Partner orbits and action differences on compact factors of the hyperbolic plane. II: Higher-order encounters

0.41 MB·English

by  Hien Minh Huynh

Partner orbits and action differences on compact factors of
the hyperbolic plane. Part II: Higher-order encounters
HIEN MINH HUYNH
6
1 Department of Mathematics,
0
2 Quy Nhon University, Binh Dinh, Vietnam;

b
Universita¨t Ko¨ln, Institut fu¨r Mathematik
e
F
Weyertal 86-90, D-50931 Ko¨ln, Germany

1
2
e-mail: [email protected]

]
h
p
- Abstract
h
t Physicistshaveargued thatperiodicorbitbunching leadstouniversal spectralfluctuations for
a
m chaoticquantumsystems. Toestablishamoredetailedmathematicalunderstanding ofthisfact,it
[ isfirstnecessarytolookmorecloselyattheclassicalsideoftheproblemanddetermineorbitpairs

3 consisting of orbits which have similar actions. We specialize to the geodesic flow on compact
v
factors of the hyperbolic plane as a classical chaotic system. The companion paper (Huynh and
9
8 Kunze,2015)provedtheexistenceofauniqueperiodicpartnerorbitforagivenperiodicorbitwith
2
a small-angle self-crossing in configuration space that is a 2-encounter and derived an estimate
4
0 for the action difference of the orbit pair. In this paper, we provide an inductive argument to
.
1 deal with higher-order encounters: we prove that a given periodic orbit including an L-parallel
0
encounter has (L 1)! 1 partner orbits; we construct partner orbits and give estimates for the
5
− −
1 actiondifferences betweenorbitpairs.
:
v
Keywords: geodesicflow,periodicsolution,partnerorbit,higher-orderencounter
i
X
r
a
1 Introduction
Inthesemi-classicallimitchaoticquantumsystemsveryoftenexhibituniversalbehavior,inthesense
that several of their characteristic quantities agree with the respective quantities found for certain
ensemblesofrandommatrices. Viatraceformulas,suchquantitiescanbeillustratedassuitablesums
over the periodic orbits of the underlying classical dynamical system. For instance, the two-point
correlatorfunctionisexpressed byadoublesumoverperiodicorbits
K(τ) = T1 AγA∗γ′e~i(Sγ−Sγ′)δ τTH − Tγ +2 Tγ′ , (1.1)
D H Xγ,γ′ (cid:16) (cid:17)E
1
where abbreviates the average over the energy and over a small time window, T denotes the
H
h·i
Heisenberg time and A , S , and T are the amplitude, the action, and the period of the orbit γ,
γ γ γ
respectively.
The diagonalapproximationγ = γ to (1.1)studiedby Hannay/Ozorio deAlmeida[9] and Berry
′
[3] in the1980’scontributesto thefirst order2τ to (1.1); see also [14]. To nextorders, as ~ 0, the
→
maintermfrom(1.1)arisesowingtothoseorbitpairsγ = γ′forwhichtheactiondifferenceSγ Sγ′ is
6 −
‘small’. Thiswas first consideredby Sieberand Richter[20, 21], who predicted thatagivenperiodic
orbit with a small-angle self-crossing in configuration space will admit a partner orbit with almost
the same action. The original orbit and its partner are called a Sieber-Richter pair. In phase space,
a Sieber-Richter pair contains a region where two stretches of each orbit are almost mutually time-
reversed and one addresses this region as a 2-encounter or, more strictly, a 2-antiparallel encounter;
the ‘2’ stands for two orbit stretches which are close in configuration space, and ‘antiparallel’means
thatthetwostretcheshaveoppositedirections(seeFigure1). Itwasshownin[21]thatSieber-Richter
pairscontributetothespectralformfactor(1.1)thesecondorderterm 2τ2,anditturnedoutthatthe
−
resultagreed withwhatis obtainedusingrandom matrixtheory[6], forcertain symmetryclasses.
x
T
1 φ
T
w 2
y
Figure1: ExampleofaSieber-Richter pair
This discovery prompted an increased research activity on the subject matter in the following
years and finallyled toan expansion
K(τ) = 2τ τ ln(1+2τ) = 2τ 2τ2 +2τ3 +...
