The case of equality in the dichotomy of Mohammadi-Oh

0.18 MB·

by  Laurent Dufloux

The case of equality in the dichotomy of
Mohammadi-Oh
Laurent Dufloux
7
1 January 18, 2017
0
2

n Abstract
a Ifn≥3 andΓ is aconvex-cocompactZariski-dense discretesubgroup
J ofSOo(1,n+1)suchthatδΓ =n−mwheremisaninteger,1≤m≤n−1,
7
weshowthatforanym-dimensionalsubgroupU inthehorosphericgroup
1
N,theBurger-RoblinmeasureassociatedtoΓonthequotientoftheframe

bundleis U-recurrent.
]
S
D 1 Introduction
.
h
t 1.1 Notations
a
m We fix once and for all an integer n≥2. Let G= SOo(1,n+1), this is
[ the group of direct isometries of the real (n+1)-dimensional hyperbolic
space Hn+1. Itsacts conformally on the boundary∂Hn+1.
1
Recall theBusemann function
v
5 b (x,y)= lim d(x,ξ )−d(y,ξ ) ξ∈∂Hn+1, x,y∈Hn+1
ξ t t
5 t→∞
5 where t7→ξ is some geodesic with positive endpoint ξ.
t
4 Fix an Iwasawa decomposition G = KAN; recall that the maximal
0 compact subgroup K is isomorphic to SO(n+1), whereas the Cartan
1. subgroup A is isomorphic to R (since G has rank 1) and the maximal
0 unipotentsubgroup N is isomorphic to Rn.
7 Denote by M the centralizer of A in K; M is isomorphic to SO(n).
1 Recall that M normalizes N and there are isomorphisms M ≃ SO(n),
: N ≃ Rn such that the operation of M on N by conjugation identifies
v
i with the natural operation of SO(n) on Rn.
X
We will always tacitly endow N with the corresponding Euclidean
r metric.
a
Let Γ be a discrete non-elementary subgroup of G. Throughout this
paperwe make thestanding assumptions that
Γ is Zariski-dense and has finite Bowen-Margulis-Sullivan
measure.
InfactexceptinthelastparagraphwewillalwaysassumethatΓisconvex-
cocompact(thisisstrongerthanfinitenessoftheBowen-Margulis-Sullivan
measure).
As usual, we denote by δ the growth exponent (also called Poincaré
Γ
exponent)of Γ
logCard{γ ∈Γ ; d(x,γx)≤R}
δ =limsup
Γ R
R→∞
1
whichdoesnotdependonthefixedpointx∈Hn+1. ThisistheHausdorff
dimension (with respect to the spherical metric on the boundary) of the
limit set
Λ =Γ·x∩∂Hn+1
Γ
(whichalsodoesnotdependonx). Bearinmindthat0<δ ≤n;inthis
Γ
paper we will be interested in the case when δ is an integer strictly less
Γ
than n.
Theboundary∂Hn+1 is endowed with thePatterson-Sullivandensity
(µx)x∈Hn+1. This is the (essentially unique since Γ has finite Bowen-
Margulis-Sullivanmeasure)familyoffiniteBorelmeasureson∂Hn+1 sat-
isfying
1. Γ-equivariance: µ isthepush-forwardofµ throughthemapping
γx x
induced by γ on ∂Hn+1;
2. δ -conformality: for anyx,y∈Hn+1, µ andµ areequivalentand
Γ x y
theRadon-Nikodymcocycle is given by
dµ
y(ξ)=e−δΓbξ(y,x)
dµ
x
almost everywhere.
This is the Patterson-Sullivan density associated to Γ. If a base point
o ∈ Hn+1 is fixed, the boundary ∂Hn+1 may be identified canonically
with the n-sphere Sn and thus endowed with the usual spherical metric.
When Γ is convex-cocompact, µ is proportional to the δ -dimensional
o Γ
Hausdorff measure on δ with respect to thespherical metric (see [16] or
Γ
[1]).
