Quasi-Periodic Schr\"odinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology

0.59 MB·English

by  Yiqian Wang

Quasi-Periodic Schro¨dinger Cocycles with Positive
Lyapunov Exponent are not Open in the Smooth
Topology
5
1
0
2 Yiqian Wang Jiangong You

n
a
J

2
2

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S
D Abstract
.
h
Oneknowsthatthesetofquasi-periodicSchro¨dingercocycleswithpositiveLya-
t
a punovexponentisopenanddenseintheanalytictopology.Inthispaper,weconstruct
m
cocycleswithpositiveLyapunovexponentwhichcanbeapproximatedbyoneswith
[
zeroLyapunovexponentinthespaceofCl(1 ≤ l ≤ ∞)smoothquasi-periodiccocy-

1 cles.Asaconsequence,thesetofquasi-periodicSchro¨dingercocycleswithpositive
v
LyapunovexponentisnotCl open.
0
8
Keywords.Lyapunovexponent;Smoothquasi-periodiccocycles;Schro¨dingeroper-
3
5 ators.
0
.
1
01 Introduction and Results
5
1
:
vLet X be a Cr compact manifold, T : X → X be ergodic with a normalized invari-
i
Xant measure µ and A(x) be a SL(2,R)-valued function on X. The dynamical system:
ar(x,w) → (T(x),A(x)w) in X × R2 is called a SL(2,R) cocycle (or cocycle for sim-
plicity) over the base dynamics (X,T). We will simply denoted it as (T,A). If the base
system is a rotation on torus, i.e., X = Tm = Rm\Zm, T = T : x → x + ω with
ω
rationalindependentω,wecall(T ,A)aquasi-periodiccocycle,whichissimplydenoted
ω
(cid:18) (cid:19)
v(x) −1
by (ω,A). If furthermore A(x) = S (x) is of the form S (x) = with
v v 1 0
v(x+1) = v(x),wecall(ω,S (x))aquasi-periodicSchro¨dingercocycle.
v
Y. Wang: Department of Mathematics, Nanjing University, Nanjing 210093, China; e-mail:
[email protected]
J. You: Department of Mathematics, Nanjing University, Nanjing 210093, China; e-mail:
[email protected]
MathematicsSubjectClassification(2010):Primary37;Secondary37D25
1
2 YiqianWang,JiangongYou
Foranyn ∈ Nandx ∈ X,wedenote
An(x) = A(Tn−1x)···A(Tx)A(x)
and
A−n(x) = A−1(T−nx)···A−1(T−1x) = (An(T−nx))−1.
If the base dynamics (X,T,µ) is fixed, the (maximum) Lyapunov exponent of (T,A) is
definedas
(cid:90) (cid:90)
1
L(A) = lim log(cid:107)An(x)(cid:107)dµ := lim L (A(x))dµ ∈ [0,∞).
n
n→∞ n n→∞
L(A)measurestheaveragegrowthrateof(cid:107)An(x)(cid:107).
The regularity and positivity of the Lyapunov exponent (LE) are the central subjects
indynamicalsystems.One isalsointerestedintheproblemwhetheror notcocycleswith
positive LE are open and dense. The problems turn out to be very subtle, which depend
onthebasedynamics(X,T)andthesmoothnessofthematrixA.
Firstly,classicalFurstenbergtheory[26]showedthatforcertainspeciallinearcocycles
overBernoullishifts,thelargestLEispositiveunderverygeneralconditions.Furstenberg
and Kifer [27] and Hennion [29] proved the continuity of the largest LE of i.i.d random
matrices under a condition of almost irreducibility. Kotani [41] showed that the LE of
Schro¨dingercocyclesS ispositiveforalmosteveryenergyE ifthepotentialv isnon-
E−v
deterministic. Viana[49] proved that for any s > 0, the set of Cs linear cocycles over any
hyperbolic ergodic transformation contains an open and dense subset of cocycles with
nonzeroLE;andtheLEiscontinuousforSL(2,R)-cocyclesoverMarkovshifts[44].For
otherrelatedresults,see[7],[11]and[50].