− −
for the orthogonal ensemble (the symmetry class relevant for time-reversal invariant systems) to all
ordersinτ,byincludingthehigher-orderencountersalso;see[10,16,17,18],andinaddition[8,15],
whichprovidemuchmorebackgroundand manyfurtherreferences.
To establish a more detailed mathematical understanding, we start on the classical side and try
to prove the existence of partner orbits and derive good estimates for the action differences of the
orbit pairs. For 2-encounters this was done in the previous work [12], where we considered the
geodesicflow oncompactfactors ofthehyperbolicplane;inthiscasetheaction ofaperiodicorbitis
proportional to its length. It was shown in [12] that a T-periodic orbit of the geodesic flow crossing
itself in configuration space at a time T has a uniquepartner orbit that remains 9 sin(φ/2) -closeto
1
| |
theoriginaloneandtheactiondifferencebetweenthemisapproximatelyequalln(1 (1+e T1)(1+
−
−
e (T T1))sin2(φ/2))) with the estimated error 12sin2(φ/2)e T, where φ is the crossing angle (see
− − −
Figure 1), and this proved the accuracy of Sieber/Richter’s prediction in [21] mentioned above. In
2
this paper, we continue considering this hyperbolic dynamical system to deal with the technically
moreinvolvedhigher-orderencounters.
In the physics community this system is often called the Hadamard-Gutzwiller model, and it has
frequentlybeen studied[5, 10, 19]; furtherrelated workincludes[8,18,22]. Forinstance,Heusleret
al. [10] identified the term 2τ3 and it was shown that there are 5 families of orbit pairs contributing
to 2τ3: 3 families of orbits with two single 2-encounters and 2 families of orbits with 3-encounters.
In that way one obtains whole bunches of periodic orbits with controlled and small mutually action
difference. Generalizing these ideas, in [17, 18] the expansion of K(τ) to all orders in τ could
be derived. Here a key point was to consider encounters where more than two orbit stretches are
involved; see also [8, 15, 22]. We speak of an L-encounter when L stretches of a periodic orbit are
mutually close to each other up to time reversal. For a precise definition one may pick one of L
encounter stretches as a reference and demand that none of the L 1 companions be further away
−
than some small distance; see [1]. In other words, all the L stretches must intersect a small Poincare´
section. Theorbitenterstheencounterregionthroughentranceportsandleavesitthroughexitports.
Using hyperbolicity, one can switch connections between entrance ports and exit ports to get new
orbitswhichstillremainclosetotheoriginalone;andtheyarecalledpartnerorbits. However,notall
the connections givegenuine periodic orbits since some of them decompose to several shorter orbits
(called pseudo-orbits;see[12]). Mu¨lleret al. [18]used combinatoricstocount thenumberofpartner
orbitsandprovidedanapproximationfortheactiondifference,butaconstructionofpartnerorbitsand
an error bound of the approximation have not been derived. Furthermore, it is necessary to arrive a
mathematicaldefinitionfor‘encounters’,‘partnerorbits’,andtointroducethenotionsof‘beginning’,
‘end’, and ‘duration’ofan encounter.
The paper is organized as follows. In Section 2, after giving some background material, we
introduceanotherversionofPoincare´ sectionsandarespectiveversionoftheAnosovclosinglemma.
In the case that the space is compact, a Poincare´ section with small radius can be identified with a
square, so everypointin aPoincare´ sectioncan berepresented by a uniquecouple(u,s) R2 called
∈
unstableand stablecoordinates. In addition,we providea ‘connecting lemma’to connect 2 orbitsin
a pseudo-orbit. In this way, one can construct a genuine periodic orbit close to a given pseudo-orbit.
In Section 3, after providing mathematically rigorous definitions of ‘encounters’, ‘partner orbits’,
‘orbit pairs’, etc, we provide an inductive argument to construct partner orbits for a given orbit with
a single L-parallel encounter. The first step of the inductive argument stated in Theorem 3.1 shows
the existence of a unique partner orbit for a given orbit with a 3-parallel encounter. The main result
of the current paper is Theorem 3.2 which proves that there exist (L 1)! 1 partner orbits for a
− −
given periodic orbit with an L-parallel encounter such that any two piercing points are not too close.
