunov exponent in this case. Theorem 3.5. If A s ∈ C ∞ ( R d / Z d , SL(2 , R )) is monotonic and C ∞ on x and s , and s α ( s ) is C ∞ , then the Lyapunov exponent of A s as a cocycle over x x + α ( s ) is a C ∞ function of s .Proof. Assume that A is w -monotonic with w a primitive vector of Z d and considerthe family A θ,s = A s ( x + θw ). The Lyapunov exponents of A θ,s and of A s (bothconsidered as cocycles over the same x x + α ( s )) are obviously equal for every θ . The result follows by the previous theorem. (cid:3) Low regularity considerations.
We return to the consideration of a fixedtranslation dynamics, but now focus on trying to obtain conclusions at low regu-larity.Consider some A ∈ C ( R d / Z d , SL(2 , R )) which is w -monotonic, and let A wθ ( x ) = A ( x + θw ). It follows directly from the definitions that ρ A wθ is an affine function of θ with negative slope deg w . In fact deg w can be explicitly given as a linear functionof w which only depends on topological data: deg w = h l, w i where l ∈ Z d is theunique integer such that A is homotopic to x R h l,x i . In particular, θ ρ A wθ is Lipschitz. More generally, we have the following result. Lemma 3.6.
Let us consider a one-parameter family A θ ∈ C ( R d / Z d , SL(2 , R )) .If for some θ , A θ is a w -monotonic cocycle, and (3.13) K = lim sup θ → θ | θ − θ | sup x k A θ ( x ) − A θ ( x ) k < ∞ then (3.14) lim sup θ → θ | θ − θ | | ρ A θ − ρ A θ | ≤ K ′ , where K ′ depends on K , the monotonicity constant of A wθ , k A θ k C and deg w .Proof. The hypothesis imply that for h close to 0 and z ∈ ∂ D , ˚ A θ + h ( x ) · z lies inthe shortest segment of ∂ D determined by ˚ A θ ( x − Chw ) · z and ˚ A θ ( x + Chw ) · z , forsome C >
0. This implies that ρ A θ h lies between ρ A θ ( · + Chw ) and ρ A θ ( ·− Chw ) , that is, in the segment [ ρ A θ + Ch deg w , ρ A θ − Ch deg w ], and the result follows. (cid:3) Thus, if A is monotonic then θ R − θ A is a monotonic family with Lipschitzrotation number. In low regularity, it may be preferable to work with this family,because it is always analytic in θ . As an application, we have the following resultwhich is a direct consequence of Theorem 2.14 (if we were to use only the family θ A wθ , we would need to assume further regularity). Theorem 3.7.
Let A ∈ C ( R / Z , SL(2 , R )) be monotonic. If L ( A ) = 0 then A is L -conjugate to a cocycle of rotations. To see this, define δ v,w ξ n = δ γ ξ n and δ v,w ζ = δ γ ζ where γ is any path homotopic to γ ( t, x ) = A ( x + v + tw ). Notice that ρ A wθ is given, up to additive constant, by δ ,θw ζ . Let us showthat δ v,w ζ = h l, w i . It is clear that δ v,w ξ n ( x, z , z ) = δ ,w ( x + v, z , z ), so that δ v,w ζ does notdepend on v . By Remark 2.2, δ v,w ζ + δ v + w,w ′ ζ = δ v,w + w ′ ζ . This shows that δ v,w ζ is a linearfunction of w . Moreover, for w ∈ Z d we have exactly δ v,w ξ n = h l, w i n , so that δ v,w ζ = h l, w i . Indeed we can construct a monotonic decreasing family ˜ A t ∈ C ( R d / Z d , SL(2 , R )), t ∈ [0 , A ( · ) = A θ ( · − Chw )), ˜ A ( · ) = A θ ( · + Chw ) and ˜ A / = A θ + h , and ˜ A t is close to A θ for every t (so that we remain in a region where there is a continuous determination of ρ ).Monotonicity of this family gives ρ ˜ A ≤ ρ ˜ A / ≤ ρ ˜ A . ONOTONIC COCYCLES 31
It also allows us to get continuity results in Lipschitz open sets of cocycles. Forthe family R − θ A , e πiθ e − πiρ ( θ ) − L ( θ ) , ℑ θ >
0, defines an univalent function from D \ { } to D \ { } (see [CJ]).This gives an harmonic conjugacy relation betweenthe Lyapunov exponent and the fibered rotation number. If the fibered rotationnumber turns out to be Lipschitz, then the Lyapunov exponent (as a function ofthe circle) has derivative in L and in fact ∂ θ L is a zero-average BMO function(since it is basically the Hilbert trasform of the derivative of the fibered rotationnumber, which is in L ∞ ). This argument also shows that the BMO norm of ∂ θ L can be bounded in terms of the Lipschitz constant of ρ . Theorem 3.8.
Let ǫ > be fixed. The Lyapunov exponent is a continuous functionof ǫ -monotonic cocycles (the frequency may be varied as well).Proof. Let A ( n ) → A be a sequence of ǫ -monotonic cocycles converging in C . Weallow A ( n ) to be regarded as cocycles over x x + α n and A over x x + α , aslong as α n → α . By the previous lemma, there exists C > θ R − θ A ( n ) is a C -Lipschitz function for all n , and by thediscussion above, θ L n ( θ ) = L ( R − θ A ( n ) ) is uniformly equicontinuous. Thus wemay assume L n → L ∞ in C ( R / Z , R ). By upper semicontinuity of the Lyapunovexponent, θ L ( R − θ A ) − L ∞ ( θ ) is a non-negative continuous function, which wemust show to be identically zero. By [AB], Z R / Z L ( R − θ A ) − L ( R − θ A ( n ) ) dθ (3.15) = Z R / Z ln k A ( x ) k + k A ( x ) k − − ln k A ( n ) ( x ) k + k A ( n ) ( x ) k − dx, so R R / Z L ( R − θ A ) − L ∞ ( θ ) dθ = lim n →∞ R R / Z L ( R − θ A ) − L ( R − θ A ( n ) ) dθ = 0. (cid:3) Minimality.