We now recall the definition of the Bowen-Margulis-Sullivan (BMS)
measure–firstontheunittangentbundle,thenontheframebundle. Let
T1Hn+1 be the unit tangent bundle over Hn+1. The Hopf isomorphism
isthebijectivemappingfromT1Hn+1 to∂2Hn+1×Rthatmapstheunit
tangent vector uwith base point x to thetriple
(ξ,η,s)=(u−,u+,bu−(x,o))
where u−,u+ respectively are the negative and positive endpoints of the
geodesic whose derivative at t=0 is u. The notation ∂2Hn+1 stands for
theset of all (ξ,η)∈∂Hn+1×∂Hn+1 such that ξ 6=η.
In thesecoordinates, theBMS measure on T1Hn+1 is given by
dm˜ (u)=eδΓ(bξ(x,u)+bη(x,u))dµ (ξ)dµ (η)ds
BMS x x
(it does not depend on thechoice of x∈Hn+1).
The BMS measure is a Radon measure that is invariant under the
geodesicflowaswellasunderthenaturaloperationofΓ. Thequotientof
this measure with respect to Γ is a Radon measure m on Γ\T1Hn+1
BMS
that is still invariant with respect to the geodesic flow. This quotient
measure may be finite or infinite; we will always assume that is is finite
and in fact we will usually assume that it is compactly supported, which
is equivalent toΓ being convex-cocompact ([12], [16]).
The Burger-Roblin (BR) measure is definedin a similar fashion:
dm˜ (u)=eδΓbξ(x,u)+nbη(x,u)dµ (ξ)dν (η)ds
BR x x
where ν is the unique Borel probability measure on ∂Hn+1 that is in-
x
variant under the stabilizer of x in G; if ∂Hn+1 is identified with Sn
accordingly, this is just theLebesgue measure on Sn.
2
Likewise, the Burger-Roblin measure is Γ-invariant and thus defines
a Radon measure on Γ\T1Hn+1. This Radon measure is always infinite,
unless Γ is a lattice.
Both these measures lift to the frame bundleover Γ\Hn+1 in the fol-
lowingway. ThehyperbolicspaceHn+1 identifieswiththequotientspace
G/M so that G identifieswith the(n+1)-frame bundleoverHn+1. The
quotient space Γ\G accordingly identifies with the (n+1)-frame bundle
over the orbifold Γ\Hn+1. There is a unique measure on Γ\G that is
(right) invariant with respect to M and projects onto the BMS measure
inΓ\G/M,wedenoteitbym aswell. SamethingfortheBRmeasure.
BMS
Thelift of the geodesic flow to Γ\G is called theframe flow.
The point in doing this is we can now let N act by translation (to
the right) on Γ\G. Let us agree that A = {a ; t ∈ R} where (a )
t t t
parametrizestheframeflowoverHn+1,insuchawaythatN parametrizes
theunstable horospheres.
Wethen have,for every h∈N,
a ha =S (h) (1)
−t t t
where S is thehomothety N →N with ratio et.
t
Wesummarize the important points in thefollowing
Lemma 1. Assume that Γ has finite BMS measure and isZariski-dense.
1. TheBMSmeasureonΓ\Gismixingwithrespect totheergodicflow.
2. TheBR measure onΓ\Gisinvariantandergodic withrespect to N.
3. If Ω⊂Γ\G has full BMS measure, then ΩN has full BR measure.
Proof. For1and2see[17]. For3comparethedefinitionsofBMSandBR
measure, taking into account the fact that N parametrizes the unstable
horospheres in the frame bundle.
1.2 Background
Thebasic motivation for this paper is thefollowing
Theorem (Mohammadi-Oh, [11], Theorem 1.1). Assume that Γ is
convex-cocompact andZariski-dense. Letm beaninteger, 1≤m≤n−1,
and U be an m-plane in N. If δ >n−m, then m is U-ergodic.
Γ BR
This result was also obtained by Maucourant and Schapira [9] under
the weaker hypothesis that Γ has finite BMS measure. The case when
δ <n−m has also been settled by these authors:
Γ
Theorem (Maucourant-Schapira, [9]). Assume that Γ is convex-
cocompact and Zariski-dense. Let m be an integr, 1≤m≤n−1, and U
be an m-plane in N. If δ <n−m, then m is totall U-dissipative. In
Γ BR
particular, it is not ergodic.