This condition is expressed by differences of stable and unstable coordinates of the piercing points
which guarantees that the encounter stretches are separated by non-vanishing loops and whence the
partnerorbitsdo notcoincide. We usecombinatoricstocount thenumberofpartnerorbitsand apply
the connecting lemma, the Anosov closing lemmas to construct partner orbits. The action difference
between the orbit pairs can be approximated by terms of coordinates of the piercing points with a
precisely estimatederror.
3
Acknowledgments: This work was initiated in the framework of thecollaborativeresearch program
SFB-TR12‘SymmetriesandUniversalityinMesoscopicSystems’fundedbytheDFG,whosefinan-
cial support is gratefully acknowledged. Special thanks to my supervisor M. Kunze for his help and
support. I enjoyed many fruitful discussions with P. Braun, K. Bieder, F. Haake, G. Knieper, and
S. Mu¨ller.
2 Preliminaries
We consider the geodesic flow on compact Riemann surfaces of constant negative curvature. In fact
this flow has had a great historical relevance for the development of the whole theory of hyperbolic
dynamical systems or Anosov systems. It is well-known that any compact orientable surface with a
metric of constant negativecurvature is isometric to a factor Γ H2, where H2 = z = x+iy C :
\ { ∈
y > 0 is thehyperbolicplaneendowedwith thehyperbolicmetricds2 = dx2+dy2 and Γ isa discrete
} y2
subgroup of the projective Lie group PSL(2,R) = SL(2,R)/ E ; here SL(2,R) is the group of
2
{± }
all real 2 2 matrices with unity determinant, and E denotes the unitmatrix. The group PSL(2,R)
2
×
acts transitively on H2 by Mo¨bius transformations z az+b. If the action is free (of fixed points),
7→ cz+d
then thefactor Γ H2 has a Riemann surface structure. Such a surface is a closed Riemann surface of
\
genusatleast2andhasthehyperbolicplaneH2 astheuniversalcovering. Thegeodesicflow(ϕXt )t R
ontheunittangentbundle = T1(Γ H2)goesalongtheunitspeedgeodesicsonΓ H2. Thismea∈ns
X \ \
gtheaotdeevsiecryonorHbi2t.uInndaedrditthieong,eeovdeerysicorflieonwted(ϕuXtn)itt∈sRpeiesdaclgoesoeddegsiecodoensiXcownhΓichH2isisthaepperroiojedcictioonrboitfoaf
\
theOgenotdheesiocthfleorwha(nϕdXt,)tht∈eRuonnitXtan=geTnt1b(Γun\dHle2)T.1(Γ H2)isisometrictothequotientspaceΓ PSL(2,R) =
\ \
Γg,g PSL(2,R) ,whichisthesystemofrightco-setsofΓinPSL(2,R),byanisometryΞ. Then
{ ∈ }
thegeodesicflow(ϕXt )t Rcanbeequivalentlydescribedasthenatural‘quotientflow’ϕXt (Γg) = Γgat
on X = Γ PSL(2,R)a∈ssociatedto theflow φ (g) = ga onPSL(2,R)by theconjugaterelation
t t
\
ϕ = Ξ 1 ϕX Ξ forall t R.
Xt − ◦ t ◦ ∈
Here a PSL(2,R) denotes the equivalenceclass obtained from the matrix A = et/2 0
t ∈ t 0 e−t/2 ∈
SL(2,R). Infact,thereareone-to-onecorrespondencesbetweenthecollectionofallp(cid:16)eriodicorbit(cid:17)sof
b(ϕyXt )t∈R),(daenndottheedcboynPjuOgaXcy),cthlaessceosllienctΓio(ndoefnoatleldorbieynCted).unTihtesppeeerdiocdloTseodfgaepoedreisoidciscoonrbYit(idnenoted
Y Γ X
CG PO
andthelengthl ofthecorrespondingclosedgeodesicin arerelatedbyT = l = 2arccosh(tr(γ)),
CGY 2
whereγ is arepresentativeoftherespectiveconjugacyclass inC .
Γ
There are some more advantages to work on X = Γ PSL(2,R) rather than on = T1(Γ H2).