Here we are interested in considering the dynamics of cocyclesfrom the topological point of view. For this, one considers the cocycle given bysome A as a map, still denoted by ( f α , A ), from R d / Z d × ∂ D → R d / Z d × ∂ D (a d + 1-dimensional torus) given by ( x, w ) ( x + α, ˚ A ( x ) · w ), where ∂ D is identifiedwith PR in the usual way. Below we consider only the case where x x + α isergodic.It can be shown (see [KKHO]) that if A ∈ C ( R / Z , SL(2 , R )) is not homotopicto the identity then for every α ∈ R \ Q , ( f α , A ) is transitive, and an adaptedargument for the multidimensional case is given in Appendix B. Though proofs oftransitivity just involve simple topological arguments, the following question seemsmuch harder: Problem . Is ( f α , A ) minimal whenever A ∈ C ( R d / Z d , SL(2 , R )) is non-homotopicto the identity?Of course the same problem still makes sense under additional smoothness as-sumptions. In this section we will show that the complexification methods allowone to address the local case, at least if one assumes enough smoothness.Let us first discuss some known results on the minimal sets of non-uniformlyhyperbolic cocycles (we follow the presentation of Herman [H], but the results aredue to Johnson [J1]). If A ∈ C ( R / Z , SL(2 , R )) and L ( A ) > u, s : R d / Z d → ∂ D (the unstable and stable directions) such that ˚ A ( θ ) · u ( θ ) = u ( θ + α ) and ˚ A ( θ ) · s ( θ ) = s ( θ + α ) and for almost every θ , for every w ∈ C , if w = s ( θ ) then | ˚ A n ( θ ) · w − u ( θ + nα ) | → w = u ( θ ) then | ˚ A n ( θ − nα ) − · w − s ( θ − nα ) | → n → ∞ . It follows (from unique ergodicity of θ θ + α ) thatthere are exactly two ergodic invariant measures on R d / Z d × ∂ D , the push-forwardsof Lebesgue measure on R d / Z d by θ ( θ, u ( θ )) and θ ( θ, s ( θ )), which we denoteby µ u and µ s . Let us denote their (compact) support by K u and K s . It followsthat any minimal set for ( f α , A ) coincides with either K u or K s . Moreover, if K u = K s then A would necessarily be uniformly hyperbolic. So assuming A to benon-homotopic to the identity with L ( A ) > K u = K s is the uniqueminimal set of ( f α , A ). Theorem 3.9.
Let A ∈ C ǫ ( R d / Z d , SL(2 , R )) be monotonic. Then ( f α , A ) isminimal.Proof. If L ( A ) = 0 then ( f α , A ) is C conjugate to a cocycle of rotations, by theargument of the proof of Theorem 2.1. For a cocycle of rotations, transitivity obviously implies minimality, so the resultfollows from Proposition B.1.Let now L ( A ) >
0. We consider the analytic case, the smooth case being analo-gous. Up to coordinate change, we may assume that A is w -monotonic where w =(1 , , ..., m : Ω + δ × R d − / Z d − → D satisfy ˚ A ( z ) · m ( z ) = m ( z + α ). Then forevery x , ..., x n and for almost every x , m ( x , ..., x n ) = lim t → m ( x + ti, x , ..., x n )exists. Since L ( A ) >
0, for almost every x ∈ R d / Z d , m ( x ) ∈ ∂ D and m | R d / Z d co-incides with either the unstable or stable directions u , s defined above. We claim that for every open set of the form J × U ⊂ R d / Z d with J ⊂ R / Z ,and any interval J ′ ⊂ ∂ D , there exists a positive measure set of x ∈ J × U suchthat m ( x ) ∈ J ′ : by the previous discussion, the unique minimal set K u = K s of( f α , A ) must intersect J × U × J ′ , and since it is arbitrary we must have K u = K s = R d / Z d × ∂ D .Suppose by contradiction that the claim does not hold. Up to a change ofcoordinates, we may assume that − ∈ J ′ . Then for almost every y ∈ U , z (1 − m ( z, y ))(1 + m ( z, y )) is a holomorphic function on Ω + δ with positive real part,bounded near J and whose non-tangential limits are purely imaginary on J ; by theSchwarz Reflection Principle there is a holomorphic extension to Ω δ \ ( R / Z \ J ). Thus x m ( x , ..., x n ) is analytic on J for almost every ( x , ..., x n ) ∈ U . By invariancewe have x m ( x , ..., x n ) analytic on R / Z for almost every ( x , ..., x n ). Thetopological degree of x m ( x , ..., x n ) is an integer valued measurable functiondeg( x , x , ..., x n ) which does not depend on x . But deg( x + α ) = deg( x ) + deg,where deg is the topological degree of x ˚ A ( x , , ..., ·
1. Since A is w -monotonicwith w = (1 , , ..., <
0. But by Poincar´e recurrence deg( x + nα ) takes thesame value infinitely many times, for almost every x , so deg = 0, contradiction. (cid:3) Premonotonic cocycles.
As remarked in the introduction, the concept ofmonotonicity is not dynamically natural. The easiest way to extend the concept of Theorem 2.1 gives a C conjugacy under a slightly stronger C ǫ condition. Under C ǫ , itstill gives a continuous invariant section m : R d / Z d → D , which implies the C conjugacy, unlessthe invariant section is real, i.e., it lies in ∂ D . However, this last possibility is impossible here,since A is non-homotopic to a constant so it can not admit invariant continuous sections in ∂ D . It is easy to see that it actually coincides with u , but this will not play a role here. ONOTONIC COCYCLES 33 monotonicity is the following. We say that a cocycle A ∈ C ( R d / Z d , SL(2 , R )) is premonotonic if some iterate is C conjugate to a monotonic cocycle: there exist n ≥ B ∈ C ( R d / Z d , SL(2 , R )) such that B ( x + nα ) A n ( x ) B ( x ) − is monotonic.This happens if and only if some iterate of ( f α , A ) is real-analytic conjugate to amonotonic cocycle (any C -perturbation of B which is real analytic will do). Noticethat premonotonic cocycles are C -stable, and there is even stability with respectto perturbations of the frequency vector defining the dynamics in the basis.Cocycles of rotations over ergodic translations which are not homotopic to aconstant provide the simplest examples of premonotonic cocycles. Given a cocycle A ∈ C ( R d / Z d , SL(2 , R )), let [ A ] be the unique cocycle homotopic to A of the form[ A ]( x ) = R h l,x i with l = l A ∈ Z d . Notice that A is not homotopic to a constant ifand only if l = 0, and in this case [ A ] is l -monotonic. Lemma 3.10.
Let A ∈ C r ( R d / Z d , SL(2 , R )) , ≤ r ≤ ∞ or r = ω , and let x x + α be ergodic on R d / Z d . If ( f α , A ) is C r -conjugate to a cocycle of rotationsthen there exists a sequence B ( n ) ∈ C r ( R d / Z d , SL(2 , R )) such that A ( n ) → [ A ] inthe C r -topology, where A ( n ) ( x ) = B ( n ) ( x + α ) A ( x ) B ( n ) ( x ) − ∈ SO(2 , R ) .Proof. By definition, we may assume that A is itself a cocycle or rotations (thehomotopy class being clearly conjugacy invariant). Thus let A ( x ) = [ A ]( x ) R φ ( x ) ,where φ : R d / Z d → R is C r . Let c = R φ ( x ) dx . Since α is irrational, we canconsider a sequence l k ∈ Z d such that h α, l k i → c mod 1.Let us consider a sequence φ ( n ) : R d / Z d → R of trigonometric polynomialsconverging to φ in C r and with R φ ( n ) ( x ) dx = c . Since x x + α is an ergodictranslation, it is easy to define, using Fourier series, trigonometric polynomials ψ ( n ) : R d / Z d → R such that φ ( n ) ( x ) = − ψ ( n ) ( x + α ) + ψ ( n ) ( x ) + c . Letting C ( n ) ( x ) = R ψ ( n ) ( x ) , we see that C ( n ) ( x + α ) R φ ( x ) C ( n ) ( x ) − is C r close to R c .Consider now B ( n ) ( x ) = R −h l kn ,x i C ( n ) ( x ). If we choose k n → ∞ very slowly, wewill have B ( n ) ( x + α ) A ( x ) B ( n ) ( x ) − → [ A ]( x ) in C r , as desired. (cid:3) While premonotonicity is only a priori invariant under C -conjugacies, we have: Theorem 3.11.