Mohammadi-OhandMaucourant-SchapirauseMarstrand’sprojection
Theorem to look at the geometry of the BMS measure along U and N.
Formore on this, see [4].
Inthispaper,weuseBesicovitch-Federer’sprojectiontheoremtostudy
thecase δ =n−m. Ourmain result is thefollowing
Γ
Theorem. Assumethat Γisconvex-cocompact andZariski-dense. Letm
be an integer, 1 ≤ m ≤ n−1. If δ = n−m, then the Burger-Roblin
Γ
measure is recurrent with respect to any m-plane U in N.
3
Whether the BR measure is ergodic with respect to U under these
hypothesesremainsan open question. Wewill seethatthereturnrateof
U-orbits is quite low (i.e. subexponential) but this does not contradict
ergodicity since BR is not finite.
LetusmentionthattheTheoremisnotempty;indeeditispossibleto
constructsomeZariski-denseconvex-cocompactgroupΓ⊂SOo(1,3)with
δ =1. StartwiththeApolloniangasketassociatedto4mutuallytangent
Γ
circles on the boundary of H3; the limit set has dimension δ > 1. Now
Γ
shrinkcontinuouslytheradiiofthecircles,thusloweringcontinuouslyδ .
Γ
The deformed group will remain Zariski-dense because the centers of the
circles are not aligned. Fordetails see [10].
With this result for δ = n−m, the situation is summarized in the
Γ
following table. We assume that Γ is Zariski-dense, has finite BMS mea-
sure,andwefixsomem-planeU inN with1≤m≤n−1. Withrespect
toU, theBMS and BR measures are:
δ < n−m δ = n−m δ > n−m
Γ Γ Γ
recurrent and
BMS dissipative [4] dissipative [4]
ergodic [9]
totally dissipative
recurrent if Γ recurrent and
BR if Γ convex-
convex-cocompact ergodic [9]
cocompact [9]
Notethat it follows immediately from thedefinitionsthat if theBMS
measure is recurrent, so is the BR measure. The other implications are
not so obvious.
We now sketch briefly our argument. In order to prove that the BR
measureisU-recurrent(whereU issomem-plane),weneedtoshowthat
theU-orbitofm -almosteveryx∈Γ\Gwillpassthroughsomecompact
BR
setKinfinitelyoften. Ifisenoughtoconstructsomesequenceh inN that
k
goestoinfinitywhilestayinguniformly closefrom U,suchthatxh ∈K;
k
indeed, if u is the orthogonal projection of h onto U, the sequence u
k k k
still goes to infinityand xu will belong to some compact K′ that is just
k
slightly bigger than K.
To show that such a sequence (h ) exists, our strategy is to prove
k k
thatanyρ-neighbourhoodofU in N hasinfinitemeasurewith respectto
theconditional measure of m along N; we then use the fact that the
BMS
support of m is a compact set. This is the main reason why we need
BMS
Γ to be convex-cocompact.
In order to prove that any “strip” along U has infinite measure, we
argue by contradiction: if some ρ-neighbourhood has finitemeasure with
respecttotheconditionalmeasureofm alongN,thenthismusthold
BMS
almost surely for any neighbourhood as large as we like (because of the
self-similarity of the conditional measures). In particular we can project
theseconditionalmeasuresontoN/U andendupwith afamily ofRadon
measures. These“transversal” Radonmeasuresmust stillhavedimension
δ =n−m(thiswasshownin[4]),andthisimpliesinturnthattheymust
Γ
be the Lebesgue measure of N/U. On the other hand, the Besicovitch-
FedererprojectionTheoremimpliesthattheprojectionoftheconditional
measures onto N/U must be singular with respect to the Lebesgue mea-
sure,becausetheconditionalmeasurearepurelyunrectifiable. Henceour
Theorem is proved.
The push-forward of the Borel measure µ through the Borel function
f is denoted by fµ; thusfµ(A)=µ(f−1(A)) for any Borel set A.
