Invariant Radon measures for horocycle flows on Abelian covers.

27 Pages·2004·0.87 MB·English

INVARIANT RADON MEASURES FOR HOROCYCLE FLOWS
ON ABELIAN COVERS
OMRISARIG
Dedicated to the memory of M. Babillot
Abstract. We classify the ergodic invariant Radon measures for horocycle
flows on Zd–covers of compact Riemannian surfaces of negative curvature,
thus proving a conjecture of M. Babillot and F. Ledrappier. An important
tool is a result in the ergodic theory of equivalence relations concerning the
reductionoftherangeofacocyclebytheadditionofacoboundary.
1. Introduction
This paper treats the problem of classifying the invariant measures of horocycle
flows on unit tangent bundles of connected hyperbolic surfaces. The first result of
this kind is due to Furstenberg [F], who showed that in the compact case, there
is just one invariant probability measure (proportional to the Liouville measure of
the geodesic flow). Dani [D] extended this to finite volume surfaces, and showed
thateveryergodicinvariantmeasureiseitherproportionaltotheLiouvillemeasure
or is supported on a periodic orbit. Ratner’s theory [R] implies that there are no
finite invariant measures in the infinite volume case, other than those supported
on periodic orbits. There are however invariant Radon measures. Burger [Bu]
classified these for a large class of geometrically finite hyperbolic surfaces.
The geometrically infinite situation is still mainly open. The only result I am
aware of is of Babillot & Ledrappier [BL], who considered regular connected Zd–
covers of compact hyperbolic surfaces. These are surfaces M for which there exists
a compact hyperbolic surface M and a regular covering (cid:102)map p : M → M such
that the group of deck transformations G is isomorphic to Zd (see §5(cid:102).2 for details).
Let h and g be the horocycle and geodesic flows on the unit tangent bundle of M.
Babi(cid:101)llot&(cid:101)Ledrappiershowedthatforeverycontinuoushomomorphismα:G→(cid:102)R,
there exists an h–ergodic invariant Radon measure (e.i.r.m.) m such that
α
(cid:101) mα◦D∗ =eα(D)mα for all D ∈G. (1)
They also showed that this measure is unique up to a multiplicative factor, and
that it is g–quasi-invariant. This gives a d–parameter family of h–e.i.r.m.’s, and it
is natural(cid:101)to ask whether there are others. It was proved in [Ba(cid:101)] and [ASS] that
this family contains (up to a constant) all e.i.r.m.’s which are g–quasi-invariant.
BabillotandLedrappier[BL]conjecturedthatallh–e.i.r.m’sareg(cid:101)–quasi-invariant,
or equivalently that every h–e.i.r.m. is proportion(cid:101)al to mα for so(cid:101)me α.
Our aim here is to prove(cid:101)this, and thus obtain the first classificationof invariant
Radon measures for the horocycle flow on a geometrically infinite surface.
Date:April7,2004.
1
2 OMRISARIG
In fact, our result is valid in a more general setting, which we now explain. Let
M bearegularZd–coverofacompactconnectedorientableC∞Riemanniansurface
M(cid:102)whosesectionalcurvaturesarenegative, andletSM beitsunittangentbundle.
Let gs be the geodesic flow on SM. Using the work o(cid:102)f Margulis [M], Marcus [M1]
cons(cid:101)tructed a continuous flow ht(cid:102):SM →SM for which (see §5.1):
(a) The h–orbit of x is equ(cid:101)al to(cid:102)Wss(x)(cid:102):={y ∈SM :d(gsx,gsy)−−−→0};
s→∞
(b) Ther(cid:101)e exists µ such that ∀s,t, g−s◦ht◦gs =h(cid:102)µst. (cid:101) (cid:101)
We regard this flow to be a generaliza(cid:101)tion o(cid:101)f th(cid:101)e hor(cid:101)ocycle flow to the variable
negative curvature case. We prove:
Theorem 1. For every homomorphism α : G → R there exists an h–invariant
ergodic Radon measure mα which satisfies (1). This measure is uniq(cid:101)ue up to a
multiplicative factor. Every h–ergodic invariant Radon measure is of this form.
(cid:101)
The proof of Theorem 1 is based on some technical results in the ergodic theory of
equivalence relations. These are presented in the next section.
Remark. TheassumptionthatGisAbeliancannotberemoved: Theorem1andthe
ergodicdecompositionimplythateveryi.r.m. whichsatisfies(1)isergodic. Butthis
is false for general finitely generated groups G even for α≡0 (Kaimanovich [Ka]).
Theergodicityofm intheAbeliancasewasfirstprovedbyBabillot&Ledrappier
α
[BL]. There are other proofs by Pollicott [Po1], Coudene [C], and Solomyak [So].
Babillot [Ba], Kaimanovich [Ka], Coudene [C], and Pollicott [Po2] give ergodicity
results for certain classes of non-Abelian G’s. It would be interesting to know if
theorem 1 extends to these cases.
Acknowledgments: The author thanks Jon Aaronson for indicating an important
gapinanearlierversionofthispaper,MaxForester,FrancoisLedrappier,MarcPol-
licott, and Rita Solomyak for useful discussions, and Benjamin Weiss for informing
me of [M] and [M1, M2].