Let A ∈ C ( R d / Z d , SL(2 , R )) be non homotopic to the identity,and let f α : x x + α be ergodic on R d / Z d . If ( f α , A ) is C -conjugate to a cocycleof rotations, then ( f α , A ) is premonotonic.Proof. Up to isometric automorphism of R d / Z d , we may assume that the firstcoordinate l of l = l A is positive.Denote by B : R d / Z d → SL(2 , R ) the C -map such that R ( · ) := B ( · + α ) A ( · ) B ( · ) − takes it values in the group of rotations. We will identify PR with R / Z so thatthe projective action of rotations corresponds to translations. Let F n : R d / Z d × PR → R d / Z d × PR be the projective action, F n ( x, y ) = ( x + nα, A n ( x ) · y ).For ( x, y ) ∈ R d / Z d × PR , let a n ( x, y ) = ∂ x ( A n ( x ) · y ), b n ( x, y ) = ∂ y ( A n ( x ) · y ), p ( x, y ) = ∂ y ( B ( x ) · y ), q n = a n p ◦ F n , q = q . We claim that b n = pp ◦ F n and(3.16) q n = n − X k =0 q ◦ F k . Indeed, from the definition of B and R , one can write for any x , B ( x + nα ) · ( A n ( x ) · y ) = R n ( x ) · ( B n ( x ) · y ) and thus taking derivatives with respect to y , p ( x + nα, A n ( x ) · y ) b n ( x, y ) = p ( x, y ) which is the relation b n = pp ◦ F n . For (3.16) we just write A n +1 ( x ) · y = A ( x + nα ) · ( A n ( x ) · y ) and take derivatives withrespect to x to get a n +1 = a ◦ F n + ( b ◦ F n ) a n ; since we have just seen that b ◦ F n = ( p ◦ F n ) / ( p ◦ F n +1 ) we have a n +1 ( p ◦ F n +1 ) = ( a p ◦ F ) ◦ F n + a n ( p ◦ F n )which obviously gives (3.16).Let e = (1 , ..., a n ( x, y ) > x, y ) is equivalent to − e -monotonicity of A n . Below we will prove that q n → ∞ , and hence a n → ∞ ,uniformly in ( x, y ), giving the premonotonicity of ( f α , A ).For x , x ∈ R d / Z d , y ∈ PR (3.17) d ǫn ( x , x, y ) = Z ǫ p ( x + nα, A n ( x + te ) · y ) a n ( x + te , y ) dt gives the oriented length of the path γ | [0 , ǫ ], where γ = γ n,x ,x,y : R → PR isgiven by γ ( t ) = B ( x + nα ) A n ( x + te ) · y (the oriented length can be defined asˆ γ ( ǫ ) − ˆ γ (0) where ˆ γ : R → R is a lift to the universal cover). Especially, for any y, y ′ ∈ PR we must have(3.18) − < d ǫn ( x , x, y ) − d ǫn ( x , x, y ′ ) < , since when y = y ′ we must have γ n,x ,x,y ( t ) = γ n,x ,x,y ′ ( t ) for every t ∈ R .Let H ( x, y ) = ( x, B ( x ) · y ) and let G = H ◦ F ◦ H − . Since G is topologicallyconjugate to F , and Lebesgue measure on ( x, y ) is invariant for G , H − ∗ Leb is aninvariant measure for F equivalent to Leb. Thus for Lebesgue almost every ( x, y ),(3.19) ˆ q ( x, y ) = lim 1 n n − X k =0 q ◦ F k ( x, y )exists. Since ˆ q is measurable, Lebesgue almost every ( x, y ) is a measurable conti-nuity point along the x direction. Especially, for almost every ( x, y ) we have(3.20) lim ǫ → ǫ lim n →∞ n d ǫn ( x, x, y ) = ˆ q ( x, y ) . Thus ˆ q ( x, y ) is almost surely independent of y , and since x x + α is ergodic,ˆ q ( x, y ) is almost surely independent of x and y .Notice that G commutes with shifts in the second coordinate T t ( x, y ) = ( x, y + t ).Thus any ergodic invariant measure µ for G gives rise to a one-parameter family ofergodic invariant measures µ t = ( T t ) ∗ µ . By unique ergodicity of x x + α , all thosemeasures project down to Lebesgue measure on x . It follows that R R / Z µ t dt = Leb.By uniqueness of the ergodic decomposition, it follows that all ergodic invariantmeasures are of the form µ t , for some t ∈ R (for any fixed µ ).Since µ t depends continuously on t ∈ R / Z (with respect to the weak- ∗ topology)and R R / Z µ t dt = Leb, the fact that ˆ q ◦ H − = lim n P n − k =0 q ◦ H − ◦ G k is almosteverywhere constant implies that R q ◦ H − dµ t is independent of t . Since q ◦ H − has constant average with respect to all ergodic invariant measures, the Birkhoffaverages of q ◦ H − converge uniformly to a constant limit. Thus q n n → R q ◦ H − d Leb uniformly.To conclude, we must show that R q ◦ H − d Leb >
0. If this is not the case, thenfor every n sufficiently large we will have a n n < . But this is impossible becauseof the identity n R a n ( t, x , ..., x n , y ) dt = l which is a positive integer. (cid:3) The definition of premonotonicity is such that results proved for monotonic cocy-cles extend easily to this larger setting. Let us comment in more detail on the results
ONOTONIC COCYCLES 35 stated in the introduction which involve premonotonicity (except for Theorem 1.5,which we discuss in the next section).Theorem 1.3 follows from Theorem 3.2 and Theorem 1.4 follows from Theo-rem 3.5 (as the Lyapunov exponent is well behaved when taking conjugacies anditerates).In order to derive Theorem 1.1 from Theorem 3.1, it is enough to notice that if a C r cocycle ( f α , A ) admits an iterate which is C r conjugate to a cocycle of rotations,then ( f α , A ) is itself C r -conjugate to a cocycle of rotations. The proof (if not the statement) of Corollary 1.2 also involves premonotonicity:it follows from Theorem 1.1 and Theorem 3.11.Theorem 1.6 follows from Theorem 3.9 (since minimality of any iterate impliesminimality).3.5.
Non-uniform hyperbolicity for typical premonotonic cocycles.
In thissection, we will only consider, for simplicity, the case of C r cocycles with r = ∞ or ω . Then the Lyapunov exponent is indeed a C r function of premonotoniccocycles (while we have only carried out the formal arguments for the dependenceof the Lyapunov exponent along one-parameter families, it is clear the estimates gothrough to the infinite dimensional parametrization). Since the Lyapunov exponent L takes non-negative values, we must have DL = 0 whenever L = 0. Here we aregoing to show that, in the case of premonotonic cocycle, if L = 0 then D L = 0.This implies that { L = 0 } is a subvariety of positive codimension in the space ofpremonotonic cocycles and completes the proof of Theorem 1.5.If B ∈ C r ( R d / Z d , SL(2 , R )), then the conjugacy operator A A ′ , A ′ ( x ) = B ( x + α ) A ( x ) B ( x ) − is a C r diffeomorphism in C r ( R d / Z d , SL(2 , R )). Since the Lyapunovexponent is clearly invariant by conjugacy, it suffices to check that any premonotoniccocycle A with L ( A ) = 0 is conjugate to some A ′ such that D L ( A ′ ) = 0. But C r premonotonic cocycles with zero Lyapunov exponent are C r conjugate to cocyclesof rotations by Theorem 1.1, and in fact, by Lemma 3.10, those may be chosenarbitrarily close to a cocycle of the form x [ A ]( x ) = R h l,x i with l = 0. Since theLyapunov exponent is C near [ A ], it suffices to show D L ([ A ]) = 0. We will in factgive a simple estimate implying the existence of cocycles near [ A ] with a quadraticlower bound on the Lyapunov exponent.For a matrix s ∈ sl(2 , R ), let s , s , s be such that s = (cid:18) s s + s s − s − s (cid:19) . Lemma 3.12.
