Deviation of ergodic averages for rational polygonal billiards

0.3 MB·English

by  Jayadev S. Athreya

DEVIATION OF ERGODIC AVERAGES FOR RATIONAL
POLYGONAL BILLIARDS
8
0
0
J.S.ATHREYAANDG.FORNI
2

n
a ABSTRACT. Weproveapolynomialupperboundonthedeviationofer-
J godicaveragesforalmostalldirectionalflowsoneverytranslationsur-

7 face, in particular, for the generic directional flow of billiards in any
1 Euclideanpolygonwithrationalangles.

]
S
D
h. 1. INTRODUCTION AND MAIN RESULTS
t
a In a celebrated paper [20], S. Kerckhoff, H. Masur and J. Smillieproved
m
theuniqueergodicityforthedirectionalflowofarationalpolygonalbilliard
[
inalmostalldirections. Inthispaperweproveapower-lawupperboundon

4 thespeed ofergodicityforweakly differentiablefunctions.
v
Similar bounds for hyperbolic systems are usually modeled on the Law
8
5 of Iterated Logarithms (or on the Central Limit Theorem) for indepen-
6
dent stochastic processes. In such cases the deviationexponent for ergodic
1
0 integrals is universal, equal to 1/2. Among well-known models of non-
7 hyperbolic behaviour, bounds on the speed of ergodicity are provided by
0
the Denjoy-Koksma inequality for rotations of the circle or, equivalently,
/
h
for linear flows on the 2-torus. In this case, for generic systems the devia-
t
a
tionofergodicintegralsisat mostlogarithmicin time.
m
There are a few results known for dynamical systems with intermediate
:
v behaviour, that is, systems which display slow (polynomial) divergence of
i
X nearby orbits. Such systems have been called parabolic (see for instance
r [17]). An important example is the horocycle flow on the unit tangent
a
bundle of a surface of constant negative curvature. M. Burger [7] proved
polynomial upper bounds for geometrically finite surfaces and found that
the deviation exponent depends on the spectrum of the Laplace-Beltrami
operator. His results were later strengthened in [12] where a quite refined
asymptotics, including lower bounds, was proved in the case of compact
surfaces. Another class of examples where polynomial upper bounds have
been proved is given by nilflows on Heisenberg manifolds [13]. In all the
2000MathematicsSubjectClassification. 32G15,37A10.
J.S.A.supportedbyNSFgrantDMS0603636.
G.F.supportedbyNSFGrantDMS0244463.
1
2 J.S.ATHREYAANDG.FORNI
above parabolic examples, the generic system is uniquely ergodic and the
polynomialupperboundsare uniformwith respect to theinitialconditions,
insharp contrasttothehyperbolicsituation.
The dynamics of a rational polygonal billiard can be viewed as the geo-
desic flow on a flat surface with trivial holonomy, called a translation sur-
face. In fact, translation surfaces arising from billiards are rather special
among higher genus translation surfaces. Flat 2-dimensional tori are the
only translation surfaces of genus 1 and correspond to integrable billiard
tables, such as rectangles or the equilateral triangle. In this case orbits do
not diverge and the system is called elliptic. In the higher genus case, the
geodesic flow is pseudo-integrable in the sense that the phase space is fo-
liated by surfaces of higher genus. Each invariant surface is determined
by fixing an angle of the unit tangent vector with respect to the horizon-
tal direction. Since the surface has a translation structure (the flat metric
has trivial holonomy) the horizontal direction is well-defined and the an-
gle is invariant under the geodesic flow. The divergence of nearby orbits is
produced by the presence of (conical) singularities which split bundles of
nearby trajectories. By the Kerckhoff-Masur-Smillie theorem, the flow is
uniquely ergodic on almost all invariant surfaces. In this paper we prove a
polynomial bound on the speed of ergodicity for a (smaller) full measure
setofinvariantsurfaces.