4
Forany set E, we denote by1 thecharacteristic function:
E
1 if x∈E
1 (x)=
E (cid:26) 0 if x∈/ E
2 Proof of the main theorem
2.1 Preliminary setup
InordertostudytheBRmeasure with respect tosomem-planeU in N,
it is useful to look at the geometry of the BMS measure with respect to
the foliation induced by U in the N-orbits (more precisely, with respect
tothe projection along this foliation).
The technical tool that allows this is disintegration of measures.
Since we are going to apply tools from classical geometric measure
theory,wewanttoworkwithmeasureslivingonN (recallthatN identifies
with the Euclidean space Rn). To m -almost every x ∈ Γ\G we are
BMS
going to associate a measure (more precisely, a projective measure, i.e. a
measure modulo a positive scalar) σ(x) on N that reflects the geometry
of m along theunstablehorosphere passing through x.
BMS
We now set up the needed formalism. The operation of N on G (on
the right) is smooth (i.e. the quotient Borel space G/N is a standard
Borel space). Lift m (which lives on Γ\G) to G; the measure we get
BMS
is a Γ-invariant Radon measure m˜ . Disintegrate this measure along
BMS
N; for almost every g ∈G we thusget a measure m supported on gN
gN
(see [12] section 3.9 for a description of thismeasure).
In general when disintegrating an infinite measure, the conditional
measures are canonically defined only up to a (non-zero) scalar; in fact
herethereisaway tonormalize theminacanonicalway (byintroducing
an appropriate measure on the space of horospheres, more precisely this
space lifted by M) but this would not beuseful for ourpurpose. Seee.g.
[13].
We now want to look at measures on N instead of measures on G.
For any g ∈ G, there is a mapping φ : N → G which parametrizes the
g
“unstablehorosphere” H+(g)=gN in theusual way: φ (h)=ghfor any
g
h∈N.
Sincem˜ is Γ-invariant,the pull-backmeasures
BMS
(φ )−1(m ), (φ )−1(m )
g gN γg γgN
(whichliveonN)areequaluptoascalarmultiple,form˜ -almostevery
BMS
g∈Gand every γ ∈Γ.
Let M (N) be the space of positive Radon measures on N and
rad
M1 (N) be the space of projective classes of Radon measures on N,
rad
that is, the quotient of M (N) by theequivalencerelation
rad
µ∼ν ⇔ν =tµ, t>0.
We define a mapping σ : Γ\G → M1 (N) by letting σ(x) be the
rad
projective class of
(φ )−1(m )
g gN
if x=Γg. This is well-defined m -almost everywhere.
BMS
Wesay that σ is obtained by disintegrating m along N.
BMS
Thisis aparticular instanceof thegeneral theory of conditional mea-
sures along a group operation, see [3] or [2] (Chapter 2).
5
Werecordthefollowing facts whichwewill usefreely throughoutthis
paper:
Lemma 2. 1. If some Borel subset Ω⊂Γ\G has full m -measure,
BMS
then for m -almost every x, the set
BMS
{h∈N ; xh∈Ω}
has full σ(x)-measure.
2. There is a Borel subset X ⊂ Γ\G of full m -measure such that
BMS
if x ∈ X and h ∈ H are such that xh ∈ X, then σ(xh ) is the
0 0 0
push-forward of σ(x) through left translation by h in N,
0
h7→h h.
0
3. For m -almost every x ∈ Γ\G, the origin of N belongs to the
BMS
support of σ(x).
4. For any t∈R and m -almost every x∈Γ\G,
BMS
σ(xa )=S σ(x)
t t
i.e. σ(xa ) is the push-forward of σ(x) through the ghomothety S :
t t
N →N.
5. For any m ∈ M, and m -almost every x ∈ Γ\G, σ(xm) is the
BMS
push-forwardofσ(x)throughthemappingh7→mhm−1. (Recallthat
theoperationofM byconjugationonN identifieswiththecanonical
operation of SO(n) on Rn.)
6. For m -almost every x∈Γ\G and σ(x)-almost every h∈N,
BMS
σ(x)(B(h,ρ)) σ(x)(B(h,ρ))
0<liminf ≤limsup <∞.