When the base dynamics is uniquely ergodic (e.g., irrational rotation or skew shift
on the torus), the positivity and continuity of the LE seem to be more sensitive to the
smoothness of the matrix-valued function A(x). The LE was proved to be discontinuous
at any non-uniform cocycles in the C0 topology by Furman [25] (Continuity at uniform
hyperbolic cocyces and cocycles with zero LE is trivial). Motivated by Man˜e´ [42, 43],
Bochi [12] further proved a stronger result that any non-uniformly hyperbolic SL(2,R)-
cocycleoverafixedergodicsystemonacompactspacecanbeapproximatedbycocycles
with zero LE in the C0 topology, which shows that any non-uniform cocycle can not be
aninnerpointofcocycleswithpositiveLEintheC0 topology.Forfurtherrelatedresults,
wereferto[9],[13],[14],[29],[36],[37],[40],[48].
On the other hand, there are tremendously many positive results in the analytic topol-
ogy. Herman [30] introduced the subharmonicity method and showed that the LE of
S is positive for |λ| > 1 and all E. Herman also proved the positivity of the
E−2λcosx
LE for trigonometric polynomials if the coupling is large enough. The generalization to
PositiveLyapunovExponentarenotOpen 3
arbitrary one-frequency non-constant real analytic potentials was obtained by Sorets and
Spencer [47]. Same results for Diophantine multi-frequency were established by Bour-
gainandGoldstein[18]andGoldsteinandSchlag[28].Zhang[55]gaveadifferentproof
of the results in [47] and applied it to a certain class of analytic Szego˝ cocycle. For more
references,onecansee[17],[23],[39].
For the continuity of the LE, Large Deviation Theorems (LDT) is an important tool,
which was first established by Bourgain and Goldstein in [18] for real analytic potentials
with Diophantine frequencies. In [28], Goldstein and Schlag proved that, by some sharp
versionofLDTandgeneralizedAvalanchePrinciple(AP),L(S )isHo¨ldercontinuous
E−v
inE ifω isaDiophantine,v(x)isanalyticandL(S ) > 0.Jitomirskaya,Kosloverand
E−v
Schulteis[32]provedthecontinuityoftheLEforaclassofanalyticone-frequencyquasi-
periodicM(2,C)-cocycles withsingularities. Wewill briefly mentionmore results along
thislineattheendofthissection.ThecontinuityoftheLEimpliesthatthecocycleswith
positiveLEareopeninanalytictopology.TogetherwiththedensenessresultbyAvila[1],
one knows that the set of quasi-periodic cocycles with positive LE is open and dense in
theanalytictopology.
WehaveseenthatthebehavioroftheLEintheC0 topologyistotallydifferentfromits
behaviorintheanalytictopology.Thesmoothcaseismoresubtle.Avila[1]proved,among
otherresults,thattheLEispositiveforadensesubsetofsmoothquasi-periodiccocycles.
Recently, with Benedicks-Carleson-Young’s method[8, 54], the authors [51] constructed
quasi-periodiccocycles(T ,A)whereT isanirrationalrotationx → x+ω onS1 withω
ω ω
ofboundedtypeandA ∈ Cl(S1,SL(2,R)),0 ≤ l ≤ ∞,suchthattheLEisnotcontinuous
at A in the Cl topology. Such an example in the Schro¨dinger class is also constructed
in [51]. For C2 cosine-like potentials, Anderson Localization and the positivity of LE
has been established by Sinai [46] and Fro¨hlich-Spencer-Wittwer [24], also see Bjerklo¨v
[10].Forthemodelin[46],WangandZhang[52]showedthecontinuityoftheLE,which
implies that non-uniform quasi-periodic cocycles can be inner points of smooth quasi-
periodiccocycleswithpositiveexponents.Aninterestingproblemiswhetherornotquasi-
periodic cocycles with positive exponent are open and dense in the smooth topology as
in the analytic topology. As we mentioned before, the denseness follows from the result
of Avila [1]. In this paper, we will prove that, different from the analytic case, the set of
smoothquasi-periodiccocycleswithpositiveexponentarenotopeninsmoothtopology.