The flow on each invariant surface, called the directional flow, is closely
related to interval exchange transformations (IET’s). In fact, all Poincaré
return maps on transverse intervals are IET’s. For a generic IET a polyno-
mialboundonthedeviationofergodicsumsforcertainpiece-wiseconstant
functionswasprovedbyZorichinhisgroundbreakingwork[35],[36],[37],
[38]. Intermsoftranslationflows,Zorich’sresultyieldsapower-lawupper
bound for the deviation in homology of the generic directional flow on the
genericsurface(seeTheorem1.2below). Asimilarpower-lawupperbound
on the deviation of ergodic integrals of weakly differentiable functions, to-
gether with results on lower order deviations and lower bounds on devia-
tions,wereprovedbythesecondauthorin [15], after conjecturesofZorich
[35](seealso[38])andKontsevich[21]. Weemphasizethatingenusg ≥ 2
all of the above mentioned results on deviations are for generic translation
surfaces and tell us absolutely nothing on rational billiards, which form a
setofmeasurezero. Thissituationisquiteunsatisfactory,sincethestudyof
billiards and of related mechanical systems is one of the main motivations
ofthetheory. Ourcurrent resultis afirst stepinaddressingthisgap.
1.1. Holomorphic Differentials. Let M be a Riemann surface of genus
g ≥ 1. Let Hol(M) denote the space of holomorphic differentials on M,
that is, the space of all tensors of the form f(z)dz in local coordinates.
DEVIATIONOFERGODICAVERAGESFORRATIONALPOLYGONALBILLIARDS 3
Any holomorphicdifferential ω ∈ Hol(M) determines a unique flat metric
withconicalsingularitiesatthezerosoftheholomorphicdifferential. Given
ω ∈ Hol(M), one obtains (via integrationof the form) an atlas of charts to
C ∼= R2,withtransitionmapsoftheformz 7→ z+c. Viceversa,givensuch
an atlas of charts, one obtains a holomorphic differential by pulling back
the form dz on C. For this reason pairs (M,ω) are also called translation
surfaces (this terminology was first introduced by E. Gutkin and C. Judge
in [19]). Any differential ω ∈ Hol(M) also determines a pair of transverse
oriented measured foliations, defined by {Re(ω) = 0} (vertical foliation),
{Im(ω) = 0} (horizontal foliation). These foliations have saddle-like sin-
gularities (possibly degenerate) at the zero of the holomorphicdifferential.
Theverticalandhorizontalflowsaretheflowsmovingwithunitspeedalong
the leaves of the vertical and horizontal foliations respectively, in the posi-
tive direction. More in general, a directional flow is the flow moving with
unit speed in a direction forming a fixed angle with the (positive) vertical
direction. We are interested in studyingthegeodesic flow of the flat metric
associated with any ω ∈ Hol(M). It is well-known that the unit cotangent
bundleof M is foliated with invariant surfaces of genus g ≥ 1 and that the
restriction of the geodesic flow to an invariant surface is isomorphic to a
directional flow on M. In fact, it is sufficient to study the vertical (or the
horizontal) flow for the one-paramer families of holomorphic differentials
onM obtainedby‘rotations’ofω ∈ Hol(M) (see below).
1.2. Moduli space. Let S = S be a compact surface of genus g ≥ 2.
g
Let Ω be the moduli space of unit-area holomorphic differentials on S.
g
That is, a point in Ω is an equivalence class of pairs (M,ω), where M
g
is a Riemann surface of genus g ≥ 1 (that is, a complex structure on S)
andω isaholomorphicdifferentialon M normalizedso thattheassociated
metric has unit total area. The equivalence relation is defined as follows:
two pairs (M ,ω ) and (M ,ω ) are equivalent if there is a biholomorphic
1 1 2 2
map f : M → M such that f∗ω = ω . For notational convenience,
1 2 1 2
we will write points in Ω simply as ω and M will denote the underlying
g ω
Riemannsurface.
The space Ω is an orbifold vector bundle over R , the moduli space of
g g
Riemann surfaces. The fiber over each point M ∈ R is the vector space
g
of holomorphic differentials on M. There is a natural stratification of Ω
g
byintegerpartitionsof2g −2: each stratumcan bedescribed as thesubset
ofΩ formed by all differentialswith a givenpattern ofzero multiplicities.
g
Strata are never compact and not always connected. However, they haveat
most finitely many connected components [22]. In our paper we will work
withoneoftheseconnected components,call itΩ.