Let l ∈ Z d \{ } , s ∈ C ( R / Z , sl(2 , R )) , and define A θ,t ∈ C ( R d / Z d , SL(2 , R )) , θ ∈ R / Z , t ∈ R by A θ,t ( x ) = R h l,x i e ts ( h l,x i− θ ) . Then (3.21) lim t → t Z R / Z L ( A θ,t ) dθ = Z R / Z s ( θ ) + s ( θ ) dθ. In particular, the limit is zero if and only if s takes values in so(2 , R ) . This is most easily seen by working with C r invariant sections (which arise from and giverise to a conjugacy to rotations in the usual way). If m ∈ C r ( R d / Z d , D ) satisfies ˚ A n ( x ) · m ( x ) = m ( x + nα ), let m j ( x ) = ˚ A j ( x − jα ) · m ( x − jα ). Then m j + n = m j and ˚ A ( x ) · m j ( x ) = m j +1 ( x ).For each x ∈ R d / Z d , let m ∗ ( x ) minimizes the sum of the squares of the hyperbolic distances (in D ) to ( m j ( x )) n − j =0 : this is a well defined C r function of x by strict convexity. Then ˚ A ( x ) · m ∗ ( x ) = m ∗ ( x + α ). Proof.
Let C t,θ ( x ) = R θ C t where C t ( x ) = R h l,x i e ts ( h l,x i ) . Notice that A θ,t ( x + θ l k l k ) = C t,θ ( x ). So L ( A θ,t ) = L ( C t,θ ). By [AB],(3.22) Z R / Z L ( C t,θ ) dθ = Z R d / Z d ln k C t ( x ) k + k C t ( x ) k − dx = Z R / Z ln k e ts ( θ ) k + k e ts ( θ ) k − dθ. On the other hand, a direct computation shows that(3.23) lim t → t Z R / Z ln k e ts ( θ ) k + k e ts ( θ ) k − dθ = Z R / Z s ( θ ) + s ( θ ) dθ. The result follows. (cid:3)
Choosing, say, s ( θ ) = cos 2 πθ , s = s = 0, we see that the family A θ,t is ananalytic family (on θ and t ) of analytic cocycles such that A ,t is constant equal to x R h l,x i . The previous lemma then implies that D L (in either setting, analyticor smooth) does not vanish on x R h l,x i , as desired.4. One-frequency cocycles: renormalization and rigidity
We continue our investigations of quasiperiodic cocycles, but now specify to thecase of one frequency. Though the number of frequencies is quite irrelevant in theanalysis of monotonic cocycles, in the one-frequency case we will be able to obtainglobal consequences from our local analysis, by means of renormalization, a toolthat is not as effective when several frequencies are involved.Below we will only consider cocycles over irrational rotations. To highlight thedependence on the base dynamics, through this section a cocycle will be specifiedby a pair ( α, A ) ∈ ( R \ Q ) × C ( R / Z , SL(2 , R )).After defining the renormalization operator, we are going to show that if ( α, A ) ∈ ( R \ Q ) × C ( R / Z , SL(2 , R )) is L -conjugate to rotations, then it admits a “renor-malization representative” ( α ′ , A ′ ) ∈ ( R \ Q ) × C ( R / Z , SL(2 , R )) (seen as a cocycleover some different irrational rotation) with A ′ C -close to x R θ +deg x for some θ (here deg is the topological degree of A ). Moreover, if A is C r , A ′ can be chosento be C r . The dynamics of A and A ′ can be related, in particular if L ( α, A ) = 0then L ( α ′ , A ′ ) = 0 and if ( α ′ , A ′ ) is C r conjugate to rotations then ( α, A ) is also C r -conjugate to rotations.Now, if A is not homotopic to a constant, deg = 0, so A ′ is monotonic. Thisleads to our main global rigidity result in the one-dimensional case, Theorem 1.8.Let us note that by our analysis of one-parameter families, Theorem 1.8 implies: Theorem 4.1.
Let ( α, A θ ) , α ∈ R \ Q , A ∈ C r ( R / Z , SL(2 , R )) , r = ∞ , ω , be a one-parameter family which is monotonic and C ǫ in θ . If the A θ are non-homotopicto a constant then for almost every θ , either L ( A θ ) > or A θ is C r conjugate torotations.Proof. By Theorem 1.7, for almost every θ with L ( α, A θ ) = 0, ( α, A ) is L conjugateto rotations. By Theorem 1.8, they must be actually C r conjugate to rotations. (cid:3) Renormalization.
In this section we recall some basic facts on renormaliza-tion. We refer to [AK] for the proofs and further details.Let ( α, A ) ∈ ((0 , \ Q ) × C r ( R / Z , SL(2 , R )) be a cocycle Let p n /q n be thecontinued fraction approximants of α and let β n = ( − n ( q n α − p n ), α n = β n /β n − .Thus α n = G n ( α ) where G ( α ) = { α − } = α − − [ α − ] is the Gauss map. ONOTONIC COCYCLES 37
Classically, the dynamical systems x x + α n can be interpreted as the sequencerenormalization of x x + α . We would like to produce, starting from ( α, A ), asequence A ( n ) ∈ C ( R / Z , SL(2 , R )) such that ( α n , A ( n ) ) can be interpreted as thesequence of renormalizations of ( α, A ). However, this can not be done canonically,and to define renormalization one must introduce commuting pairs .Fixing x ∗ ∈ R / Z , we associate to ( α, A ) a sequence of pairs ( A ( n, , A ( n, ) ∈ C ( R , SL(2 , R )), by(4.1) A ( n, ( x ) = A ( − n − q n − ( x ∗ + β n − x ) , (4.2) A ( n, ( x ) = A ( − n q n ( x ∗ + β n − x ) . We should regard A ( n, and A ( n, as defining cocycles over the dynamics on R given by x x + 1 and x x + α n . It is easy to see that A ( n, ( x + 1) A ( n, ( x ) = A ( n, ( x + α n ) A ( n, ( x ), which expresses the commutation of the cocycles. We call((1 , A ( n, ) , ( α n , A ( n, )) the n -th renormalization of ( α, A ) around x ∗ .The dynamics of A ( n, (and of A ( n, as well) is trivial, since all orbits goto infinity. In fact we can always define ([AK], Lemma 4.1) a (non-canonical) normalizing map associated to (1 , A ( n, ), that is, some B ( n ) ∈ C ( R , SL(2 , R ))such that B ( n ) ( x + 1) A ( n, ( x ) B ( n ) ( x ) − = id.Because of the commutation relation, it follows that if B ( n ) is a normalizing mapfor (1 , A ( n, ), then A ( n ) ( x ) = B ( n ) ( x + α n ) A ( n, ( x ) B ( n ) ( x ) − satisfies A ( n ) ( x +1) = A ( n ) ( x ). Thus A ( n ) can be seen as an element of C ( R / Z , SL(2 , R )), and ( α n , A ( n ) )is called a representative of the n -th renormalization of ( α, A ).Of course, choosing a different normalizing map ˜ B ( n ) leads to a possibly different˜ A ( n ) . But it is easy to see that C = ˜ B ( n ) ( B ( n ) ) − is 1-periodic, which implies that˜ A ( n ) ( x ) = C ( x + α n ) A ( n ) ( x ) C ( x ) − , expressing the fact that all renormalizationrepresentatives are conjugate, and in fact any element of the conjugacy class of A ( n ) arises as a renormalization representative.Now, if A is C r , 1 ≤ r ≤ ∞ or r = ω , the normalizing maps may be chosen tobe C r as well ([AK], Lemma 4.1). Hence we may restrict considerations to renor-malization representatives obtained by the use of a C r normalizing map, which wecall C r -renormalization representatives. Such C r -renormalization representativesare defined up to C r -conjugacy.The dynamics of ( α, A ) and of its renormalization representatives are of courseintimately related. For instance: Proposition 4.2.