2. Cocycle Reduction
Let (X,F) be a polish space together with its Borel σ–algebra. A countable
Borel equivalence relationonX isanequivalencerelationG∈F⊗F whoseequiva-
lence classes are countable. A G–holonomy is a bi-measurable bijection κ:A→B
with A,B ∈ F s.t. for all ω ∈ X, ω,κ(ω) ∈ G. A measure m on (X,F) is
called G–invariant, if every G–holonom(cid:0)y satisfi(cid:1)es m◦κ| =m| . A mea-
dom(κ) dom(κ)
sure m is called (G,Ψ)–conformal for some Ψ : G → R, if for every G–holonomy,
m◦κ| ∼ m| and dm◦κ(x) = expΨ(x,κx). A measure m on (X,F)
dom(κ) dom(κ) dm
is called G–ergodic if every f ∈ L1 which is invariant under all G–holonomies
is constant. We say that a Borel property P(x,y) holds a.e. in G if the set
{x:P x,y holds for all y equivalent to x} has full measure. Such sets are always
(cid:0) (cid:1)
measurable for countable Borel equivalence relations [FM].
Let (cid:104)G,+(cid:105) be a polish Abelian group with a norm (cid:107)·(cid:107) such that every bounded
subset of G is precompact (e.g. Rd or Zd with the euclidean norm). A G–valued
G–cocycle is a function Φ:G→G for which
(ω ,ω ),(ω ,ω )∈G=⇒Φ(ω ,ω )+Φ(ω ,ω )=Φ(ω ,ω ).
1 2 2 3 1 2 2 3 1 3
INVARIANT RADON MEASURES FOR HOROCYCLE FLOWS ON ABELIAN COVERS 3
Every G–valued G–cocycle determines an equivalence relation G on X×G via
Φ
G :={ (x,ξ),(y,η) ∈(X×G)2 :(x,y)∈G and η−ξ =Φ(x,y)}.
Φ
(cid:0) (cid:1)
It turns out that the orbit equivalence relation of a suitable Poincar´e section of
h is of this form, and this leads us to consider the general problem of describing
(cid:101)GΦ–invariant measures in terms of G and Φ.
Let m be a G –ergodic invariant measure on X ×G, and assume m is locally
Φ
finite: m(X×K)<∞ for all compact K ⊆G. Define
H :={a∈G:m◦Q ∼m} where Q (x,ξ)=(x,ξ+a).
m a a
It follows from [ANSS] that this is a closed subgroup of G, and that
Lemma 1. If H =G, then there exists a continuous homomorphism α :G→R
m
such that dm = e−α(ξ)dν (x)dm (ξ) where ν is a finite (G,α ◦ Φ)–conformal
α G α
measure on X, and m is a Haar measure of G.
G
This gives a classification of locally finite G –ergodic invariant measures, when
Φ
H = G. The problem is, of course, that H can be much smaller than G. The
m m
following observation is very useful in this respect (for a proof see §3.1):
Lemma 2. Let H be a closed subgroup of G and suppose ∃u : X → G Borel s.t.
Φ (x,y):=Φ(x,y)+u(x)−u(y)∈H a.e. in G . Set ϑ (x,ξ):= x,ξ−u(x)−c .
u Φ c
Then ∃c ∈ G s.t. m◦ϑ−1 is a G –ergodic invariant measure(cid:0)on X ×H, an(cid:1)d
Hm◦ϑ−c1 =Hm. If esssupc(cid:107)u(cid:107)<∞,Φtuhen m◦ϑ−c1 is locally finite.
Cocyclesoftheformu(x)−u(y)arecalledcoboundries. Twococycleswhichdiffer
byacouboundrya.e. inGaresaidtobecohomologous mod m. Thefunctionu(x)
is called the transfer function. The process of reducing the range of a cocycle by
adding a coboundry is called cocycle reduction. The lemma suggests using cocycle
reduction in cases when H (cid:54)=G. The crux of this paper is that this is possible:
m
Theorem 2 (CocycleReductionTheorem). LetGbeacountableBorelequivalence
relation on a polish space X, and let Φ:G→R be a Borel cocycle. For every G –
Φ
ergodic invariant locally finite measure m on X ×R, there exists a Borel function
u:X →R such that Φ(x,y)+u(x)−u(y)∈H m–a.e. in G .
m Φ
Weshowlaterthattherangeofthecocyclecannotbereducedanyfurther(§3.1,
Lemma 8). It is not true in general that esssup(cid:107)u(cid:107) < ∞ or that the cohomology
holds at every point (see §6). This is however the case for the cocycles used in the
proof of Theorem 1. We describe these cocycles.
Let (Σ+,T) be a one–sided subshift of finite type with finite set of states S and
transition matrix A=(tij)S×S, tij ∈{0,1}, i.e.,
Σ+ ={(x ,x ,...)∈SN∪{0} :∀i, t =1}
0 1 xixi+1
with the metric d(x,y) := 1(1−δ ), and the map T : Σ+ → Σ+ given
(cid:80)i≥0 2i xiyi
by T(x ,x ,...) = (x ,x ,...). Let φ : Σ+ → Rd be a function with summable
0 1 1 2
variations: var φ<∞ where var φ=sup{(cid:107)φ(x)−φ(y)(cid:107) :xn−1 =yn−1}.
n n 2 0 0
The tail r(cid:80)elation and grand tail relation of (Σ+,T) are defined, respectively, by:
T := {(x,y)∈Σ+×Σ+ :∃n s.t Tnx=Tny}
T := {(x,y)∈Σ+×Σ+ :∃p,q s.t. Tpx=Tqy}.