\ X \
Onecan calculateexplicitlythestableand unstablemanifoldsat apointxtobe
Ws(x) = θX(x),t R and Wu(x) = ηX(x),t R ,
X { t ∈ } X { t ∈ }
where(θtX)t R and(ηtX)t R arethehorocycleflowandconjugatehorocycleflowdefinedbyθtX(Γg) =
Γgb and ηX∈(Γg) = Γg∈c ; here b ,c PSL(2,R) denote the equivalence classes obtained from
t t t t t ∈
4
Bt = 10 1t , Ct = 1t 01 ∈ SL(2,R). The flow (ϕXt )t∈R is hyperbolic,that is, for every x ∈ X
theree(cid:16)xistsan(cid:1)orthogo(cid:16)naland(cid:17)(ϕXt )t R-stablesplittingofthetangentspace TxX
∈
T X = E0(x) Es(x) Eu(x)
x
⊕ ⊕
suchthatthedifferentialoftheflow(ϕXt )t R isuniformlyexpandingonEu(x),uniformlycontracting
∈
on Es(x) andisometricon E0(x) = dϕX(x) . Oncecan choose
hdt t |t=0i
d d
Es(x) = span θX(x) and Eu(x) = span ηX(x) .
dt t dt t
t=0 t=0
n (cid:12) o n (cid:12) o
General references forthissection(cid:12)are[2,7,13],andtheseworksmaybec(cid:12)onsultedfortheproofs
(cid:12) (cid:12)
to all results which are stated above. In what follows, we will drop the superscript X from (ϕXt )t R
∈
tosimplifynotation.
2.1 Poincare´ sections, stable and unstable coordinates
RecallthattheRiemannsurfaceΓ H2 iscompactifandonlyifthequotientspaceX = Γ PSL(2,R)
\ \
is compact. Then there is σ0 > 0 such that dG(g,γg) σ0 holds for all g G = PSL(2,R) and
≥ ∈
γ Γ e . In the whole paper, we assume that X is compact. First we recall the definition of
∈ \ { }
Poincare´ sectionsin Part I[12].
Definition 2.1. Let x X andε > 0. The Poincare´ sectionof radiusεat xis
∈
(x) = Γ(gc b ) : u < ε, s < ε ,
ε u s
P { | | | | }
where g Gissuchthatx = Γg.
∈
Lemma 2.1. Let X be compact and ε ]0,σ /4[. For every y (x) there exists a unique couple
0 ε
∈ ∈ P
(u,s) ] ε,ε[ ] ε,ε[suchthat
∈ − × −
y = Γ(gc b )
u s
foranyg Gsatisfyingx = Γg.
∈
Proof. Bydefinitionsuchacouple(u,s)doesexist. Toshowitsuniqueness,supposethatx = Γg =
1
Γg and y = Γg b c = Γg b c for g ,g G and (u ,s ),(u ,s ) ] ε,ε[ ] ε,ε[. Then
2 1 s1 u1 2 s2 u2 1 2 ∈ 1 1 2 2 ∈ − × −
thereare γ,γ Γ such that
′
∈
γg = g and γ g c b = g c b .
1 2 ′ 1 u1 s1 2 u2 s2
Therefore
dG(γ−1γ′g1cu1,g1cu1) = dG(γ−1g2cu2bs2 s1,g1c1) = dG(g1cu2bs2 s1,g1cu1)
− −
= dG(bs2−s1,cu1−u2) ≤ dG(bs1−s2,e)+dG(cu2−u1,e)
s s + u u < 2ε+2ε < σ .
1 2 1 2 0
≤ | − | | − |
5
From the property of σ , it implies that γ 1γ = e, so that γ = γ . Then g c b = γg c b =
0 − ′ ′ 2 u2 s2 1 u1 s1
g c b yieldsc = b ,and consequentlys = s ,u = u by consideringmatrices. (cid:3)
2 u1 s1 u2 u1 s1 s2 1 2 1 2
− −
Thusfor ε ]0,σ /4[ themapping
0
∈
(x) ] ε,ε[ ] ε,ε[, y (u,s),
ε
P → − × − 7→
such thaty = Γ(gc b )defines abijection.
u s
y u
ϕ (x)
t
x
s
( x )
ε
P
Figure2: CoordinatizationofPoincare´ section
Definition2.2. Thenumbersu = u(y)ands = s(y)arecalledtheunstablecoordinateandthestable
coordinateof y,respectively,and wewritey = (u,s) (seeFigure2).
x
We can alsodefine Poincare´ sectionsas thefollowing.