If a C r -renormalization representative ( α n , A ( n ) ) is C r conju-gate to rotations, then ( α, A ) is C r conjugate to rotations.Proof. Let B ( n ) be a C r -normalizing map for (1 , A ( n, ) such that we have, for every x ∈ R , B ( n ) ( x + α n ) A ( n, B ( n ) ( x ) − ∈ SO(2 , R ), and let B ′ ( x ∗ + β n − x ) = B ( n ) ( x ).Note that A ( n, q n − ( x + q n ) A ( n, q n ( x ) = id so that B ( n ) ( x + q n + α n q n − ) B ( n ) ( x ) − ∈ SO(2 , R ) for every x ∈ R . Writing ˜ A (0) ( x ) = B ′ ( x + 1) B ′ ( x ) − and using thatthat β n − = q n + α n q n − , we see that ˜ A (0) ( x ) ∈ SO(2 , R ) for every x ∈ R . Ananalogous argument shows that ˜ A (1) ( x ) = B ′ ( x + α ) A ( x ) B ′ ( x ) − ∈ SO(2 , R ) forevery x ∈ R . As remarked in the beginning of the proof of Lemma 4.4 of [AK],a simpler version of Lemma 4.1 of [AK] shows the existence of an SO(2 , R )-valued C r -normalizing map ˜ B for (1 , ˜ A (0) ). Then B ( x ) = ˜ B ( x ) B ′ ( x ) is 1-periodic and B ( x + α ) A ( x ) B ( x ) − ∈ SO(2 , R ) for every x ∈ R . (cid:3) Convergence of renormalization.
A weak version of convergence of renor-malization can be stated as follows:
Theorem 4.3.
Let ( α, A ) be a C r cocycle, ≤ r ≤ ∞ or r = ω , over an irrationalrotation, and let deg be the topological degree of A . If ( α, A ) is L -conjugate to ro-tations then there exist a sequence of C r -renormalization representatives ( α n , A ( n ) ) and θ n ∈ R , such that R − θ n − ( − n deg x A ( n ) ( x ) → id in C r . Proof of Theorem 1.8.
If ( α, A ) is non-homotopic to a constant, then deg = 0.By Theorem 4.3, it admits a monotonic C r -renormalization representative, whichis C r -conjugate to rotations by Theorem 2.1. By Proposition 4.2, ( α, A ) is C r -conjugate to rotations as well. (cid:3) We call the convergence given by Theorem 4.3 weak because it does not sayanything about the normalizing map leading to the “nice” renormalization repre-sentative. The strong form of convergence is the following:
Theorem 4.4.
Let ( α, A ) be a C r cocycle, ≤ r ≤ ∞ or r = ω , over an irrationalrotation. If ( α, A ) is L -conjugate to rotations then for almost every x ∗ ∈ R / Z there exists B ( x ∗ ) ∈ SL(2 , R ) , and a sequence of affine functions with boundedlinear coefficients φ ( n, , φ ( n, : R → R such that (4.3) R − φ ( n, ( x ) B ( x ∗ ) A ( n, ( x ) B − ( x ∗ ) → id and (4.4) R − φ ( n, ( x ) B ( x ∗ ) A ( n, ( x ) B − ( x ∗ ) → id , in C r . In [AK] is is shown that if A is C r , then there exists x ∗ and B ( x ∗ ) ∈ SL(2 , R )such that B ( x ∗ ) A ( n,i ) ( x ) B − ( x ∗ ), i = 0 ,
1, approaches SO(2 , R )-valued functionsin the C r topology for r = ∞ , ω , or C r − if 1 ≤ r < ∞ . While computations in[AK] are “local”, the more precise version obtained, based on the recent work [A1],takes into account global aspects of the (asymptotically) holomorphic extensions ofmatrix products. This complex variables proof turns out to be simpler and morepowerful than our original real variables approach, which shows that if A is C thenthe oscillations of the derivative of B ( x ∗ ) A ( n,i ) ( x ) B − ( x ∗ ) become less pronouncedas n → ∞ (due to cancellations appearing through the Ergodic Theorem).We will prove Theorem 4.4 in the next section. For the moment, we will justrelate it to Theorem 4.3. Proof of Theorem 4.3.
Let B ( x ∗ ), φ ( n, = a n, x + b n, and φ ( n, = a n, x + b n, be as in Theorem4.4. Let n be large and let ˜ B ( x ) = R − ( a n, x − x + b n, x ) B ( x ∗ ). Then ˜ A ( x ) = ˜ B ( x +1) A ( n, ( x ) ˜ B ( x ) − is C r -close to the identity and ˜ B ( x + α n ) A ( n, ( x ) ˜ B ( x ) − is C r -close to R ψ ( n ) ( x ) , where ψ ( n ) ( x ) = ( − a n, α n + a n, ) x + ( b n, − α n b n, − a n, α n − α n ).By Lemma 4.1 of [AK], there exists C ∈ C r ( R , SL(2 , R )) which is C r -close to theidentity such that C ( x + 1) ˜ A ( x ) C ( x ) − = id. Set B ( n ) = C ˜ B . Then B ( n ) is a C r normalizing map for A ( n, and A ( n ) ( x ) = B ( n ) ( x + α n ) A ( n, B ( n ) ( x ) − is a C r renormalization representative close to R ψ n ( x ) . Since ( α n , A ( n ) ) is a renormaliza-tion representative of ( α, A ), the topological degree of A ( n ) is ( − n deg (compute In fact, as the proof will show the convergence holds uniformly on any compact subsets oflarger and larger complex strips.
ONOTONIC COCYCLES 39 directly the degree of an n -th renormalization representative of ( α, R deg x ), whichwill be automatically homotopic to ( α, A ( n ) ), or see [AK], Appendix A). Thus thelinear coefficient of ψ n must be close to ( − n deg and A ( n ) ( x ) must be C r -close R θ n +( − n deg x for some θ n ∈ R . (cid:3) Proof of Theorem 4.4.