(cid:101)
4 OMRISARIG
Let Per be a collection of periodic points in Σ+ which intersects every periodic
orbit of T at exactly one point. If x is preperiodic, set n(x):=inf{k :Tkx∈Per}
and u (x):= n(x)−1φ(Tkx). Set φ :=φ+φ◦T +...+φ◦Tn−1, and define
Per (cid:80)k=0 n
Φ:T→Rd and Φ:T→Rd by
Φ(x,y) := (cid:101) (cid:101) [φ(Tny)−φ(Tnx)]
(cid:80)n≥0
φ (y)−φ (x) x,y not preperiodic and Tpx=Tqy, (2)
Φ(x,y) := (cid:40) q p
u (y)−u (x) x,y are preperiodic.
(cid:101) Per Per
Two sided versions of these cocycles are considered in [KS].
We show in §5.3 that there is a correspondence between h–ergodic invariant
Radon measures and T –ergodic invariant locally finite meas(cid:101)ures, for a suitable
Φ(cid:101)
choice of Σ+ and φ. T(cid:101)his reduces the proof of theorem 1 to the classification of
T –ergodic invariant locally finite measures. Set Fix(Tn):={x:Tnx=x} and
Φ(cid:101)
(cid:101)
H := (cid:104)φ (x)−φ (y):x,y ∈Fix(Tn),n∈N(cid:105),
φ n n
H := (cid:104)φ (x):x∈Fix(Tn),n∈N(cid:105).
φ n
The following sta(cid:101)tements are proved in §4:
Proposition 1. ∃u : Σ+ → Rd with summable variations and c ∈ Rd such that
φ φ
φ+u −u ◦T +c ∈ H , φ+u −u ◦T ∈ H , and H = H +c Z. H
φ φ φ φ φ φ φ φ φ φ φ
(resp. H ) is contained in any closed group H for(cid:101)which ∃u(cid:101)continuous such that
φ
φ+u−(cid:101)u◦T +c∈H for some c (resp. φ+u−u◦T ∈H).
Proposition 1 generalizes Proposition 5.1 in [PS]. In what follows we use the
abbreviation e.i.l.f. for ‘ergodic, invariant, and locally finite’.
Theorem 3. Let m be a measure on Σ+×Rd.
(1) If m is T –e.i.l.f., then Φ(x,y)+u (x)−u (y)∈H everywhere in T, and
Φ φ φ m
H =H .
m φ
(2) If m is T –e.i.l.f., then Φ(x,y)+u (x)−u (y)∈H everywhere in T, and
Φ(cid:101) φ φ m
H =H(cid:101) . (cid:101) (cid:101)
m φ
(cid:101)
The description of H in terms of weights of periodic orbits generalizes Bowen’s
m
descriptionoftheratiosetoftheAnosovfoliationsw.r.t. thevolumemeasure[B2]
(see also Series [S]). Theorem 3 and Lemma 2 lead to
Theorem 4. Define for c∈Rd,ϑ (x,ξ)=(x,ξ−u (x)−c), and let m , m be
c φ Hφ H(cid:101)φ
the Haar measures on H ,H . Let m be a measure on Σ+×Rd.
φ φ
(1) If m is T –e.i.l.f.,(cid:101)then dm◦ϑ−1(x,ξ) = e−(cid:104)a,ξ(cid:105)dν (x)dm (ξ) for some
Φ c a Hφ
c,a ∈ Rd and νa finite s.t. dνdaν◦aT ∝ exp[(cid:104)a,φ+uφ −uφ ◦T(cid:105)]. Such a
measure exists for all a∈Rd, and is unique up to a constant factor.
(2) If m is T –e.i.l.f., then dm◦ϑ−1(x,ξ) = e−(cid:104)a,ξ(cid:105)dν (x)dm (ξ) for some
Φ(cid:101) c a H(cid:101)φ
cm,eaas∈urRe(cid:101)dexaisntds iνffa Pfinit(e(cid:104)αs,.tφ.(cid:105))d=νdaν◦a0T, a=ndewxph[e(cid:104)na,iφt e+xiustφs−ituisφu◦nTiq(cid:105)u].e uSpuctho aa
top
constant factor.
Part (1) generalizes Theorem 2.2 in [ANSS], by removing the assumptions of
finite memory and aperiodicity. Part (2) generalizes Theorem 3.1 in [ASS], by
removing the assumptions that m is quasi–invariant under σs(x,t,ξ)=(x,t+s,ξ)
INVARIANT RADON MEASURES FOR HOROCYCLE FLOWS ON ABELIAN COVERS 5
and that φ is non-arithmetic. Theorem 1 follows from Theorem 4, Part (2) and a
symbolic dynamics argument. This is explained in detail in §5.
3. Proof of the Cocycle Reduction Theorem
3.1. Preliminaries on Equivalence Relations. Throughoutthissection,Gisa
countableBorelequivalencerelationonapolishspace(X,d),Gisalocallycompact
Abelian polish group with norm (cid:107)·(cid:107) such that bounded sets are precompact, and
Φ:G→G is a Borel cocycle. We also fix some G –ergodic invariant locally finite
Φ
measure m on X×G. The lemmas in this section are standard.
Lemma 3. If κ is a G–holonomy, then κ (x,ξ) = (κx,ξ +Φ[x,κx]) is a G –
Φ Φ
holonomy, and these holonomies generate G : (x,ξ) ∼ (y,η) iff ∃ a G–holonomy
Φ
κ for which (y,η)=κ (x,ξ).
Φ
In what follows, κ and κ are always assumed to be related in this way.