The LE of quasi-periodic Schro¨dinger cocycles have attracted so much attention not
only because of its importance in dynamical systems, but also due to its close relation
with quasi-periodic Schro¨dinger operators. The latter has strong background in physics.
TheLEofSchro¨dingiercocyclescomingfromtheeigenvalueequationsofquasi-periodic
Scho¨dinger operators encodes enormous information on the spectrum. It is known from
KotanitheorythatpositiveLEimpliessingularspectrum,andtypicallyAndersonlocaliza-
4 YiqianWang,JiangongYou
tion,see[31,38,45];whilezeroLyapunovspectrumusuallyimpliescontinuous,typically
absolutelycontinuousspectrum.ThepositivityoftheLEisalsoastartingpointformany
otherproblemsindynamicalsystemsandspectraltheory,suchasho¨ldercontinuityofLE,
continuityandtopologicalstructureofspectrumset.Therecentdevelopedmethods,such
asGreen’sfunctionestimatesandAvalanchePrinciple,etc.(see[16]),dependcruciallyon
thepositivityoftheLE.
Another related interesting question is the robustness of Anderson localization. i.e.,
wether or not the perturbations of a Schro¨dinger operator exhibiting Anderson localiza-
tion still have Anderson localization? The answer is affirmative in the analytic category
since the LE is continuous and thus the positivity of the LE is kept under perturbations.3
We are interested in the question in smooth case, which is closely related to the problem
whetherornotthepositivityoftheLEiskeptunderperturbationsinthesmoothcategory,
equivalentlywhetherornotthereexistsmoothnon-uniformlyhyperbolicSchro¨dingerco-
cycleswhichcanbeaccumulatedbyoneswithzeroLEinCl topology(l = 1,2,··· ,∞).
Ifitisthecase,thenatureofthespectrumofSchro¨dingeroperatorsmightexhibitdramat-
icallychangesundersmallperturbationsofthepotentialinsmoothtopology.
Thefollowingisthemainresultofthispaper.
Theorem 1. Consider quasi-periodic Schro¨dinger cocycles over S1 with ω being a fixed
irrationalnumberofbounded-type.4 Forany0 ≤ l ≤ ∞,thereexistsaSchro¨dingercocy-
cle S with arbitrarily large Lyapunov exponent and a sequence of Schro¨dinger cocycles
v
S withzeroLyapunovexponentsuchthatv (x) → v(x)intheCl topology.Asaconse-
vn n
quence, the set of quasi-periodic Schro¨dinger cocycles with positive Lyapunov exponent
isnotCl open.
Theorem 1 can be obtained from Theorem 2 in the same way as in [51] to derive
examples in Schro¨dinger cocycles from examples in SL(2,R) cocycles. Thus we only
needtoproveTheorem2.
Theorem 2. Consider quasi-periodic SL(2,R) cocycles over S1 with ω being a fixed
irrational number of bounded-type. For any 0 ≤ l ≤ ∞, there exists a cocycle D ∈
l
Cl(S1,SL(2,R)) with arbitrarily large Lyapunov exponent and a sequence of cocycles
C ∈ Cl(S1,SL(2,R)) with zero Lyapunov exponent such that C → D in the Cl topol-
k k l
ogy. As a consequence, the set of SL(2,R)-cocycles with positive Lyapunov exponent is
notCl open.
Remark 1.1. Completely different from the result in Theorem 1, Bonatti, Go´mez-Mont
and Viana [15] proved that there exist Ho¨lder continuous cocycles over Bernoulli shift
3Moreprecisely,itistrueforallalmostallfrequencies.
4Boundedtypemeans pk,thebestapproximationofω,satisfiesq ≤Mq forsomeM >0.
qk k+1 k
PositiveLyapunovExponentarenotOpen 5
with positive LE which can be approximated by continuous cocycles with zero LE, but
not by Ho¨lder ones, which shows that the base dynamics plays an important role in the
regularityproblemoftheLE.