ρ→0 ρδΓ ρ→0 ρδΓ
Proof. Statements 1, 2 and 3 are clear. Statement 4 holds because of
invariance of m with respect to the geodesic flow and formula (1).
BMS
Statement 5 holds because m is M-invariant by definition. State-
BMS
ment6holdsbecauseΓisconvex-cocompactandσ(x)isequivalenttothe
Patterson-Sullivan measure; see [1], Proposition 7.4 and [12], section 3.9
Notation. If µ is a Borel measure or projective measure on N, the sup-
port of which contains the origin on N, we let
µ
µ∗ =
µ(B )
1
i.e. µ∗ is the measure colinear to µ that gives measure 1 to the unit ball
B .
1
We also denote by S∗µ the measure (S µ)∗.
t t
In particular, since for m -almost every x ∈ Γ\G, the origin of N
BMS
belongs to the support of σ(x), we denote by σ∗(x) the Radon measure
onN thatbelongstotheprojectiveclassσ(x)andsuchthattheunitball
B ⊂N has measure 1:
1
σ∗(x)(B )=1.
1
WedenotebyDirac(x)theDirac mass at x, i.e. theprobability mea-
sure giving measure 1 to {x}. Associated to m is the following prob-
BMS
ability measure on the space of Radon measures on N:
P = dm (x) Dirac(σ∗(x)). (2)
BMS
Z
Γ\G
6
Recall that we assume that Γ is Zariski-dense and has finite BMS mea-
sure, so that P is an Ergodic Fractal Distribution (EFD) in the sense of
Hochman(see[5],Definition1.2,and[4],Lemma5.3foraproofthatPis
indeed an EFD).
2.2 Unrectifiability of the limit set
Recall that a Radon measure µ on the Euclidean space Rn is said to be
purely m-unrectifiable if for any Lipschitz mapping f : Rm → Rn, the
range f(Rm) has measure zero with respect to µ.
Assumethat thegrowth exponentδ isan integer<n. Thefact that
Γ
the limit set of Γ is purely δ -unrectifiable when Γ is convex-cocompact
Γ
andZariski-dense(thelatterhypothesisisobviouslynecessary)isprobably
well-known, and certainly very intuitive. Wegive a full proof of this fact
as it is pivotal in our argument.
Proposition 3. Assume that Γ is convex-cocompact and Zariski-dense.
Ifδ isaninteger strictly smallerthan n, the conditional measure σ(x)is
Γ
almost surely purely δ -unrectifiable.
Γ
Proof. Let Ω betheset of all x∈Γ\G such that
1 T
Dirac(S∗σ(x))dt
T Z t
0
converges weakly to P (recall equation (2)) as T → +∞. This set has
full BMS measure ([4], Lemma 5.4). Now fix some x ∈ Ω such that for
0
σ(x )-almost every h∈N, x h∈Ω (see Lemma 2).
0 0
We argue by contradiction. Assume that some subset L ∈ N is the
image of a Lipschitz mappingRδΓ →N and satisfies
σ(x )(L)>0.
0
Note that the restriction σ(x )|L, which we denote by σ (x ), is δ -
0 L 0 Γ
rectifiable, and satisfies
σ (x )(B(h,ρ)) σ (x )(B(h,ρ))
0<liminf L 0 ≤limsup L 0 <∞
ρ→0 ρδΓ ρ→0 ρδΓ
forσ (x )-almosteveryh(Lemma2). Byvirtueof[8],Theorem16.7and
L 0
Lemma 14.5, for σ (x )-almost every h, there is a δ -plane V(h) such
L 0 Γ
that
S∗σ(x h)
t 0
converges weakly to the Haarmeasure on V(h) as t→∞
Recall that for σ(x )-almost every h,
0
1 T
Dirac(S∗σ(x h))dt
T Z t 0
0
also converges weakly to P as T goes to infinity.
We thus see that P-almost every µ is the Haar measure on some δ -
Γ
plane. In other words, for m -almost every x the conditional measure
BMS
atx,σ(x),isconcentratedonsomeδ -planeofN;thiscontradictsthefact
Γ
that the support of σ(x) must be Zariski-dense, since Γ is Zariski-dense.