Φ
Lemma 4. If m is G –ergodic, and A,B ⊆ X ×G are measurable with positive
Φ
measure, then there exists a G –holonomy κ such that m[A∩κ (B)] > 0. In
Φ Φ Φ
particular, almost every x∈A has a holonomy κ such that κ (x)∈B.
Φ Φ
Proof. The equivalence classes of G are countable. By a theorem of Feldman and
Φ
Moore [FM], there is a countable group of transformations G acting on X×G s.t.
(x,ξ),(y,η) ∈G iff ∃S ∈G s.t. y =S(x).
Φ
(cid:0) (cid:1)
ThesetE := S(B)ismeasurable,becauseGiscountable. ItisG –invariant,
by choice of G(cid:83).S∈SGince m is ergodic, E = X ×Gmodm, whence m(EΦ∩A) > 0.
The countability of G implies that ∃S ∈ G s.t. m[A∩S(B)] > 0, and this S is a
holonomy by definition.
The second statement of the Lemma follows from this, because if E :={x∈A:
S(x) (cid:54)∈ B for all S ∈ G}, then m(E) = 0, otherwise ∃S ∈ G,x ∈ E s.t. S(x) ∈ B
in contradiction to the definition of E. (cid:3)
Lemma 5. For every B ∈ F of positive measure, set G[B] := G∩(B×B). If m
is G–ergodic, then m| is G[B]–ergodic.
B
Proof. Suppose f : B → R is G[B]–invariant. Define F : X → R by F(x) = f(y)
where y ∈ B satisfies (x,y) ∈ G. It is easy to see that this is a proper definition,
and the previous lemma says that F is defined almost everywhere. This function
is measurable, because if G={Si}i≥1 is as in the previous proof, then
∞
[F <t]= S−1[f <t].
(cid:91) i
i=1
ItisobviousthatF isG–invariant,andthereforeF isequala.e. toaconstant. But
F| =f, so f is also equal a.e. to constant. (cid:3)
B
Lemma 6. Suppose G = Rd, H ≤ G is closed, and µ is G–ergodic. If ∃u (x)
0
measurable s.t. Φ(x,y) + u (x) − u (y) ∈ H a.e. in G[B] for B with positive
0 0
measure, then ∃u(x) measurable s.t. Φ(x,y)+u(x)−u(y)∈H a.e. in G.
Proof. DefineU :X →G/HbyU(x)=u (y)+Φ(y,x)+Hwhenevery ∈B,(x,y)∈
0
G. Such a y exists for a.e. x, because of the ergodicity of G. This is a proper
definition, because for a.e. x, if y,y(cid:48) ∈B and (x,y),(x,y(cid:48))∈G then
[u (y)+Φ(y,x)]−[u (y(cid:48))+Φ(y(cid:48),x)]=Φ(y,y(cid:48))+u (y)−u (y(cid:48))∈H.
0 0 0 0
6 OMRISARIG
For a.e. x∈X, ∃y ∈B s.t. (x,y)∈G. For this y and all x(cid:48) s.t. (x,x(cid:48))∈G,
U(x)−U(x(cid:48))+Φ(x,x(cid:48))=u (y)+Φ(y,x)−u (y)−Φ(y,x(cid:48))+Φ(x,x(cid:48))+H=H.
0 0
SinceH≤G=Rdisclosed,∃c:Rd →Rdmeasurables.t. ∀ξ ∈Rd, c(ξ)+H=ξ+H
and ξ−ξ(cid:48) ∈H⇒c(ξ)=c(ξ(cid:48)). The function C(ξ+H):=c(ξ)+H is well–defined
and Borel measurable. A standard calculation demonstrates that u(x):=C(U(x))
satisfies our requirements. (cid:3)
Lemma 7. Let m be a σ–finite G–invariant Borel measure on X. Then there
exists a standard probability space (Y,D,π) and a family {µ : y ∈ Y} of σ–finite
y
G–ergodic invariant Borel measures on X such that:
(1) µ ⊥µ whenever y (cid:54)=y ;
y1 y2 1 2
(2) y (cid:55)→µ (E) is D–measurable for every E ⊆X Borel;
y
(3) m(E)= µ (E)dπ(y) for all E ⊆X Borel.
(cid:82)Y y
Proof. Since G is a Borel equivalence relation with countable equivalence classes,
there exists a countable group of bi-measurable invertible Borel maps G such that
(x,y) ∈ G ⇔ ∃S ∈ G s.t. y = Sx ([FM], Theorem 1). The Lemma now follows
from the ergodic decomposition for countable group actions ([Sch], Cor. 6.9). (cid:3)
Proof of Lemma 2. Suppose Φ (x,y) := Φ(x,y) + u(x) − u(y) ∈ H a.e. in
u
G and set ϑ (x,ξ) := x,ξ −u(x)−c . One checks that κ ◦ϑ−1 = ϑ−1 ◦κ .
Consequentlyc, ifmisG(cid:0) –ergodicinvar(cid:1)iant, thenm◦ϑ−1 isGΦ –ecrgodicicnvariaΦnut.