4 J.S.ATHREYAANDG.FORNI
There isanaturallydefined actionofthegroupSL(2,R) of2×2 matri-
ceswithdeterminant1onthemodulispaceΩ ofholomorphicdifferentials.
g
In fact, SL(2,R) acts linearly on R2 ≡ C and its action on Ω can be de-
g
fined by post-composition with holomorphic differentials or, equivalently,
with charts in any atlas for the corresponding translation structure. Given
any translation surface (M,ω), post-composition with any A ∈ SL(2,R)
uniquely defines a translation surface (M ,ω ). The SL(2,R) action pre-
A A
servesthemultiplicitiesofzeroesoftheholomorphicdifferentials(orequiv-
alentlythe angles ofthe conical singularities),hence it preserves each stra-
tumofthemodulispace.
Weareinterestedinthedynamicsoftheverticalflowforalmostallholo-
morphic differentials in every orbit of the standard maximal compact sub-
groupSO(2,R),
cosθ sinθ
SO(2,R) = r = : θ ∈ S1 .
θ −sinθ cosθ
(cid:26) (cid:18) (cid:19) (cid:27)
The action of SO(2,R), known as the circle flow, preserves the under-
lying holomorphic structure (as well as the flat metric), so it acts as the
identitywhenprojectedtoR . Fromageometricalpointofview,thematrix
g
r simplyrotatestheverticaldirectionby angleθ ∈ S1.
θ
1.3. Main Theorem. Before we state our main result, we fix some more
notation. If we fix ω ∈ Ω, let ϕ denote the vertical flow. Let ϕ denote
t θ,t
thevertical flowassociatedto r ω (thisis simplythedirectionalflowin the
θ
direction at angle θ ∈ S1 with the positivevertical direction of ω). We say
a point x ∈ S is non-singular for θ ∈ S1 if it is not on a singular leaf of
the vertical foliation associated to r ω. Let A denote the area form on S
θ ω
associatedtoω (notethat A = A ).
rθω ω
Theorem 1.1. There is an α = α(Ω) > 0 such that the following holds.
For all ω ∈ Ω there is a measurable function K : S1 → R+ such that for
ω
almostallθ ∈ S1 (with respecttoLebesguemeasure), forallfunctionsf in
thestandardSobolevspaceH1(S) andforallnon-singularx ∈ S,
T
(1.1) f(ϕ (x))dt−T fdA ≤ K (θ)kfk T1−α.
θ,t ω ω H1(S)
(cid:12)Z0 Z (cid:12)
(cid:12) (cid:12)
We also(cid:12)have the following result for t(cid:12)he growth of homology classes:
(cid:12) (cid:12)
Fix ω ∈ Ω, and let x ∈ S be non-singular. For T > 0, define h (T) :=
ω,x
[γ¯ (T)] ∈ H (S,R), where γ¯ (T) is the closed curve given by taking
ω,x 1 ω,x
thepieceofleafγ (T) := {ϕ (x)}T,and‘closing’itupbyconnectingthe
ω,x t 0
endpoints by any given curve of bounded length. We remark that h (T)
ω,x
depends on the choice of the curve joining the endpoints, but it is uniquely
defined upto additionofauniformlyboundedterm.
DEVIATIONOFERGODICAVERAGESFORRATIONALPOLYGONALBILLIARDS 5
Theorem1.2. Thereisanα = α(Ω) > 0(thesameasinTheorem1.1)and
for all ω ∈ Ω there is a measurablefunction K′ : S1 → R+, such that for
ω
almost all θ ∈ S1, there is a homology class h ∈ H (S,R) so that for all
θ 1
non-singularx ∈ S,
(1.2) |h (T)−h T| ≤ K′(θ)T1−α.
rθω,x θ ω
Interestingly,whilegenericallythedeviationexponentinTheorem1.1or
Theorem1.2ispositive[15],therearehighergenussurfacesforwhichitcan
be zero, and in [16], an example is given of a surface of genus 3 where the
exponent is zero in almost all directions. Thus, we cannot prove a general
non-trivial lower bound for the deviation exponent. Unpublished work of
the second author suggests that such non-generic examples do not exist in
genus2. Moreprecisely,itcanbeprovedthatingenus2the(upper)second
Lyapunov exponent of the so-called Kontsevich-Zorich cocycle (see §3.1
and§4.1)isstrictlypositiveforallholomorphicdifferentialsandforalmost
all angles. In particular, the second Lyapunov exponent is strictly positive
with respect to all SL(2,R)-invariant measures. Recently, M. Bainbridge
[5] has derived explicit numerical values for the (second) exponent, with
respect to all SL(2,R)-invariant measures, from a formula found by M.