Definition 2.3. Let x X andε > 0. The Poincare´ sectionof radiusεat xis
∈
(x) = Γ(gb c ) : s < ε, u < ε
Pε′ { s u | | | | }
where g Gissuchthatx = Γg.
∈
Notethatb and c are reversedas compared to (x). Wealso havethefollowingresult.
s u ε
P
Lemma 2.2. Let X be compact and ε ]0,σ /4[. For every y (x) there exists a unique couple
∈ 0 ∈ Pε′
(s,u) ] ε,ε[ ] ε,ε[suchthat
∈ − × −
y = Γ(gb c )
s u
foranyg Gsatisfyingx = Γg.
∈
Definition 2.4. Again the numbers s = s(y) and u = u(y) are called the stable coordinate and the
unstablecoordinateof y,respectively,and wewritey = (s,u) .
′x
6
Thefollowingresultshowsthatiftwo trajectoriesoftheflow(ϕt)t R intersect aPoincare´ section
∈
and are sufficiently close in the past time (the future time, respectively), then the stable coordinates
(unstable coordinates, respectively) of the piercing points are nearly equal. A similar approximation
was usedin [18]butleft proven.
Theorem 2.1. For every ρ > 0, there exists ε = ε(ρ) > 0 satisfying the following property. If
δ ]0, σ0[ and x,x X are such that x (x) with x = (u ,s ) , for i = 1,2, then for any
∈ 8 i ∈ i ∈ Pδ i i i x
T > 0,
(a) ifd (ϕ (x ),ϕ (x )) < εfort [0,T]then u u < ρe T;
X t 1 t 2 1 2 −
∈ | − |
(b) ifd (ϕ (x ),ϕ (x )) < εfort [ T,0] then s s s s (u u ) < ρe T.
X t 1 t 2 1 2 1 2 1 2 −
∈ − | − − − |
(c) ifd (ϕ (x ),ϕ (x )) < εfort [ T,T] then u u < ρe T and s s < 3ρe T.
X t 1 t 2 ∈ − | 1 − 2| − | 1 − 2| 2 −
Proof. Let ρ > 0 be given. According to Lemma 2.4 below, we can select ε = ε (ρ) > 0 so that
1 1
dG(u,e) < ε1 and u = π(U) ∈ G for U = uu1211 uu1222 ∈ SL(2,R) yield |u12| + |u21| < ρ. Let
ε = min σ0 ,ε . (cid:16) (cid:17)
2+√2 1
(a)&(b) Writex = Γg and x = Γgc b for g G. Denotec (t) = gc b a G. By thedefinition
(cid:8) (cid:9) i ui si ∈ i ui si t ∈
ofdX, forevery t [ T,T] thereis γ(t) Γ so that dX(ϕt(x1),ϕt(x2)) = dG(c1(t),γ(t)c2(t)) < ε.
∈ − ∈
Using the property of σ , we can show that γ(t) = γ(0) for all t [ T,T]. In addition, since
0
∈ −
dG(gcu1bs1,gcu2bs2) = dG(cu1bs1,cu2bs2) ≤ |u1|+|u2|+|s1|+|s2| < 4δ < σ20, wehave
dX(ϕ0(x1),ϕ0(x2)) = dX(x1,x2) = dG(cu1bs1,cu2bs2) = dG(c1(0),c2(0)).
Thismeansthatwecantakeγ(0) = eandhenceγ(t) = eforallt [ T,T]. ThendG(c1(t),c2(t)) <
∈ −
ε forallt [ T,T], i.e.,
∈ −
dG(cu1bs1at,cu2bs2at) < ε ≤ ε1 forall t ∈ [−T,T],
orequivalently,
dG(a−tb−s2cu1−u2bs1at,e) < ε1 forall t ∈ [−T,T].