Remark1.2. AvilaandKrikorian[6]showedthattheLEissmoothinthespaceofsmooth
monotonicquasi-periodiccocycles.Ourresultshowsthatthemonotonicityassumptionin
[6] is necessary, and behavior of the LE in smooth quasi-periodic Schro¨dinger cocycles
homotopic to the identity are completely different from its behavior in the class of mono-
tonecocycles.
The proof of Theorem 2 is constructive. Recall in [51], we have constructed a smooth
cocycles D with positive LE and a smooth cocycle A in 1 -neighborhood of D in the
l 1 2k l
Cl topology for any given k > 0 such that the finite LE of A , defined by L (A ) =
1 n1 1
1 (cid:82) log(cid:107)An1(x)(cid:107)dx, is smaller that (1 − δ )L(D ) for a fixed number δ > 0. As a
n1 S1 1 2 l 2
consequence of subadditivity of finite LE, L(A ) < (1−δ )L(A). It follows that the LE
1 2
isdiscontinuousatD .However,theconstructionin[51]didnottellushowsmallL(A )
l 1
can be. In this paper we will define a new A somehow different from the one in [51] but
1
satisfies the same property stated as above. Then we further locally modify A such that
1
themodifiedcocycle,sayA ,satisfies(cid:107)A −A (cid:107) < 1 andL (A ) < (1−δ )L (A ).
2 2 1 Cl 4k n2 2 2 n1 1
ItfollowsthatA isintheδ-neighborhoodofAandL(A ) < (1−δ )2L(A).Inductively,
2 2 2
welocallymodifyA suchthatthemodifiedcocycle,sayA ,satisfies(cid:107)A −A (cid:107) <
i i+1 i+1 i Cl
1 andL (A ) < (1−δ )L (A ),wheren → ∞willbe specifiedlater.It follows
2ik ni+1 i+1 2 ni i i
that all A are in the 1-neighborhood of D and L(A ) < (1−δ )iL(D ). It is easy to
i k l i+1 2 l
seethatA hasalimit,sayC ,withL(C ) = 0.Moreover,(cid:107)C −D (cid:107) < 1.Theorem2
i k k k l Cl k
isthusprovedsincek isarbitrary.
WeremarkthatD andC weconstructedareoftheformsΛR andΛR where
l k φ(x) φ (x)
k
Λ = diag{λ,−λ},λ (cid:29) 1 with L(D ) ∼ lnλ and L(C ) = 0. Moreover, φ (x) is an
l k k
arbitrarily small modification of φ(x) in an arbitrarily small neighborhood of two special
points(calledcriticalpoints).Soasmallchangemakesabigdifference!ForSchro¨dinger
cocycles, we actually construct, for arbitrarily large but fixed λ, smooth v(x) and v¯(x)
which are arbitrarily close to each other and slightly different only at the neighborhood
of two critical points such that L(S ) is very big while L(S ) = 0. The result is
λv(x) λv¯(x)
surprising as we have even not seen any example of smooth Schro¨dinger cocycles of the
formS withλ (cid:29) 1suchthatL(S ) = 0.
λv¯(x) λv¯(x)
Fromourconstruction,onecanseehowandwheretomodifyacocyclesoastocontrol
theLE.Thismightbeusefulforotherproblems.
More results on the continuity of the LE in the analytic topology. When the base dynam-
icsisashiftorskew-shiftofahigherdimensionaltorus,thelog-continuityoftheLEwas
proved in [19] by Bourgain, Goldstein and Schlag. Recently, the result of [32] was gen-
6 YiqianWang,JiangongYou
eralized by Jitomirskaya and Marx [33] for all non-trivial singular analytic quasiperiodic
cocycleswithone-frequencywithapplicationtotheextendedHarper’smodel[34].