Hencetheproposition.
Corollary 4. Under the same hypotheses, the limit set Λ is purely δ -
Γ Γ
unrectifiable.
7
Recall that the limit set Λ is the set of accumulation points of Γ in
Γ
Hn+1∪∂Hn+1. It is locally bilipschitz equivalent to thesupport of σ(x)
for m -almost every x, so that the corollary follows readily from the
BMS
proposition.
2.3 The conditional measures are transversally
singular
Proposition 5. Assume that Γ is Zariski-dense and convex-cocompact
and that δ = n−m where m is an integer, 1 ≤ m ≤ n−1. Fix some
Γ
m-plane U in N.
For m -almost every x∈Γ\G, the push-forward of the conditional
BMS
measure σ(x) through the canonical projection N →N/U is singular with
respect to the Lebesgue measure on N/U.
Recall that a measure µ is singular with respect to a measure ν if it
gives full measure to a ν-negligible set.
Proof. For any m-plane V, denote by π the canonical projection N →
V
N/V.
We will show that there exists an m-plane U such that for almost
0
every x, the push-forward of σ(x) through N → N/U is singular with
0
respect to the Lebesgue measure on N/U. Since the BMS measure is
M-invariant, this implies that the same statement holds for any other
m-planeU.
AccordingtoLemma6andthepreviousPropostion,form -almost
BMS
everyx thereis a sequenceof Borel sets (A ) such that
k k
• ∪ A has full σ(x)-measure,
k k
• each A has finite(n−m)-dimensional Hausdorff measure,
k
• and each A is purely (n−m)-unrectifiable.
k
ByvirtueoftheBesicovitch-Federerprojectiontheorem([8],Theorem
18.1 (2)), the image of ∪ A in N/V is Lebesgue-negligible for almost
k k
everym-planeV (withrespecttotheHaarmeasureontheGrassmannian
of m-planes in N). This shows that for almost every m-plane V, the
push-forwardofσ(x)throughπ issingularwithrespecttotheLebesgue
V
measure.
This holds for almost every x. A standard application of Fubini’s
theorem now yields that thereexists an m-planeU such that for almost
0
everyx,thepush-forwardofσ(x)throughπ issingularwith respectto
U0
theLebesgue measure. The proposition is thusproved.
Lemma 6. Assume that Γ is convex-cocompact. For m -almost every
BMS
x∈Γ\G, σ(x) is supported by a countable union of δ -sets.
Γ
RecallthatE isaδ-setifitsδ-dimensionalHausdorffmeasureisfinite
and non-zero.
Proof. Itiswell-known(see[15],Theorem7)thatthelimitsetΛ isaδ -
Γ Γ
set. Sinceitis(almostsurely)locallybilipschitz-equivalenttothesupport
of σ(x),thelemma follows.
8
2.4 Conditional measure of strips
If U is any m-plane in N (1 ≤ m ≤ n−1), we denote by BT(U) the
ρ
ρ-neighbourhood of U in N, that is the set of all h∈N such that
d(h,U)<ρ.
Whenitisclearfromthecontextwhichm-planewearetalkingabout,we
dispenseourselves with theletter U in the notation.
Proposition 7. Assume that Γ is convex-cocompact and Zariski-dense
and that δ = n−m where m is an integer, 1 ≤ m ≤ n−1. Fix some
Γ
m-plane U in N. For m -almost every x∈Γ\G and any ρ>0,
BMS
σ(x)(BT)=∞.
ρ
Proof. Itisenoughtoshowthatforanyρ>0,andalmosteveryx∈Γ\G,
σ(x)(BT)=∞(seelemma2). Wearguebycontradictionandassumethat
ρ
theset of those x such that
σ(x)(BT)<∞
ρ
haspositive BMS measure; it mustthen havefull measure since m is
BMS
mixingand because of Lemma 2.4.
It is easy to see then that for m -almost every x∈Γ\G,
BMS
σ(x)(BT)<∞
ρ
for any ρ>0.