Kontsevich[21](seealso[15]foracompleteproofoftheformula). Itturns
outthatthevaluesareequalto1/3forallmeasuressupportedinthestratum
corresponding to a double zero of the holomorphic differential and to 1/2
for all measures in the stratum corresponding to two simple zeros. These
valueswereconjecturedbyKontsevichandZorich(seeforinstance[21]for
the case of the absolutely continuous invariant measures on each stratum)
buttheircalculationshaveremainedunpublished.
1.4. Rational Polygonal Billiards. A special class ofholomorphicdiffer-
entials(ortranslationsurfaces)isgivenbyrationalpolygonalbilliards. Let
P ⊂ R2 be a Euclidean polygon and let G(P) ⊂ O(2,R) be the subgroup
generated by all reflections with axis parallel to an edge of P (and passing
through the origin). The polygon P is called rational if G(P) is finite. A
necessarycondition,whichisalsosufficientifP issimplyconnected,isthat
the angles of P belong to πQ (see, for example, the excellent survey [26],
§1.3). The billiard flow on P is a discontinuousLagrangian (Hamiltonian)
flow on the unit tangent bundle T (P) ≡ P × S1. The trajectory of any
1
(x,v) ∈ T (P) moves with unit speed along a straight line in the direction
1
v ∈ S1 up to the boundary ∂P where it is reflected according to the law
of geometric optics (angle of incidence equal angle of reflection), which
followsfromtheassumptionthatcollisionswiththeboundaryare elastic.
6 J.S.ATHREYAANDG.FORNI
ThebilliardflowonarationaltableP leavesinvarianttheanglefunction
Θ : P ×S1 → R obtainedas acompositionofthecanonical projections
Θ : P ×S1 → S1 → S1/G(P)
onto the quotient S1/G(P), which can be identified to a compact interval
I(P) ⊂ R. It follows that the phase space P × S1 is foliated by the level
surfaces S = {(x,v) ∈ P × S1|Θ(x,v) = θ} which are invariant un-
P,θ
der the billiard flow and have natural translation structures induced by the
translation structure on P ⊂ R2. By following the unfolding construction
ofZemljakov-Katok[34],itispossibletoshowthatallinvarianttranslation
surfacesS forθ 6∈ Θ−1(∂I(P))canbeidentifiedwithrotationsofafixed
P,θ
translationsurfaceS ofgenusg(P) ≥ 1 sothatthebilliardflow restricted
P
to S can be identified with the directional flow on S in the direction at
P,θ P
angle θ ∈ S1 from the vertical (see the original work by E. Gutkin [18] or
thesurveybyH. Masurand S. Tabachnikov[26], §1.5).
IfP isrationalbilliardtable,letA denotetheareaformontheinvariant
P,θ
translationsurface S forallθ ∈ S1. Wehavethefollowingcorollary:
P,θ
Corollary 1.3. Let P ⊂ R2 be a rational polygon. For any θ ∈ S1, let
ψ be the restriction of the billiard flow to the invariant surface S ⊂
t,θ P,θ
P × S1. There exist an α = α(P) > 0 (depending only on the shape of
P, in particular on the stratum arising from the unfolding procedure) and
a measurablefunction K : S1 → R+ such that, for almost all θ ∈ S1, for
P
all f in the standard Sobolev space H1(S ), and all x ∈ S for which
P,θ P,θ
ψ (x) is definedfor allt > 0,
θ,t
T
(1.3) f(ψ (x))dt−T fdA ≤ K (θ)kfk T1−α.