An arithmetic version of large deviations and inductive scheme were developed by
BourgainandJitomirskayain[20]allowingtoobtainjointcontinuityoftheLEforSL(2,C)
cocycles,infrequencyandcocyclemap,atanyirrationalfrequencies.Thisresulthasbeen
crucialinmanyfurtherimportantachievements,suchastheproofoftheTenMartiniprob-
lem [4], Avila’s global theory of one-frequency cocycles [2, 3]. It was extended to multi-
frequency case by Bourgain [17] and to general M(2,C) case by Jitomirskaya and Marx
[34].Morerecently,acompletelydifferentmethodwithoutLDTorAPwasdevelopedby
Avila, Jitomirskaya and Sadel [5] and was applied to prove the continuity of the LE in
M(d,C),d ≥ 2.Forfurtherworks,see[21],[22],[35],[39],[53].
2 The construction of D
l
We consider the case m = 1. We say a SL(2,R)-matrix A is hyperbolic if (cid:107)A(cid:107) > 1.
A quasi-periodic cocycle (ω,A(x)) of degree d is defined by a matrix function A(x) =
R · Λ(x) · R on R, with Λ(x + 1) = Λ(x) = diag{(cid:107)A(cid:107), 1 }, ψ(x + 1) =
ψ(x) φ(x) (cid:107)A(cid:107)
(cid:18) (cid:19)
cosθ −sinθ
2πd + ψ(x),φ(x + 1) = 2πd + φ(x) where R = . It is easy to
θ sinθ cosθ
see that (φ(x) + ψ(x − ω)) is uniquely determined by A(x) up to 2πZ and L(A) =
L(Λ(x)·R )asAisconjugatedtoΛ(x)·R .
φ(x)+ψ(x−ω) φ(x)+ψ(x−ω)
Let Λ = diag{λ, 1} with λ (cid:29) 1. In this section, we will construct a sequence of
λ
smooth cocycles B of the form Λ · R , converging in Cl such that L(limB ) > 0.
k ξ (x) k
k
Moreover ξ (x) will be specially designed so that, in the next section, we can further
k
constructed cocycles C with zero Lyapunov exponent in any small neighborhood of B .
k k
When λ is big, we will see that the Lyapunov exponent of B crucially depends on the
k
local behavior, more precisely the degeneracy, of ξ (x) at the critical points {c : ξ (c) =
k k
π (mod π)} due to the cancelation. The construction in this section is in principle along
2
the line of the construction in [51], the difference is in this paper, we use the decomposi-
tion of a matrix instead of the most expended and contracted direction of a matrix which
makestheproofmoretransparent.
Let ω be a fixed irrational number and pk be its best approximation. Throughout the
q
k
paper, we assume that ω is of the bounded type, i.e., q ≤ Mq ; (cid:15) > 0 is small. l is a
k+1 k
fixedpositiveintegerreflectingthesmoothnessofcocycles.LetλandN arelargeenough
sothat
∞
(cid:88) logq
10l k+1 ≤ (cid:15), λ−1 (cid:28) q−2. (2.1)
q N
k
k=N
PositiveLyapunovExponentarenotOpen 7
We define the decaying sequence {λ } inductively by logλ = logλ − 10llogqk
k k k−1 q
k−1
whereλ = λ (cid:29) 1.Itiseasytoseethatλ convergestoλ withλ > λ1−(cid:15).
N k ∞ ∞
(cid:8) (cid:9)
For k ≥ N, let C = 0, 1 , I = [− 1 , 1 ], I = [1 − 1 , 1 + 1 ] and I =
0 2 k,1 q2 q2 k,2 2 q2 2 q2 k
k k k k
I (cid:83)I .ForC ≥ 1,wedenoteby Ik,1 = [− 1 , 1 ], Ik,2 = [1 − 1 , 1 + 1 ],andby
k,1 k,2 C Cq2 Cq2 C 2 Cq2 2 Cq2
k k k k
Ik the set Ik,1 ∪ Ik,2. Denote Lebesgue measure of I by |I |. For each x ∈ I , let r+(x)
C C C k k k k
(respectivelyr−(x))bethesmallestpositiveintegersuchthatTr+(x)(x) ∈ I (respectively
k k k
T−r−(x)(x) ∈ I ). Let r± = min r±(x) and r = min{r+,r−}. Obviously, r ≥ q .