It is easy to see thatΦϑ−1 commutes with Q (x,ξ) =c (x,ξ+Φua) for every a ∈ G.
c a
It fWolelowclsaitmhatthaHtm∃◦cϑ−c∈1G=fHormw.hich m is carried by X×H. Consider F :X×G→
G/H given by F(x,ξ) := ξ −u(x)+H. This function is G –invariant, whence
Φ
constant. Therefore, ∃c s.t. F(x,ξ) = c+H m–a.e. Therefore, m is carried by
{(x,ξ):ξ−u(x)∈c+H}=ϑ−1(X×H).
c
Now suppose that m is locally finite and (cid:107)u(cid:107) ∈ L∞(m). For every K ⊆ G
compact, ϑ−1(X ×K) ⊆ X ×{ξ ∈ G : dist(ξ,K) ≤ (cid:107)c(cid:107)+ esssup(cid:107)u(cid:107)}modm.
c
By assumption bounded subsets of G are precompact and m is locally finite, so
m◦ϑ−1(X×K)<∞. (cid:3)
c
Lemma 8. Suppose m is G –ergodic and invariant. Every closed group H ≤ G
Φ
such that Φ is cohomologous mod m to an H–valued cocycle must contain H .
m
Proof. Letϑ beasinLemma2. Thenm◦ϑ−1 issupportedinX×H,andtherefore
c c
m◦ϑ−c1◦Qa ⊥m◦ϑ−c1 for all a(cid:54)∈H. It follows that H⊇Hm◦ϑ−c1 =Hm. (cid:3)
3.2. Square holes and the proof of Theorem 2. Let G be a countable Borel
equivalence relation on a polish space (X,d), and suppose G is a normed locally
compact polish Abelian group.
Definition 1. A Borel measure m on X ×G is said to have square holes if there
exists B ⊆X Borel, ∅(cid:54)=U ⊆G open s.t. m(B×G)>0 and m(B×U)=0.
Proposition 2. Let Φ:G→G be a Borel cocycle, and suppose m is a G –e.i.l.f.
Φ
measure on X×G. If m has no square holes, then H =G.
m
INVARIANT RADON MEASURES FOR HOROCYCLE FLOWS ON ABELIAN COVERS 7
Proof. Assume by way of contradiction that m (cid:54)∼ m◦Q for some a ∈ G. Then
a
necessarily m(cid:54)∼m◦Q−a. By Lemma 3, Qa commutes with all GΦ–holonomies, so
m◦Q±a are GΦ–ergodic invariant. Ergodic measures are equivalent or mutually
singular, so m⊥m:=m◦Qa+m◦Q−a.
Choose f : X ×G → R uniformly continuous with the following properties:
+
fdm=1, fdm< 1, and m[f >0]<∞. Since f is uniformly continuous,
(cid:82) (cid:82) 4
d(x,y)<δ 1
∃δ >0 s.t. (cid:107)ξ−η(cid:107)<δ (cid:27)=⇒|f(x,ξ)−f(y,η)|< 4m[f (cid:54)=0].
We use the no square holes condition to construct a G–holonomy κ such that
dom(κ)×G=X×Gmodm for which ∀x∈dom(κ),
(H1) d(x,κx)<δ;
(H2) min (cid:107)Φ(x,κx)−a(cid:107),(cid:107)Φ(x,κx)+a(cid:107) <δ.
(cid:8) (cid:9)
Before constructing κ, we show how it can be used to complete the proof. The
inverse of κ satisfies (H1),(H2) for all x∈im(κ), so
1
min{|f ◦κ−Φ1−f ◦Qa|,|f ◦κ−Φ1−f ◦Q−a|}< 4m[f (cid:54)=0].
Since dom(κ )=X×Gmodm, we have
Φ
1=(cid:90) fdm=(cid:90) fdm◦κΦ =(cid:90) f ◦κ−Φ1dm=(cid:90) f ◦κ−Φ1dm
κΦ[f(cid:54)=0]
1 1 1
≤(cid:90) (cid:18)f ◦Qa+f ◦Q−a+ 4m[f (cid:54)=0](cid:19)dm≤(cid:90) fdm+ 4 < 2.
κΦ[f(cid:54)=0]
This contradiction proves that m∼m◦Q for all a, whence H =G.
a m
Construction of κ: Letα beacountablepartitionofX intopairwisedisjoint Borel
sets with diameter less than δ, and choose some Borel probability measure P ∼m
2
(such a measure exists because m is σ–finite1).
Fix A ∈ α, and let H[A] be the collection of G–holonomies κ(cid:48) with (H2) s.t.
dom(κ(cid:48)),im(κ(cid:48)) ⊆ A. This collection is not empty, because it contains the empty
function. There is a natural partial order on H[A]:
κ (cid:52)κ iff dom(κ )⊆dom(κ ) and κ | =κ .
1 2 1 2 2 dom(κ1) 1
Choose some (cid:15) ↓ 0, and define by induction H [A] ⊆ H[A], s ≥ 0, κ ∈
n n n n
H [A] as follows: If n = 0, H [A] := H[A], κ ∈ H [A] is arbitrary, and s :=
n 0 0 0 0
sup{P[dom(κ(cid:48))×G]:κ(cid:48) ∈H [A]}. If n>1, then
0
Hn[A] := {κ(cid:48) ∈Hn−1[A]:κ(cid:48) (cid:60)κn−1}
s := sup{P[dom(κ(cid:48))×G]:κ(cid:48) ∈H [A]}
n n
and choose some κ ∈H [A] for which P[dom(κ )×G]>s −(cid:15) .
n n n n n
By construction, κ (cid:52) κ . Let κ : dom(κ) → im(κ) be their common exten-
n n+1
sion. This extension obviously satisfies (H2). We claim that κ is maximal in the
sense that κ ∈ H[A],κ (cid:60) κ ⇒ P[dom(κ)×G] = P[dom(κ)×G]. This is because
κ,κ∈H [A] for all n, so
n
s −(cid:15) ≤P[dom(κ )×G]≤P[dom(κ)×G]≤P[dom(κ)×G]≤s
n n n n
1Gisσ–compact,becausetheidentityhasacountablesystemofcompactneighborhoodsand
Gisseparable. Sincemislocallyfinite,itmustbeσ-finite.