θ,t P,θ P H1(SP,θ)
(cid:12)Z0 Z (cid:12)
(cid:12) (cid:12)
The fi(cid:12)rst estimates on the speed of erg(cid:12)odicity for billiards in polygons
(cid:12) (cid:12)
were obtained by Ya. B. Vorobets in [32], [33]. For rational polygonal
billiards his bound (which holds for Lipschitz functions vanishing on the
boundary of the billiard table) is much weaker than ours. In fact, it is far
from being polynomial. However, he can control explicitly the mean over
the angle of the deviation of ergodic averages in terms of the shape of the
billiard table, hence his result yields ergodicity for an explicit class (mea-
sure zero, but topologically large) of non-rational polygonal billiards by
the approximation method of Katok and Zemljakov [34] (see also [20]).
Vorobetsmethodsare notbased on Teichmüllertheory. We remark that the
dependence of the exponent in our Corollary 1.3 with respect to the shape
ofthebilliardtableisnotsufficientlyexplicittoderivebyapproximationan
effectiveergodicityresultfornon-rationalpolygonalbilliards.
DEVIATIONOFERGODICAVERAGESFORRATIONALPOLYGONALBILLIARDS 7
Therestofthispaperisstructuredasfollows: Insection2,wediscussthe
Teichmüllergeodesicflowandthedictionarybetweenergodicpropertiesof
foliations (or flows) on surfaces and recurrence properties of Teichmuüller
orbits, as well as recalling the results from [15] for deviations of generic
vertical flows. In section 3, we discuss the Kontsevich-Zorich cocycle, a
symplectic cocycle over the flow, and distributional generalizations intro-
duced in [15]. In section 4, we prove the key lemma for our theorems, an
estimateonthespectralgapoftheKontsevich-Zorichcocycleandofthedis-
tributionalcocycle, which combines explicit formulas from [15] with large
deviationsestimatesontheTeichmüllerflowfrom[2]. Finally,insection5,
webringtheseresultstogetherinorderto proveTheorems1.1and 1.2.
2. A DICTIONARY
Let
et 0
(2.1) A = g = : t ∈ R .
t 0 e−t
(cid:26) (cid:18) (cid:19) (cid:27)
The action of A on a stratum Ω is known as the Teichmüller geodesic
flow,sincetheprojectionofanyA-orbityieldsageodesicintheTeichmüller
metriconR (andinfact,allTeichmüllergeodesicsarisethisway). Interms
g
offoliations,ifwe writeω = (Re(ω),Im(ω)),wehave
g ω = (etRe(ω),e−tIm(ω)).
t
EachstratumΩ,whilenon-compact,isendowedwithacanonicalabsolutely
continuous, ergodic, A-invariant (in fact, SL(2,R)-invariant) probability
measure µ = µ [23]. In any stratum Ω of surfaces of genus g ≥ 2 the set
Ω
ofpointsarisingfromrationalbilliardshasµ-measure0. Asaconsequence,
resultsforµ-genericpointswhichareeasiertoobtainbymethodsofergodic
theory, do not directly apply to billiards. As we have remarked above, this
isaseriousdifficultyinthestudyofrationalpolygonalbilliards.
There is a well-studied dictionary between the ergodic properties of the
vertical flow (or foliation) associated to ω ∈ Ω, and the recurrence proper-
tiesoftheforwardgeodesictrajectory{g ω} (asimilardiscussioncanbe
t t≥0
had about the horizontal flow and the backward trajectory {g ω} ). The
t t≤0
first mainresultinthisdictionaryisknownas Masur’sCriterion:
Theorem 2.1. (Masur [24]) If the vertical foliation {Re(ω) = 0} is non-
uniquelyergodic(that is, there is more than one transverseinvariant prob-
abilitymeasure), {g ω} isdivergent inΩ.
t t≥0
Combining this with the fact that {g } is ergodic with respect to µ [23,
t
30], one obtains that for µ-almost every ω ∈ Ω, the vertical foliation is
8 J.S.ATHREYAANDG.FORNI
uniquelyergodic. However,since(thesetofsurfacesarisingfrom)billiards
havemeasurezero, thisisnot usefulforbilliards.