Writea b c b a = [H(t)] with
t s2 u1 u2 s1 t
− − −
1 s (u u ) e t(s s s s (u u ))
H(t) = − 2 1 − 2 − 1 − 2 − 1 2 1 − 2 SL(2,R).
et(u u ) 1+s (u u ) ∈
(cid:18) 1 − 2 1 1 − 2 (cid:19)
Whencethedefinitionofε leadsto
1
et u u +e t s s s s (u u ) < ρ for t [ T,T].
1 2 − 1 2 1 2 1 2
| − | | − − − | ∈ −
In particular,t = T andt = T imply(a) and (b), respectively.
−
(c)This followsfrom(a) and (b). (cid:3)
We alsohavea reversestatement.
7
Theorem 2.2. Forε ∈]0, σ40[. Assumethatx,x1,x2 ∈ X aresuchthatxi ∈ P5ε(x)andxi = (ui,si)x
fori = 1,2. Then foranyT > 0,
(a) if u u < εe T thend (ϕ (x ),ϕ (x )) < εforallt [0,T];
| 1 − 2| 2 − X t 1 t 2 ∈
(b) if s s s s (u u ) < εe T thend (ϕ (x ),ϕ (x )) < ε forall t [ T,0];
| 1 − 2 − 1 2 1 − 2 | 2 − X t 1 t 2 ∈ −
(c) if u u + s s < εe T thend (ϕ (x ),ϕ (x )) < εforallt [ T,T].
| 1 − 2| | 1 − 2| 2 − X t 1 t 2 ∈ −
Proof. (a) Let T > 0begivenand fix t [0,T]. Wehave
∈
dX(ϕt(x1),ϕt(x2)) = dX(Γgcu1bs1at,Γgcu2bs2at) ≤ dG(gcu1bs1at,gcu2bs2at)
= dG(c(u1−u2)etbs1e−t,bs2e−t) ≤ dG(c(u1−u2)etbs1e−t,e)+dG(bs2e−t,e)
≤ dεG(c(u1−u2)εet,e)+dG(bs1e−t,e)+dG(bs2e−t,e) ≤ |u1 −u2|et +(|s1|+|s2|)e−t
< et T +2 e t < ε.
− −
2 5
(b)First, wewritec b = b c a with
ui si s˜i u˜i τ˜i
s
i
τ˜ = 2ln(1+u s ), u˜ = u (1+u s ), s˜ = .
i i i i i i i i
− 1+u s
i i
Fort [ T,0], analogouslyto (a), weobtain
∈ −
dX(ϕt(x1),ϕt(x2)) ≤ dG(cu1bs1at,cu2bs2at) = dG(bs˜1cu˜1aτ˜1at,bs˜2cu˜2aτ˜2at)
= dG(b(s˜1 s˜2)e−tcu˜1etaτ˜1,cu˜2etaτ˜2)
−
≤ dG(b(s˜1−s˜2)e−t,e)+dG(cu˜1et,e)+dG(cu˜2et,e)+dG(aτ˜1,e)+dG(aτ˜2,e)
1
s˜ s˜ e t +( u˜ + u˜ )et + ( τ˜ + τ˜ )
1 2 − 1 2 1 2
≤ | − | | | | | √2 | | | |
4
s s s s (u u ) e t +( u˜ + u˜ )et + ( u s + u s )
1 2 1 2 1 2 − 1 2 1 1 2 2
≤ | − − − | | | | | √2 | | | |
ε 2ε 4 2ε2
< e T t + + < ε.
− −
2 5 √2 · 25
(c)Itfollowsfrom u u + s s < εe T that u u < εe T and s s s s (u u ) <
| 1− 2| | 1− 2| 2 − | 1− 2| 2 − | 1− 2− 1 2 1− 2 |
εe T which prove(c) by(a) and (b). (cid:3)
2 −
2.2 Shadowing lemma, Anosov closing lemmas, and connecting lemma
Werecall theshadowinglemmaand reformulatetheAnosovclosinglemmain Part I[12];
Denote by Ws (x) = Γ(gb ) : t < ε and Wu (x) = Γ(gc ) : t < ε for x = Γg thelocal
X,ε { t | | } X,ε { t | | }
stableand local unstablemanifoldofx ofsizeε, respectively.