k k k x∈Ik k k k k k k
Moreover,fromthesymmetrybetweenI andI ,wehaver = r+ = r−.
k,1 k,2 k k k
Wedefineξ onI = I (cid:83)I = {x : |x| ≤ 1 }(cid:83){x : |x− 1| ≤ 1 }by
0 1 2 2q2 2 2q2
N N
(cid:40)
ξ (x), |x| ≤ 1 ;
ξ (x) = 01 2qN2 (2.2)
0 −ξ (x)(orξ (x)), |x− 1| ≤ 1
02 02 2 2q2
N
where
1 1
ξ (x) = sgn(x)|x|l+1, ξ (x) = sgn(x− )|x− |l+1. (2.3)
01 02
2 2
ξ(x)isaliftofa1-periodicCl functionsatisfying
(cid:40)
ξ (x), |x| ≤ 1 ;
ξ(x) = 01 2qN2 (2.4)
−ξ (x)(orπ +ξ (x)), |x− 1| ≤ 1 ,
02 02 2 2q2
N
and |ξ(x)(mod π)| > 1 for any x(mod 1) ∈/ I. See Figures 1 and 2 for the picture of
2q2
N
ξ(x).
ξ(x) ξ(x)
2π
π π
x x
0 1 1 0 1 1
2 2
−π
Figure1:homotopictoidentity Figure2:nonhomotopictoidentity
In the following, we will use c, C, C(l), etc, to denote universal positive constants
independent of iterative steps. For any cocycle A(x), n ∈ Z+ and x ∈ I, we decompose
An(x) as R · Λ (x) · R when An(x) is hyperbolic in I and decompose
ψA,n(x) A,n φA,n(x)
An(T−nx)asR ·Λ (x)·R whenAn(T−nx)ishyperbolicinI.
ψA,−n(x) A,−n φA,−n(x)
Letξ (x) = ξ(x)definedabove.DefineB (x) = ΛR .
N N π2−ξN(x)
8 YiqianWang,JiangongYou
Proposition 2.1. There are Cl functions ξ (x) (k = N + 1,N + 2,···) constructed
k
inductivelysuchthat
1.|ξ (x)−ξ (x)| ≤ C(l)·λ−2rk ·|I |−l2. (2.5)
k k−1 Cl k k
2.LetB (x) = ΛR .Foreachx ∈ I ,wehave
k π−ξ (x) k
2 k
r±(x) r±(x)
(cid:107)B k (x)(cid:107) ≥ λ k . (2.6)
k k
3.Forx ∈ I ,wehave
k
(1) ψ (x)+φ (x)− π = ξ (x) on Ik;
k Bk,−rk− Bk,rk+ 2 0 10
(2) |ψ (x)+φ (x)− π| ≥ 1 , x ∈ I \Ik,
k Bk,−rk− Bk,rk+ 2 (20qk2)l+1 k 10
whereξ (x)isdefinedin(2.2)and(2.3).
0
Remark 2.1. It is easy to see from (2.5) that B converges to a limit D in Cl -topology.
k l
Moreover,from(2.5)and(2.6)aswellasTheorem3in[51],wehaveL(D ) ≥ (1−(cid:15))lnλ.
l
ToproveProposition2.1,wefirstgivethefollowingLemma2.1.