This difficulty was resolved by Kerckhoff, Masur and Smillie [20], who
analyzed the recurrence behavior of {g r ω} for a fixed ω and θ ∈ S1
t θ t≥0
varying. Themainresult in[20]isas follows:
Theorem 2.2. (Kerckhoff, Masur and Smillie [20]) For any ω ∈ Ω, the set
ofθ ∈ S1 forwhich{g r ω} is divergenthasmeasure0.
t θ t≥0
Thus, by Masur’s criterion, for all translation surfaces (and in particular
forallrationalpolygonalbilliards),andinalmostalldirections,thevertical
flowis uniquelyergodic.
As mentioned above, the first results on the deviation of ergodic aver-
ages for such systems were proved by Zorich [37], who proved a result
equivalent to Theorem 1.2 for generic interval exchange transformations.
Zorich [35], [38] and Kontsevich [21] conjectured precise power-laws for
the deviations of ergodic averages of smooth functions for generic interval
exchange transformations and translations flows. The Kontsevich-Zorich
conjectures, including Theorem 1.1 in the generic case, were proved by
the second author [15], with exception of the simplicity of the deviation
(Lyapunov)spectrum, recently provedby A. Avilaand M. Viana [4]. Once
again,noneoftheseresultsapplyto rationalpolygonalbilliards.
The results of [15] were obtained by careful study of the Lyapunov ex-
ponents of the Kontsevich-Zorich cocycle and of its distributional exten-
sions over the Teichmüller flow, which we discuss in the next section. In
particular,theseresultsare basedon therecurrence behaviorofgenericTe-
ichmüller trajectories to compact sets given by ergodicity. In [2] the first
author, buildingon work ofEskin-Masur[10], analyzed in depth the recur-
rence to a certain exhaustion by compact subsets of the moduli space for
almost all geodesic trajectories in every orbit of the group SO(2,R). We
willusethefollowingkeyresult:
Theorem 2.3. For any η > 0, there is a compact set C = C(η) ⊂ Ω such
thatforallω ∈ Ωand almostallθ ∈ S1,
1
limsup |{0 ≤ t ≤ T : g r ω ∈/ C}| ≤ η.
t θ
T
T→∞
Remark: This is essentially Corollary 2.4 from [2]. However, there the
compact set C depends on the basepoint ω ∈ Ω . We thank Barak Weiss
g
forpointingoutthattheproofofthisresultin[2]showsthatforanystratum
Ω ⊂ Ω , theset C can bepickedindependentlyofω ∈ Ω.
g
3. COCYCLES OVER THE TEICHMÜLLER FLOW
DEVIATIONOFERGODICAVERAGESFORRATIONALPOLYGONALBILLIARDS 9
3.1. The Kontsevich-Zorich cocycle. The Kontsevich-Zorich cocycle is
a multiplicative symplectic cocycle over the Teichmüller geodesic flow on
themodulispaceofholomorphicabeliandifferentialsoncompactRiemann
surfaces. This cocycle appears in the study of the dynamics of interval
exchange transformationsand of translationflows on surfaces, for which it
represents a renormalization dynamics, and of the Teichmüller flow itself.
In fact, the study of the tangent cocycle of the Teichmüller flow can be
reduced tothat oftheKontsevich-Zorichcocycle.
Let Ω˜ be the Teichmüller space of abelian differentials on a closed sur-
g
face S of genus g ≥ 1. Since points in Ω˜ are pairs (M,ω), with M a
g
marked Riemann surface, the trivial cohomology bundleΩ˜ ×H1(S,R) is
g
well-defined. TheKontsevich-Zorichcocycle{Φt}t∈R can bedefinedas the
projectionofthetrivialcocycle
(3.1) g ×id : Ω˜ ×H1(S,R) → Ω˜ ×H1(S,R)
t g g
ontotheorbifoldvectorbundleH1(S,R)overΩ defined as
g g
(3.2) H1(S,R) := Ω˜ ×H1(S,R) /Γ .
g g g
The mapping class group Γ acts(cid:0)naturally on th(cid:1)e bundle Ω˜ × H1(S,R)
g g
sinceit actsnaturallyon thereal cohomologyH1(S,R) bypull-back.