8 OMRISARIG
whence 0≤P[dom(κ)×G]−P[dom(κ)×G]≤(cid:15) −−−−→0.
n
n→∞
We use this to show that either dom(κ)×G = A×Gmodm, or im(κ)×G =
A×Gmodm. Define A := A\dom(κ), A := A\im(κ), and suppose by way of
1 2
contradictionthatP[A ×G]>0 and P[A ×G]>0. LetU :={ξ ∈G:(cid:107)ξ(cid:107)< δ},
1 2 1 2
U :={ξ ∈G:(cid:107)ξ−a(cid:107)< δ}. Since m has no square holes,
2 2
m(A ×U )>0 and m(A ×U )>0.
1 1 2 2
Ergodicity implies that ∃E ⊆ A ×U of positive measure (i = 1,2) with a G
i i i Φ
holonomy E →E (Lemma 4). W.l.o.g. this holonomy is of the form
1 2
(x,ξ)(cid:55)→ κ(cid:48)x,ξ+Φ(x,κ(cid:48)x) , where κ(cid:48) is a G-holonomy.
(cid:0) (cid:1)
(If not, use Lemma 3.)
Let µ be the Borel measure on X given by µ(A) := P(A × G). The set
proj(E ):= {x ∈ X : ∃ξ ∈ G s.t. (x,ξ) ∈ E } is analytic. The universal measura-
1 1
bilitytheorem(seee.g. [Ar],Theorem3.2.4)saysthat∃A(cid:48),A(cid:48)(cid:48) ⊆X Borelsuchthat
1 1
A(cid:48) ⊆proj(E )⊆A(cid:48)(cid:48)andµ(A(cid:48)(cid:48)\A(cid:48))=0. Clearlyµ(A(cid:48))=P(A(cid:48)(cid:48)×G)≥P(E )>0,
1 1 1 1 1 1 1 1
and if κ(cid:48)(cid:48) :A(cid:48) →κ(cid:48)(cid:48)(A(cid:48)) is the restriction of κ(cid:48) to A(cid:48), then
1 1 1
(1) dom(κ(cid:48)(cid:48))⊆A =A\dom(κ) and im(κ(cid:48)(cid:48))⊆A =A\im(κ),
1 2
(2) ∀x ∈ dom(κ(cid:48)(cid:48)), (cid:107)Φ(x,κ(cid:48)(cid:48)x)−a(cid:107) < δ, since ∃ξ such that (x,ξ) ∈ A ×U
1 1
and κ(cid:48)(cid:48)x,ξ+Φ(x,κ(cid:48)(cid:48)x) ∈A ×U .
2 2
(cid:0) (cid:1)
Note that κ,κ(cid:48)(cid:48) have disjoint domains and images. It follows that the following
map is well-defined, and belongs to H[A]:
κ(x) x∈dom(κ)
κ(x):=(cid:40)
κ(cid:48)(cid:48)(x) x∈dom(κ(cid:48)(cid:48)).
But this contradicts the maximality of κ, because κ(cid:60)κ and
P[dom(κ)×G]=P[dom(κ)×G]+P[A(cid:48) ×G]>P[dom(κ)×G].
1
This contradiction proves that
dom(κ)×G=A×Gmodm or im(κ)×G=A×Gmodm.
In the first case, define κ :=κ and in the second define κ :=κ−1. This gives
A A
us κ ∈ H[A] which is defined a.e. in A. Now define κ : A → A
by κA(x) = κ (x) whenever x ∈ A. This is a well-defined bi(cid:83)jecAt∈ioαn, beca(cid:83)usAe∈αthe
A
domains and images of κ are pairwise disjoint. This bijection is a holonomy,
A
and its domain is equal to X modm. It satisfies (H1) because for every x ∈ A,
d(x,κ(x))≤diam(A)<δ, and it satisfies (H2) because κ| =κ do for all A. (cid:3)
A A
Proposition 3. Suppose Φ : G → R is a Borel cocycle, and m is a G –e.i.l.f.
Φ
measure on X × R. If m has a square hole, then ∃u : X → R Borel and Φ–
measurable such that Φ(x,y)+u(x)−u(y)∈H m–a.e. in G .
m Φ
Proof. Fix p : X ×R → (0,∞) Borel s.t. pdm < ∞, and define dP := pdm.
(cid:82)X×R
LetµbethemeasureonX givenbyµ(E):=P(E×R). Thereexistsa measurable
map x(cid:55)→P ∈P({x}×R) s.t.
x
∀f ∈L1(P), (cid:90) fdP =(cid:90) (cid:18)(cid:90) fdPx(cid:19)dµ(x).
X R
Claim 1. Almost every x ∈ X has the following property: every G holonomy κ
Φ Φ
s.t. x∈dom(κ) satisfies P ◦κ ∼P .
κ(x) Φ x
INVARIANT RADON MEASURES FOR HOROCYCLE FLOWS ON ABELIAN COVERS 9
Proof. A calculation shows that for every G –holonomy κ , P ◦ κ ∼ P
Φ Φ κ(x) Φ x
for almost every x ∈ dom(κ). Since G is a countable equivalence relation, it is
Φ
generated by a countable group of holonomies [FM], and the claim follows.