The complex variables proof given below followsbasically [A1], which uses “renormalization in parameter space” as an approach tothe local distribution of zeros of orthogonal polynomials, originally treated in [ALS]with a different technique. We translate the argument of [A1] to the usual renor-malization operator in the analytic case, and then use asymptotically holomorphicextensions to address the non-analytic case.We consider first the analytic case. Let B : R / Z → SL(2 , R ) be a measurablemap with k B k ∈ L and for any x ∈ R / Z , B ( x + α ) A ( x ) B ( x ) − ∈ SO(2 , R ). Let S ( x ) = sup n ≥ n P n − k =0 k B ( x + kα ) k , which is finite almost everywhere by theMaximal Ergodic Theorem. Assume that A has a holomorphic extension which isLipschitz in Ω δ . Lemma 4.5.
There exists
C > such that if x ∈ R / Z then for every k ≥ and z ∈ Ω δ we have (4.5) k A k ( x ) − ( A k ( x ) − A k ( x )) k ≤ e C k B ( x k S ( x ) k | x − x | − . Proof.
The proof is the same as the proof of Lemma 3.1 of [AK] (there only thecase x ∈ R / Z is considered, but the proof works equally well for the complexextension). (cid:3) Suppose now that x ∗ is a measurable continuity point of S and B (this means that x ∗ is a Lebesgue density point of {| S ( x ) − S ( x ∗ ) | < ǫ } and of {k B ( x ) − B ( x ∗ ) k < ǫ } for every ǫ > k A ( − n q n ( x ) k ≤ inf x ′ − x ∗ ∈ [ − dqn , + dqn ] C ( x ∗ ) e C ( x ∗ ) q n | x − x ′ | , for every d >
0, as long as n > n ( d ). The argument is as in Lemma 3.3 of[AK]: if n is large, the measurable continuity hypothesis implies that for every x ′ ∈ [ x ∗ − dq n , x ∗ + dq n ] we can locate x with | x ′ − x | ≤ q n and such that B ( x ) , B ( x + β n ) are close to B ( x ∗ ) and S ( x ) , S ( x + β n ) are close to S ( x ∗ ), andthen apply Lemma 4.5 to estimate either k A q n ( x ) − ( A q n ( x ) − A q n ( x ) k (if n iseven), or k A − q n ( x )( A − q n ( x ) − − A − q n ( x ) − ) k (if n is odd), using also the bound k A ( − n q n ( x ) k ≤ k B ( x ) kk B ( x + β n − ) k .This estimate implies, since q n − < q n < β − n − ,(4.7) k A ( n,i ) ( x ) k ≤ inf x ∈ [ − d,d ] Ce C | x − x | , i = 0 , , x ∈ Ω δ/β n − , x ∈ [ − d, d ] . It follows that the sequences A ( n,i ) are precompact in C ω , and the limits are entirefunctions ˜ A with k ˜ A ( z ) k ≤ Ce C |ℑ z | . We now show that B ( x ∗ ) ˜ A ( x ) B ( x ∗ ) − mustbe of the form R ˜ φ ( x ) with ˜ φ affine with bounded linear coefficient.Indeed, Lemma 3.4 of [AK] shows that limits ˜ A of the A ( n,i ) satisfy B ( x ∗ ) ˜ A ( x ) B ( x ∗ ) − ∈ SO(2 , R ), x ∈ R . It follows that we can write ˜ A ( z ) = B ( x ∗ ) − R ˜ φ ( x ) B ( x ∗ ) for someentire function ˜ φ : C → C , satisfying the estimate |ℑ ˜ φ ( z ) | ≤ C + C |ℑ z | (since ℑ ˜ φ = 0 on the real axis). This implies that ˜ φ is affine with bounded linear coeffi-cient. We consider now the C r case, 1 ≤ r < ∞ , since it implies the C ∞ case. Consideran r -asymptotically holomorphic extension of A to some Ω δ , and let x ∗ be selectedas in the analytic case. The asymptotically holomorphic extension is in particularLipschitz in Ω δ , thus estimate (4.7) still holds. For 0 < ǫ ≤ δ , let us denote by k · k C r − ǫ the C r − norm of the restriction to Ω ǫ of a function defined on Ω δ . Lemma 4.6.
Suppose that x ∈ R / Z and k ≥ satisfy S ( x ) , k B ( x ) k , k B ( x + kα ) k ≤ C . Then there exists C > (depending on C and k A k C rδ , but not on x ),such that if z ∈ Ω ǫ with < ǫ ≤ δ then (4.8) max ≤ s ≤ r − k D s ∂ z A k ( z ) k ≤ Ck r e Ck | z − x | k ∂ z A k C r − ǫ ( D stands for the full derivative).Proof. The proof is the same as that of Lemma 3.2 of [AK] which estimates the realderivatives of matrix products: the consideration of the complex extension is againharmless, and the incorporation of a ∂ z in the estimates is straightforward. (cid:3) By the same measurable continuity argument given above, we obtain(4.9)max ≤ s ≤ r − k D s ∂ z A ( − n q n ( z ) k ≤ inf x − x ∗ ∈ [ − dqn , dqn ] Cq rn e Cq n | z − x | k ∂ z A k C r − ǫ , z ∈ Ω ǫ , which yields(4.10) max ≤ s ≤ r − k D s ∂ z A ( n,i ) ( z ) k ≤ inf x ∈ [ − d,d ] Ce C | z − x | k ∂ z A k C r − ǫ , z ∈ Ω ǫ/β n − , This implies that we can write A ( n,i ) = A ( n,i ) c + A ( n,i ) h where each term is definedin an increasing sequence of disks D n with ∪ D n = C , A ( n,i ) h are matrix valued (notnecessarily SL(2 , C )) holomorphic functions and form precompact sequences withlimits satisfying k ˜ A ( z ) k ≤ Ce C | z | , and A ( n,i ) c are C r matrix valued functions with C r norm going to 0. It follows that A ( n,i ) are precompact in C r and the limits are entirefunctions (necessarily SL(2 , C ) valued now) satisfying k ˜ A ( z ) k ≤ Ce C |ℑ z | . By thesame argument of the analytic case, the limits have the form B ( x ∗ ) − R ˜ φ ( z ) B ( x ∗ ),where ˜ φ is affine with bounded linear coefficient. Appendix A. Conformal barycenter
Let M be the set of probability measures on D , and for µ ∈ M , let Φ( µ ) = R D −| z | dµ ( z ). For w ∈ D , let Φ w ( µ ) = Φ( µ ′ ) where µ ′ is the pushforward of µ bysome Moebius transformation of D taking w to 0. Notice that if Φ( µ ) < ∞ thenΦ w ( µ ) < ∞ for every w . For every 1 ≤ K < ∞ , let M K = { µ ∈ M , Φ( µ ) ≤ K } ,and let M ∞ = ∪M K . Notice that M K is compact in the weak-* topology forevery K < ∞ .The next proposition can be proved using the conformal barycenter of Douady-Earle [DE]. The construction is sufficiently simple for us to give the details here. Proposition A.1.
There exists a Borelian function B : M → D , equivariant withrespect to M¨oebius transformations of D and such that Φ( δ B ( µ ) ) ≤ Φ( µ ) . ONOTONIC COCYCLES 41
Proof.