Lemma 2.1. For any function σ(x) defined on S1, let d (σ) = min {|σ(x)|}. Assume
k x(cid:54)∈I
k
thatforanyx ∈ I ,
k
log(cid:107)Ark(x)(cid:107) (cid:29) −logd , (2.7)
k+1
where d = d (φ (x)+ψ (x)− π). Furthermore assume that, for i ≤ l and
k+1 k+1 A,r+ A,−r− 2
k k
m± = r±(x),
k
 (cid:12) (cid:12) (cid:12) (cid:12)
 (cid:12)(cid:12)ddxiiφA,m+(x)(cid:12)(cid:12), (cid:12)(cid:12)ddxiiψA,−m−(x)(cid:12)(cid:12) ≤ C(i)·d−k+i1 (1)k
(cid:12) (cid:12)
 (cid:12)di(cid:107)A±m(x)(cid:107)(cid:12)·(cid:107)A±m(x)(cid:107)−1 ≤ C(i)·d−i . (2)
(cid:12) dxi (cid:12) k+1 k
Thenfori ≤ l,x ∈ I andmˆ± = r± (x)itholdsthat
k+1 k+1
 (cid:12) (cid:12) (cid:12) (cid:12)
 (cid:12)(cid:12)ddxiiφA,mˆ+(x)(cid:12)(cid:12),(cid:12)(cid:12)ddxiiψA,−mˆ−(x)(cid:12)(cid:12) ≤ C(i)·d−k+i1, (1)k+1
(cid:12) (cid:12)
 (cid:12)di(cid:107)A±mˆ(x)(cid:107)(cid:12)·(cid:107)A±mˆ(x)(cid:107)−1 ≤ C(i)·d−i . (2)
(cid:12) dxi (cid:12) k+1 k+1
Moreover,foranyi ≥ 0,x ∈ I ,itholdsthat
k+1
(cid:12) (cid:12)
(cid:12) di (φ (x)−φ (x))(cid:12) ≤ C(i)·(cid:107)Ar+(cid:107)−2 ·d−i,
(cid:12)dxi A,r+ A,r+ (cid:12) k k
(cid:12) k+1 k (cid:12) (2.8)
(cid:12) di (ψ (x)−ψ (x))(cid:12) ≤ C(i)·(cid:107)Ar−(cid:107)−2 ·d−i.
(cid:12)dxi A,−r− A,−r− (cid:12) k k
k+1 k
PositiveLyapunovExponentarenotOpen 9
TheproofofLemma2.1willbegivenintheAppendix.
ProofofProposition2.1. Foreachk ≥ N andx ∈ I ,since
k
ˆ
f (x) := (ψ (x)+φ (x))−(ψ (x)+φ (x))
k B ,−r− B ,r+ B ,−r− B ,r+
k−1 k k−1 k k−1 k−1 k−1 k−1
usually does not vanish on I and thus ψ (x)+φ (x)− π (cid:54)= ξ (x) on I . To
k−1 Bk,−rk− Bk,rk+ 2 0 k
guarantee (1) in Proposition 2.1, we modify ξ (x) on I as ξ (x) = ξ (x)+f (x),
k k−1 k k k−1 k
whereCl periodicfunctionf (x)isdefinedasfollows
k
 fˆ(x) x ∈ Ik
 k 10



f (x) = h±(x), x ∈ I \Ik
k k k 10




 0, x ∈ S1\I
k
whereh±(x)isapolynomialofdegree2l+1restrictedineachintervalofI \Ik satisfying
k k 10
djh±k (± 1 ) = djfˆk(± 1 )
dxj 10q2 dxj 10q2
k k
djh±k (± 1 ) = 0, i = 1,2, 0 ≤ j ≤ l.
dxj q2
k
From(2.8)inLemma2.1,wehave
|(ψ (x)+φ (x))−(ψ (x)+φ (x))| ≤ C(l)·λ−2rk·|I |−l2,
Bk−1,−rk− Bk−1,rk+ Bk−1,−rk−−1 Bk−1,rk+−1 Cl k k
(2.9)
where (2.7) is fulfilled by conclusion 2 and 3 of the induction assumption for the case
k −1.