Claim 2. ∃a(cid:48) <b(cid:48) <α<β <a<b, |α−β|< 1min{|a(cid:48)−b(cid:48)|,|b(cid:48)−α|,|β−a|,|a−b|}
2
and ∃B ⊂X measurable s.t. µ(B)>0 and for all x∈B,
(i) P ({x}×[a(cid:48),b(cid:48)])=0,P ({x}×[α,β])>0,P ({x}×[a,b])=0;
x x x
(ii) if κ is a G –holonomy and x∈dom(κ), then P ∼P ◦κ .
Φ Φ x κ(x) Φ
Proof. By assumption, ∃a < b and B ∈ X such that m(B × R) > 0 and
0 0 0 0
m(B0×[a0,b0]) = 0. Fix (cid:15) < b0−6a0. Since m[B0×R] > 0, ∃s ∈ R\[a0,b0] such
thatm[B ×B (s)]>0. Supposew.l.o.gthats<a (thecases>b issymmetric).
0 (cid:15) 0 0
Let B ⊆ B be measurable s.t. m[B ×R] > 0, m(B ×[a ,b ]) = 0 and for
1 0 1 1 0 0
which ∀x∈B :
1
(1) P ∼P ◦κ for every G –holonomy κ ;
x κ(x) Φ Φ Φ
(2) P {x}×[a ,b ] =0;
x 0 0
(cid:0) (cid:1)
(3) P {x}×B (s) >0.
x (cid:15)
(cid:0) (cid:1)
Such a set exists, because the last two properties cannot fail on a set of full µ–
measure, and the first is guaranteed by claim 1.
Consider t := s−(a0+2b0 −s) = 2s− a0+2b0. We claim that m[B1×B(cid:15)(t)] = 0.
Otherwise,byLemma4,P–almostevery(x,ξ)∈B ×B (t)hasaG –holonomyκ
1 (cid:15) Φ Φ
such that κ (x,ξ)∈B ×B (s). This κ satisfies κ(x)∈B ,Φ(x,κx)∈B (s−t)
Φ 1 (cid:15) Φ 1 2(cid:15)
for all x∈B . Therefore,
1
Px|{x}×B(cid:15)(s) ∼Pκx◦κΦ|{x}×B(cid:15)(s) (cid:28)Pκx|{κx}×B3(cid:15)(2s−t)◦κΦ ≡0,
(because B3(cid:15)(2s−t)=B3(cid:15)(a0+2b0)⊂[a0,b0]). But this contradicts property (3).
Choose B ⊆ B of full µ–measure such that ∀x ∈ B , P {x}×B (t) = 0.
1 1 x (cid:15)
Evidently, m[B × B (t)] = 0, m[B × B (s)] > 0, m(B × [a(cid:0),b ]) = 0, a(cid:1)nd by
(cid:15) (cid:15) 0 0
construction t+(cid:15)<s−(cid:15)<a <b .
0 0
Finally,seta(cid:48) =t−(cid:15),b(cid:48) =t+(cid:15),a=a ,b=b andchooseα,β ∈B (s)sufficiently
0 0 (cid:15)
close so that m(B×[α,β]) > 0, |α−β| < 1min{|a(cid:48)−b(cid:48)|,|b(cid:48)−α|,|β−a|,|a−b|}
2
and b(cid:48) <α<β <a .
0
Claim3. ∃u :B →Rmeasurables.t. ∀ (x,ξ),(x(cid:48),η) ∈G B×[α,β] ,Φ(x,x(cid:48))=
0 Φ
u (x(cid:48))−u (x) and ξ =u (x) for a.e. (x(cid:0),ξ)∈B×[α(cid:1),β]. (cid:2) (cid:3)
0 0 0
Proof. Set
U(x) := inf{b(cid:48) ≤t≤b:P {x}×(t,b] =0},
x
(cid:0) (cid:1)
L(x) := sup{a(cid:48) ≤t≤a:P {x}×[a(cid:48),t) =0}.
x
(cid:0) (cid:1)
If x∈B,x(cid:48) =κ(x)∈B, and |Φ(x,x(cid:48))|<|α−β|, then
U(x(cid:48)) = U κ(x) =inf{b(cid:48) ≤t≤b:P {κ(x)}×(t,b] =0}
κ(x)
(cid:0) (cid:1) (cid:0) (cid:1)
= inf{α≤t≤a:P {κ(x)}×(t,a+b] =0}
κ(x)(cid:0) 2 (cid:1)
= inf{α≤t≤a:P ◦κ {x}×(t−Φ(x,x(cid:48)),a+b −Φ(x,x(cid:48))] =0}
κ(x) Φ(cid:0) 2 (cid:1)
= inf{α≤t≤a:P {x}×(t−Φ(x,x(cid:48)),a] =0}
x
(cid:0) (cid:1)
= inf{α−Φ(x,x(cid:48))≤s≤a−Φ(x,x(cid:48)):P {x}×(s,a] =0}+Φ(x,x(cid:48))
x
(cid:0) (cid:1)
≥ inf{b(cid:48) ≤s≤b:P {x}×(s,a] =0}+Φ(x,x(cid:48))
x
(cid:0) (cid:1)
= inf{b(cid:48) ≤s≤b:P {x}×(s,b] =0}+Φ(x,x(cid:48))≡U(x)+Φ(x,x(cid:48))
x
(cid:0) (cid:1)
10 OMRISARIG
In the same way one shows that if x ∈ B,x(cid:48) = κ(x) ∈ B, and |Φ(x,x(cid:48))| < |α−β|,
then L(x(cid:48))≤L(x)+Φ(x,x(cid:48)).