Following an idea of Yoccoz, let us define a pairing D × D → D by setting z ∗ w as the midpoint of the hyperbolic geodesic passing through z and w if z = w ,and z ∗ z = z . This pairing is continuous and equivariant, and we have(A.1) u s ( z, w ) ≡ Φ s (cid:18) δ z + δ w (cid:19) − Φ s ( δ z ∗ w ) ≥ , with equality if and only if z = w . Notice that(A.2) u s ( z, s ) = (2Φ s ( δ z ∗ s ) − s ( δ z ∗ s ) − ≥ Φ s ( δ z ∗ s ) − . Extend the pairing ∗ to M × M → M linearly. Thus(A.3) µ ∗ ν = Z D × D δ z ∗ w dµ ( z ) dν ( w ) . If µ, ν ∈ M ∞ then(A.4) u s ( µ, ν ) ≡ Φ s ( 12 ( µ + ν )) − Φ s ( µ ∗ ν ) = Z D × D u s ( z, w ) dµ ( z ) dν ( w ) ≥ , with equality if and only if µ = ν is a Dirac mass. Notice that u s : M×M → [0 , ∞ ]is lower semicontinuous, so if µ k → µ and u s ( µ k , µ k ) → µ is a Dirac mass.If µ k → δ s we have(A.5) lim sup k →∞ u s ( µ k , µ k ) ≥ lim sup k →∞ u s ( µ k , δ s ) ≥ lim sup k →∞ Z D Φ s ( δ z ∗ s ) − dµ k ( z ) , and in particular if additionally lim u s ( µ k , µ k ) = 0 then lim Φ s ( µ k ∗ δ s ) = 1.Given µ ∈ M , define µ ( k ) inductively by µ (0) = µ and µ k = µ ( k − ∗ µ ( k − . If µ ∈ M ∞ then µ ( k ) ∈ M ∞ and we have Φ( µ ( k +1) ) = Φ( µ ( k ) ) − u ( µ ( k ) , µ ( k ) ). Thus u s ( µ ( k ) , µ ( k ) ) →
0, and any limit of µ ( k ) (which exists by compactness) must be aDirac mass. Moreover, if µ ( n k ) → δ s then Φ s ( µ ( n k ) ∗ δ s ) →
1, so Φ s ( µ ( n ) ∗ δ s ) → δ s must be the unique limit of µ ( n ) . Now we can set B ( µ ) = s , whichis clearly Borelian. (cid:3) The estimates above allow us to obtain compactness result for invariant sectionsof cocycles. For instance, we have the following.
Proposition A.2.
Let f k : X → X be a sequence of homeomorphisms of X pre-serving a probability measure µ and converging uniformly to a homeomorphism f : X → X . Let A k ∈ C ( X, Υ) be a sequence converging to A ∈ C ( X, Υ) . As-sume there exists measurable m k : X → D satisfying A k ( x ) · m k ( x ) = m k ( f k ( x )) ,such that (A.6) H ≡ lim inf K,k →∞ Z X min { K, − | m k ( x ) | } dµ ( x ) < ∞ . Then there exists a measurable m : X → D such that A ( x ) · m ( x ) = m ( f ( x )) and R X −| m ( x ) | dµ ( x ) ≤ H .Proof. Let X K,k = { x ∈ X, −| m k ( x ) | < K } , and let ν K,k = R X K,k δ m k ( x ) dµ ( x ).Let ν be any limit of ν K,k along a sequence K i → ∞ , k i → ∞ attaining thelim inf in (A.6). Then ν is a probability measure which projects onto µ andsatisfies R X × D −| z | dν ( x, z ) ≤ H . Let ν x , x ∈ R / Z be a desintegration of ν : Although we do not need this fact, it is easy to see that B is continuous in each M K ,1 ≤ K < ∞ . R X × D φ ( x, z ) dν ( x, z ) = R X ( R D φ ( x, z ) dν x ( z )) dµ ( x ). Then ν f ( x ) is the pushforwardof ν x by w ˚ A ( x ) · w , and ν x ∈ M ∞ for µ -almost every x . Let m ( x ) = B ( ν x ). Then m ( f ( x )) = ˚ A ( x ) · m ( x ) and we have R −| m ( x ) | dµ ( x ) ≤ R R −| z | dν x ( z ) dµ ( x ) ≤ H . (cid:3) Appendix B. Transitivity of the projective action
We follow the notation of section 3.3. Our goal is to show the transitivity of theprojective action of multidimensional quasiperiodic cocycles which are not homo-topic to the identity. The one-dimensional case was considered in [KKHO], and infact the topological ideas that make the one-dimensional argument work are easilyimplemented in the multidimensional case as well. Let f α : x x + α be an ergodictranslation in R d / Z d . Proposition B.1.
Let A ∈ C ( R d / Z d , SL(2 , R )) be non-homotopic to the identity.If f α : x x + α be an ergodic translation in R d / Z d then ( f α , A ) is transitive on R d / Z d × ∂ D .Proof. Up to change of coordinate, we may assume that x A ( x , ..., x d ) haspositive degree deg ≥
1. To prove transitivity, it is enough to show that for anyopen set U ⊂ R d / Z d × ∂ D , the set ∪ k ≥ ( f α , A ) k ( U ) is dense in R d / Z d × ∂ D .We will actually show a stronger statement. Let Π : R d / Z d × ∂ D → R / Z ,Π : R d / Z d × ∂ D → R d − / Z d − and Π : R d / Z d × ∂ D → ∂ D be given byΠ ( x , ..., x d , z ) = x , Π ( x , ..., x d , z ) = ( x , ..., x d ) and Π ( x , ..., x d , z ) = z .Let 0 < ǫ < /
10. Let us say that a point x ∈ R d / Z d is ǫ -short if there exists asequence of paths γ n : [0 , → R d / Z d × ∂ D such that Π ◦ γ n ( t ) converges uniformlyto Π ( x ) + ǫt , Π ◦ γ n ( t ) converges uniformly to Π ( x ), and Π (( f α , A ) k ( γ n ( t ))) has(algebraic) length at most 2 π − /
10 for every k ≥
0. It is clear that the set of ǫ -short x is forward invariant and closed, so for each ǫ , either every point is ǫ -shortor no point is ǫ -short.If for every ǫ > ǫ -short, then for any ( x, z ) =( x , ..., x d , z ) and for every δ >
0, there exists k ≥ J δ ( x, z ) =[ x , x + δ ] ×{ ( x , ..., x d , z ) } , we have | Π ( f α , A ) k ( J δ ( x, z )) | > π − δ . It follows thatfor every δ , the closure of ∪ k ≥ ( f α , A ) k ( J δ ( x, z )) contains some circle { y } × ∂ D .Since this set is also forward invariant, it must contain also the circles { y + lα } × ∂ D for every l ≥
0, and hence, by minimality of x x + α , the whole R d / Z d × ∂ D .Since ( x, z ) and δ > ǫ > ǫ -short. We mayassume that ǫ = k for some k ≥
2. Let γ n,i , 1 ≤ i ≤ k , n ≥
1, be the sequencesof paths associated to ( i − k , , ..., γ n : R / Z → R d / Z d × ∂ D so that(1) ˜ γ n | [( i − /k, (3 i − / k ] is given by ˜ γ n ( t ) = γ n,i (3 kt − i + 3),(2) ˜ γ n | [(3 i − / k, (3 i − / k ] is such that Π ◦ ˜ γ n and Π ◦ ˜ γ n are constantand Π ◦ ˜ γ n is a homeomorphism,(3) the diameter of the image of ˜ γ n | [(3 i − / k, i/k ] converges to 0.One readily checks that these properties imply that for every l ≥