Inviewofthedefinitionoff (x)weobtain
k
|f | ≤ C(l)·λ−2rk ·|I |−l2. (2.10)
k Cl k k
LetB (x) = Λ·R ,thenwehave
k π−ξ (x)
2 k
Lemma2.2. Forx ∈ I ,itholdsthat
k
r+(x) r+(x)
B k (x) = B k (x)·R
k k−1 −fk(x)
and
Brk−(x)(T−rk−(x)x) = Brk−(x)(T−rk−(x)x).
k k−1
Proof.ObviouslyTix ∈ S1\I forx ∈ I and1 ≤ i ≤ r+(x)−1.SinceB (x) = B (x)
k k k k k−1
forx ∈ S1\I ,wehavethat
k
Brk+(x)(x) = Brk+(x)(x)·(B−1 (x)B (x)), x ∈ I .
k k−1 k−1 k k
10 YiqianWang,JiangongYou
Fromthedefinition,wehaveB (x) = B (x)·R ,whichimpliesB−1 (x)B (x) =
k k−1 ξk−1(x)−ξk(x) k−1 k
R . Thus we obtain the first equation in Lemma 2.2. Similarly we can prove
ξ (x)−ξ (x)
k−1 k
thesecondone. (cid:116)(cid:117)
Lemma2.3. Itholdsthat
f (x) = (ψ (x)+φ (x))−(ψ (x)+φ (x)), x ∈ I .
k B ,−r− B ,r+ B ,−r− B ,r+ k
k−1 k k−1 k k k k k
Proof. Since a rotation does not change the norm of a vector, for a hyperbolic matrix A
andarotationmatrixR ,wehave
θ
φ = φ +θ. (2.11)
A·R A
θ
FromLemma2.2,wehave
φ (x) = φ (x)−f (x), ψ (x) = ψ (x).
B ,r+ B ,r+ k B ,−r− B ,−r−
k k k−1 k k k k−1 k
Thus
f (x) = (ψ (x)+φ (x))−(ψ (x)+φ (x)), x ∈ I ,
k B ,−r− B ,r+ B ,−r− B ,r+ k
k−1 k k−1 k k k k k
whichconcludestheproof. (cid:116)(cid:117)
Proofof(1) and(2) Fromthedefinitionoff (x),wehavef (x) = (ψ (x)+
k k k k B ,−r−
k−1 k
φ (x))−(ψ (x)+φ (x)) on Ik, which together with Lemma 2.3
Bk−1,rk+ Bk−1,−rk−−1 Bk−1,rk+−1 10
impliesthatforeachx ∈ Ik,
10
ψ (x)+φ (x) = (ψ (x)+φ (x))−f (x) = ψ (x)+φ (x).
B ,−r− B ,r+ B ,−r− B ,r+ k B ,−r− B ,r+
k k k k k−1 k k−1 k k−1 k−1 k−1 k−1
Since ψ (x) + φ (x) = ξ (x) on Ik−1 by induction assumption (1) ,
Bk−1,−rk−−1 Bk−1,rk+−1 0 10 k−1
weobtain(1) inproposition2.1.
k
Obviouslyλqk−1 (cid:29) q2l.Hence(2) inProposition2.1canbeobtainedfromtheinduc-
k k k
tionassumption(2) and(2.10).
k−1
Proofofconclusion1ofProposition2.1. Conclusion1canbeobtainedfrom(2.9).
Proof of conclusion 2 of Proposition 2.1. For x ∈ I , let i (x) < i (x) < ··· <
k 1 2
i (x) ≤ r be the returning times of I less than r . Since |I | ≤ 1|I | (we
j(x) k k−1 k k 4 k−1
can make a slight modification of the definition of I if necessary), from the symme-
k
try between I and I , we have that for any x ∈ I , we have Trkx ∈ I . Then
k,1 k,2 k k−1
we have that i (x) = r . Since Tis(x)x (cid:54)∈ I for s < j(x), |θ − π| ≥ 1 , where
j(x) k k s 2 q2l
k
θ = φ (Tis(x)x)+ψ (Tis−1(x)x).Togetherwiththeconclusion
s Bk,is+1(x)−is(x) Bk,is(x)−is−1(x)
3 of the induction assumption for (k − 1)-th step we have that |θ˜ − π| ≥ 1 , where
s 2 2q2l
k
θ˜ = φ (Tis(x)x) + ψ (x). Thus from the definition of λ , we obtain
s Bk,is+1(x)−is(x) Bk,is(x) k
theconclusion2fork-thstepbyrepeatedapplicationsofLemmaA.1.