It follows that if (x,ξ),(x(cid:48),η) ∈G ∩ B×[α,β] 2, then
Φ
(cid:0) (cid:1) (cid:0) (cid:1)
L(x(cid:48))−L(x)≤Φ(x,x(cid:48))≤U(x(cid:48))−U(x),
whence (L−U)(x(cid:48))≤(L−U)(x). By symmetry, the inequality is an equality and
thus (x,ξ),(x(cid:48),η) ∈G ∩ B×[α,β] 2 ⇒(L−U)(x(cid:48))=(L−U)(x).
Φ
(cid:0) (cid:1) (cid:0) (cid:1)
Sincem(B×[α,β])>0andG isergodic,G ∩ B×[α,β] 2 isergodic(Lemma
Φ Φ
5), whence L−U =const. This proves that (cid:0) (cid:1)
(x,ξ),(x(cid:48),η) ∈G ∩ B×[α,β] 2 ⇒Φ(x,x(cid:48))=U(x(cid:48))−U(x).
Φ
(cid:0) (cid:1) (cid:0) (cid:1)
It follows that F(x,ξ)=ξ−U(x) is G ∩ B×[α,β] 2–invariant, and therefore
Φ
ξ−U(x)=c a.e. in B×[α,β]. Setting u :=(cid:0) U +c w(cid:1)e see that
0 0 0
B×[α,β]⊆(B×R)∩[ξ =u (x)]modm,
0
and Φ(x,x(cid:48))=U(x(cid:48))−U(x)=u (x(cid:48))−u (x).
0 0
Claim 4. Define u (x) := sup{t ≥ u (x) : P {x}×(u (x),t) = 0} ≤ ∞. There
1 0 x 0
exists c ∈(0,∞] such that u =u +c µ–a.e(cid:0). in B. (cid:1)
1 1 0 1
Proof. Suppose (x,ξ),(x(cid:48),η) ∈G B×[α,β] andletκ besomeholonomysuch
Φ Φ
that (x(cid:48),η)=κ(cid:0)(x,ξ). If u (x(cid:1)(cid:48))<∞(cid:2), then (cid:3)
Φ 1
Px(cid:48) {x(cid:48)}× u0(x(cid:48)),u1(x(cid:48)) = 0,
(cid:2) (cid:0) (cid:1)(cid:3)
Px(cid:48) {x(cid:48)}×[u1(x(cid:48)),u1(x(cid:48))+(cid:15)) > 0 for all (cid:15)>0.
(cid:2) (cid:3)
We know that Px(cid:48) =Pκx ∼Px◦κ−Φ1, so
P {x}×(u (x(cid:48))−Φ(x,x(cid:48)),u (x(cid:48))−Φ(x,x(cid:48))) = 0,
x 0 1
(cid:0) (cid:1)
P {x}×[u (x(cid:48))−Φ(x,x(cid:48)),u (x(cid:48))−Φ(x,x(cid:48))+(cid:15)) > 0 for all (cid:15)>0.
x 1 1
(cid:0) (cid:1)
But u (x(cid:48))−Φ(x,x(cid:48))=u (x), so u (x)<∞ and
0 0 1
u (x)=u (x(cid:48))−Φ(x,x(cid:48))=u (x(cid:48))−u (x(cid:48))+u (x).
1 1 1 0 0
This shows that u −u is G B ×[α,β] –invariant. By Lemma 5, this relation
1 0 Φ
is ergodic, so either u = ∞ a(cid:2)lmost ever(cid:3)ywhere, or u = u +const. We write
1 1 0
u =u +c with c ∈[0,∞].
1 0 1 1
Weclaimthatc >0. Itisenoughtoconsiderthecasewhenc isfinite. Observe
1 1
that u1(x)≥β ≥u0(x) a.e. in B, because Px|{x}×[α,β] (cid:28)δ(x,u(x0)) for a.e. x∈B.
It is enough to show that one of these inequalities is strict on a set of positive
measure, because this implies that c =u −u >0.
1 1 0
Assume by way of contradiction that u (x) = β = u (x) a.e. in B. The sets
1 0
B × {β} and B × (β,β + δ) must have positive P–measure for all δ > 0. By
Lemma 4, ∃B(cid:48) ⊆ B and ∃ a holonomy κ defined on B(cid:48) s.t. P(B(cid:48) ×{β}) > 0,
κ (B(cid:48)×{β})⊆B×(β,β+δ). It follows that 0<Φ(x,κx)<δ for µ–a.e. x∈B(cid:48).
Φ
Fixing 0 < δ < β − α, we see that P ◦ κ B(cid:48) × (α,β) ≥ P κ(B(cid:48)) × {β} =
Φ
P κ(B(cid:48))×[α,β] = P ({x}×[α,β])dµ((cid:0)x)>0. Sinc(cid:1)e κ is(cid:0)P–nonsingula(cid:1)r,
(cid:0) (cid:1) (cid:82)κ(B(cid:48)) x Φ
P B(cid:48)×(α,β) >0.
(cid:0) (cid:1)
Since B(cid:48) ⊆ B, P B×(α,β) > 0. It follows that {(x,ξ) ∈ B×[α,β] : ξ < u (x)}
0
(cid:0) (cid:1)
has positive measure, a contradiction to Claim 3.