The cohomology bundleH1(S,R) can be endowed with the structure of
g
a smooth euclidean bundle with respect to the Hodge inner product. In
fact, by the Hodge theory on Riemann surfaces ( [11], III.2), any real co-
homology class c ∈ H1(M,R) can be represented as the real (or imagi-
nary)part ofa holomorphicdifferentialh ∈ Hol(M) onaRiemann surface
M. Let ω ∈ Ω and let M be the underlying Riemann surface. For any
g ω
c ∈ H1(M ,R),itsHodgenormis defined as
ω
i
(3.3) kck2 := h∧h if c = [Re(h)], h ∈ Hol(M ).
ω 2 ω
ZS
Weremark thattheHodgenormisdefined intermsoftheRiemann surface
M but itisotherwiseindependentofthedifferentialω ∈ Ω .
ω g
Real cohomologyclassesonS can alsoberepresented intermsofmero-
morphic functions in L2(S) (see [15], §2). In fact, any holomorphic dif-
ω
ferential ω ∈ Ω induces an isomorphism between the space Hol(M ) of
g ω
all holomorphicdifferentials on M and the subspace M of meromorphic
ω ω
functions in L2(S). Such a subspace can be characterized as the space of
ω
all meromorphic functions with poles at the zeros of ω of orders bounded
intermsoftheorderofthecorrespondingzero [14], [15].
Any holomorphicdifferential h ∈ Hol(M ) can be written in terms of a
ω
meromorphicfunctionm ∈ L2(S) as follows:
ω
(3.4) h := mω, m ∈ M .
ω
10 J.S.ATHREYAANDG.FORNI
Thefollowingrepresentationofreal cohomologyclassestherefore holds:
(3.5) c ∈ H1(S,R) ⇐⇒ c = [Re(mω)], m ∈ M .
ω
Themapc : M → H1(S,R)givenbytherepresentation(3.5)isbijective
ω ω
and it is in fact isometric if the space M is endowed with the Euclidean
ω
structure induced by L2(S) and H1(S,R) with the Hodge norm k · k in-
ω ω
troducedin (3.3). In fact, thefollowingidentityholds:
(3.6) kc (m)k2 := |m|2dA , forall m ∈ M .
ω ω ω ω
ZS
The Kontsevich-Zorich cocycle was introduced in [21] as a continuous-
time version of the Zorich cocycle. The Zorich cocycle was introduced
earlier by A. Zorich [36], in order to explain polynomial deviations in the
homological asymptotic behavior of typical leaves of orientable measured
foliationsoncompactsurfaces,aphenomenonhehaddiscoveredinnumer-
ical experiments [35]. We recall that the real homology H (S,R) and the
1
real cohomologyH1(S,R) of an orientable closed surface S are (symplec-
tically)isomorphicbyPoincaré duality.
Zorich proved in [36] that the integrability condition of Oseledec’s mul-
tiplicative ergodic theorem is satisfied for a suitable acceleration of the
Rauzy-Veech induction, now often called the Zorich induction. The inte-
grabilityconditionisimmediatefortheKontsevich-Zorichcocycle. Infact,
thelogarithmoftheHodgenormofthecocycleisaboundedfunctiononthe
modulispace,henceitisintegrablewithrespecttoanyprobabilitymeasure.
Since the Kontsevich-Zorich cocycle is symplectic, its Lyapunov spec-
trum is symmetric with respect to the origin. Hence for any g -invariant
t
ergodicprobabilitymeasureµonΩ ,theLyapunovspectrumover(g ,µ)is
g t
equalto an ordered setoftheform:
(3.7) λµ ≥ ··· ≥ λµ ≥ λµ = −λµ ≥ ··· ≥ λµ = −λµ.
1 g g+1 g 2g 1
A crucial property of the cocycle is the following ‘spectral gap’ theorem,
proved by W. Veech [31] for a class of measures satisfying certain integra-
bilityconditions(includingthecanonicalabsolutelycontinuousg -invariant
t
measure on any connected component of any stratum of the moduli space)
andgeneralized by thesecondauthor[15], [16].
Theorem 3.1. For any g -invariant ergodic probability measure µ on the
t
modulispaceΩ ,thefollowinginequalityholds:
g
(3.8) λµ < λµ = 1 .
2 1
We recall that Zorich [35], [36], [38] and Kontsevich [21] conjectured
that,ifµis hecanonical absolutelycontinuousg -invariantmeasureon any
t