This implies that the push-forward of σ(x) through the projection
π :N →N/U is a projective Radon measure.
U
Now consider thedistribution
PT = dm(x)Dirac((π σ(x))∗)
U
Z
on the space of Radon measures on N/U. It is straight-forward to check
that PT is an Ergodic Fractal Distribution (see [4], Lemma 5.3). Since
PT hasdimension n−m (see [4], Theorem 4.1) this is possible only if
PT =Dirac(Haar )
N/U
i.e. PT is theDirac mass at the Haar measure of N/U.
We are using the fact that a Fractal Distribution of dimension d on
some Euclidean space Rd has to be the only one we can think of, i.e.
Dirac(HaarRd). In essence, this fact goes back to Ledrappier-Young([6],
Corollary G). In the setting of Fractal Distributions it was proved by
Hochman in [5], Proposition 6.4 (see also [7]).
Now we end up with the conclusion that for m -almost every x ∈
BMS
Γ\G, thepush-forward of σ(x)through π is theHaar measureon N/U;
U
thiscontradicts Proposition 5. Hencetheproposition is proved.
Remark. Propositions 3, 5 and 7 admit obvious counter-examples when
Γ is not Zariski-dense: take some lattice Γ ⊂ SOo(1,m+1) and look at
the image of Γ through the embedding
SOo(1,m+1)→SOo(1,n+1).
9
2.5 Recurrence of the Burger-Roblin measure
We are now ready to prove our main theorem. We use the following
consequenceof proposition 7.
Lemma 8. Assume that Γ is Zariski-dense and convex-cocompact and
thatand δ = n−m. Fix an m-plane U in N. Let Ω be the support of
Γ Γ
theBowen-Margulis-SullivanmeasureinΓ\G. Foralmosteveryx∈Γ\G,
and any ρ>0, the set of all h∈BT(U) such that xh∈Ω is unbounded.
ρ Γ
Proof. By construction of the disintegration mapping σ, the support of
σ(x), supp(σ(x)), is almost surely the set of all h ∈ N such that xh
belongs to Ω . Since the Radon measure σ(x) gives infinite measure to
Γ
BT(U), theintersection BT ∩supp(σ(x)) must be unbounded;hence the
ρ ρ
lemma.
Proposition 9. Assume that Γ is Zariski-dense and convex-cocompact
and that δ =n−m. Fix some m-plane U in N. For BMS-almost every
Γ
x, there is a compact K ⊂Γ\G such that
1 (xu)du=∞.
K
Z
U
Furthermore, if W is any neighbourhood of Ω , K may be chosen inside
Γ
W.
Of course U is endowed with theHaar measur in this formula.
Proof. First of all, recall that Ω is a compact subset of Γ\G since Γ is
Γ
convex-cocompact.
Foranyρ>0,letK bethesetofallxhwherex∈Ω andhbelongs
ρ Γ
to the closed ρ-ball centered at the origin in N. This is again a compact
set. If ρ is small enough, K is a subset of W. Fix such a ρ.
ρ
Bylemma8,wemayfindasequence(h ) ofelementsofBT(U)that
k k ρ
goes to infinity and such that xh ∈ Ω for any k; if we let h = u v
k Γ k k k
where u ∈U and v is orthogonal to U, we have
k k
xu ∈K
k ρ
for any k, and the sequence(u ) goes toinfinity.
k k
According to lemma 11, we may thicken K to get a compact set
ρ
K ⊂W,such that theconclusion of theproposition holds.
Remark. It is necessary to consider a compact set K that is slightly
bigger than Ω in this lemma, since by virtue of Proposition 3, one has
Γ
1 (xu)du=0
Z ΩΓ
U
for BMS-almost every x.
Corollary 10. Under the same hypothesis, for BR-almost every x there
is a compact K such that
1 (xu)du=∞.
K
Z
U
In particular, the BR measure is recurrent with respect to U.
Proof. The set of all x ∈ Γ\G that satisfy the conclusion is obviously
N-invariant;sinceithasfullBMSmeasure,itmusthavefullBRmeasure
as well.
10