8
Ws(x )
ε 1
x
1
x
x
2
Wu(x )
ε 2
Figure3: Shadowinglemma
Theorem 2.3 (Shadowinglemma). Ifε > 0, x ,x X, andx Ws (x ) Wu (x ), then
1 2 ∈ ∈ X,ε 1 ∩ X,ε 2
d (ϕ (x ),ϕ (x)) < εe t forall t [0, [
X t 1 t −
∈ ∞
and
d (ϕ (x ),ϕ (x)) < εet for all t ] ,0].
X t 2 t
∈ −∞
Theorem 2.4 (Anosov closing lemma I). Suppose that ε ]0,min 1, σ0 [, x X, T 1, and
∈ {4 8 } ∈ ≥
ϕ (x) (x). Let ϕ (x) = (u,s) , inthenotationfromDefinition2.2. Then thereare x (x)
T ∈ Pε T x ′ ∈ P2ε
with x = (σ,η) and T R so that
′ x ′
∈
ϕT′(x′) = x′ and dX(ϕt(x),ϕt(x′)) < 2 u + η < 4ε for all t [0,T].
| | | | ∈
Furthermore,
T T
′ − ln(1+us) < 5 us e T,
−
2 − | |
(cid:12) (cid:12)
eT(cid:12)′/2 +e T′/2 = eT/2 +(cid:12)e T/2 +useT/2,
(cid:12) − (cid:12)−
and
σ < 2 u e T, η s < 2s2 u +2 s e T.
− −
| | | | | − | | | | |
Using the version of Poincare´ sections in Definition 2.3, we have a respective statement for the
Anosovclosinglemmawhich willbealso usefulafterwards.
Theorem 2.5 (Anosov closing lemma II). Suppose that ε ]0,min 1, σ0 [, x X, T 1, and
∈ {4 8 } ∈ ≥
ϕT(x) ∈ Pε′(x). Let ϕT(x) = (s,u)′x, inthenotationfromDefinition2.4. Then thereare x′ ∈ P2′ε(x)
with x = (η,σ) andT R sothat
′ ′x ′ ∈
ϕT′(x) = x and dX(ϕt(x),ϕt(x′)) 2 u + η < 4ε forall t [0,T]. (2.1)
≤ | | | | ∈
9
Furthermore,
T T
′ < 4 us e T, (2.2)
−
2 − 2 | |
eT′/2 +e(cid:12)(cid:12)T′/2 = eT(cid:12)(cid:12)/2 +e T/2 +use T/2, (2.3)
− − −
(cid:12) (cid:12)
and
η s 2 s e T, σ < 2 u e T. (2.4)
− −
| − | ≤ | | | | | |
x
x ′
ϕ (x)
T
T
′
T
Figure4: Anosovclosinglemma
Proof. See Figure 4 for an illustration. By assumption, there are s,u ] ε,ε[ and g G such that
∈ − ∈
Γg = x and ϕ (x) = Γgb c . Then there is ζ Γ such that ζga = gb c or ζ = gb c a g 1. The
T s u T s u s u T −
∈ −
equation
ue Tη2 ((1+su)e T 1)η s = 0
− −
− − −
has asolutionη R satisfying η s < 2 s e T as wellas η < 2 s . Then
−
∈ | − | | | | | | |
u
σ :=
1+(s η)u ηu eT
− − −
iswell-defined and σ < 2 u e T.Put g = gb c Gand x = Γg to obtainx (x). Defining
| | | | − ′ η σ ∈ ′ ′ ′ ∈ P2′ε
T = T 2ln(1+(s η)u), (2.5)
′
− −
wehave
T T
′ = ln(1+(s η)u) 2 s η u < 4 us e T,
−
2 − 2 | − | ≤ | − || | | |
(cid:12) (cid:12)
which is(2.2). Simila(cid:12)rlyto the(cid:12) proofof theAnosovclosinglemmaI, wecan check that ζgbηcσaT′ =
(cid:12) (cid:12)
gbηcσ and hence ϕT′(x′) = x′ and we also have the latter of (2.1). Due to ζ = gbscua Tg−1 =
−
gbηcσa T′c σb ηg−1, this implies (2.3); and (2.4) can be done analogously to the Anosov closing
lemma−I. − − (cid:3)
10