0, if n issufficiently large, then Π ◦ ( f α , A ) l ◦ ˜ γ n has topological degree 1, Π ◦ ( f α , A ) l ◦ ˜ γ n is homotopic to a constant and Π ◦ ( f α , A ) l ◦ ˜ γ n has topological degree deg l,n satisfying | deg l,n | ≤ k −
1. But deg l +1 ,n = deg l,n + deg ≥ deg l,n +1 for every l ONOTONIC COCYCLES 43 and n , since A is not homotopic to the identity. Thus for large n we have bothdeg k,n − deg ,n ≥ k and | deg k,n | , | deg ,n | ≤ k −
1, a contradiction. (cid:3)
References [Am] Amor, Sana Hadj H¨older continuity of the rotation number for quasi-periodic co-cyclesin SL(2 , R ∼ avila/).[AB] Avila, Artur; Bochi, Jairo A formula with some applications to the theory of Lyapunovexponents. Israel J. Math. 131 (2002), 125–137.[AFK] Avila, Artur; Fayad, Bassam; Krikorian, Rapha¨el A KAM scheme for SL(2 , R ) cocycleswith Liouvillean frequencies. Geometric and Functional Analysis 21 (2011), 1001-1019.[AJ] Avila, Artur; Jitomirskaya, Svetlana Almost localization and almost reducibility. Journalof the European Mathematical Society 12 (2010), 93-131.[AK] Avila, Artur; Krikorian, Rapha¨el Reducibility and non-uniform hyperbolicity for one-dimensional quasiperiodic Schr¨odinger cocycles. Ann. Math. 164 (2006), 911-940.[ALS] Avila, Artur; Last, Yoram; Simon, Barry. Bulk universality and clock spacing of zerosfor ergodic Jacobi matrices with ac spectrum. Analysis & PDE 3 (2010), 81-118.[BC] Benedicks, Michael; Carleson, Lennart The dynamics of the H´enon map. Ann. of Math.(2) 133 (1991), no. 1, 73–169.[Bj1] Bjerkl¨ov, K. Positive Lyapunov exponent and minimality for a class of one-dimensionalquasi-periodic Schr¨odinger equations, Ergodic Theory Dynam. Systems 25 (2005), no. 4,1015–1045.[Bj2] Bjerkl¨ov, K. Dynamics of the quasi-periodic Schr¨odinger cocycle at the lowest energy inthe spectrum, Comm. Math. Phys. 272 (2007), 397–442.[BjJ] Bjerkl¨ov, K.; Johnson, R. Minimal subsets of projective flows. Discrete Contin. Dyn.Syst. Ser. B 9 (2008), no. 3-4, 493–516.[Bo] Bochi, Jairo Genericity of zero Lyapunov exponents. Ergodic Theory Dynam. Systems22 (2002), no. 6, 1667–1696.[B] Bourgain, J. Positivity and continuity of the Lyapounov exponent for shifts on T d witharbitrary frequency vector and real analytic potential. J. Anal. Math. 96 (2005), 313–355.[BJ] Bourgain, J.; Jitomirskaya, S. Absolutely continuous spectrum for 1D quasiperiodicoperators. Invent. Math. 148 (2002), no. 3, 453–463.[CJ] De Concini, Corrado; Johnson, Russell A. The algebraic-geometric AKNS potentials.Ergodic Theory Dynam. Systems 7 (1987), no. 1, 1–24.[DeS] Deift, P.; Simon, B. Almost periodic Schr¨odinger operators, III. The absolutely contin-uous spectrum in one dimension, Commun. Math. Phys. 90 (1983), 389–411.[DS] Dinaburg, E. I.; Sinai, Ja. G. The one-dimensional Schr¨odinger equation with quasiperi-odic potential. Funkcional. Anal. i Prilozen. 9 (1975), no. 4, 8–21.[DE] Douady, Adrien; Earle, Clifford J. Conformally natural extension of homeomorphismsof the circle. Acta Math. 157 (1986), no. 1-2, 23–48.[E] Eliasson, L. H. Floquet solutions for the 1-dimensional quasi-periodic Schrdinger equa-tion. Comm. Math. Phys. 146 (1992), no. 3, 447–482.[GS] Goldstein, Michael; Schlag, Wilhelm H¨older continuity of the integrated density of statesfor quasi-periodic Schr¨odinger equations and averages of shifts of subharmonic functions.Ann. of Math. (2) 154 (2001), no. 1, 155–203.[H] Herman, Michael-R. Une m´ethode pour minorer les exposants de Lyapounov et quelquesexemples montrant le caract`ere local d’un th´eor`eme d’Arnold et de Moser sur le tore dedimension 2. Comment. Math. Helv. 58 (1983), no. 3, 453–502.[HPS] Hirsch, M.W.; Pugh, C.C.; Shub, M. Invariant manifolds. Lecture Notes in Mathematics,Vol. 583. Springer-Verlag, Berline-New York, 1977. ii+149 pp.[J1] Johnson, Russell A. Ergodic theory and linear differential equations. J. Differential Equa-tions 28 (1978), no. 1, 23–34.[J2] Johnson, R. Two-dimensional, almost periodic linear systems with proximal and recur-rent behavior. Proc. Amer. Math. Soc., 82 (1981), 417–422. [Jo] Journ´e, J.-L. A regularity lemma for functions of several variables. Rev. Mat. Iberoamer-icana 4 (1988), 187-193.[KS] Khanin, K. M.; Sinai, Ya. G. A new proof of M. Herman’s theorem. Comm. Math. Phys.112 (1987), no. 1, 89–101.[KKHO] Kim, J.-W.; Kim, S.-Y.; Hunt, B.; Ott, E. Fractal properties of robust strange nonchaoticattractors in maps of two or more dimensions. Phys. Rev. E 67, 036211 (2003).[Ko] Kotani, S. Ljapunov indices determine absolutely continuous spectra of stationary ran-dom one-dimensional Schr¨odinger operators. Stochastic analysis (Katata/Kyoto, 1982),225–247, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984.[K1] Krikorian, Rapha¨el, Global density of reducible quasi-periodic cocycles on T × SU (2),– Annals of Mathematics , 269-326, (2001).[K2] Krikorian, Rapha¨el Reducibility, differentiable rigidity and Lyapunov exponents forquasi-periodic cocycles on T × SL(2 , R CNRS UMR 7586, Institut de Math´ematiques de Jussieu - Paris Rive Gauche, BˆatimentSophie Germain, Case 7012, 75205 Paris Cedex 13, France & IMPA, Estrada Dona Cas-torina 110, 22460-320, Rio de Janeiro, Brazil
E-mail address : [email protected] Laboratoire de Probabilit´es et Mod`eles al´eatoires, Universit´e Pierre et MarieCurie–Boite courrier 188, 75252–Paris Cedex 05, France
E-mail address ::