Geometric and spectral characterization of Zoll manifolds, invariant measures and quantum limits
Emmanuel Humbert
∗Yannick Privat
†Emmanuel Tr´ elat
‡Abstract
We provide new geometric and spectral characterizations for a Riemannian manifold to be a Zoll manifold, i.e., all geodesics of which are periodic. We analyze relationships with invariant measures and quantum limits.
1 Introduction and main results
Let (M, g) be a closed connected smooth Riemannian manifold of finite dimensionn∈IN∗, endowed with its canonical Riemannian measuredxg.
We denote by T∗M the cotangent bundle ofM and by S∗M the unit cotangent bundle, en- dowed with the Liouville measureµL, and we denote byω=−dµLthe canonical symplectic form on T∗M. We consider the Riemannian geodesic flow (ϕt)t∈IR, where, for every t ∈ IR, ϕt is a symplectomorphism on (T∗M, ω) which preservesS∗M. A geodesic is a curvet7→ϕt(z) onS∗M for somez∈S∗M. Throughout the paper, we denote byπ:T∗M →M the canonical projection.
The same notation is used to designate the restriction ofπtoS∗M. A (geodesic) rayγis a curve onM that is the projection ontoM of a geodesic curve onS∗M, that is,γ(t) =π◦ϕt(z) for some z∈S∗M. We denote by Γ the set of all geodesic rays.
Given anyk∈IN∗, we denote bySk(M) the set of classical symbols of order k onM, and by Ψk(M) the set of pseudodifferential operators of order k (see [12, 21]). Choosing a quantization Op onM (for instance, the Weyl quantization), given any a∈ Sk(M), we have Op(a)∈Ψk(M).
Any element of Ψ0(M) is a bounded endomorphism ofL2(M, dxg).
Throughout the paper, we denote by h , i the scalar product in L2(M, dxg) and by k k the corresponding norm.
We consider the Laplace-Beltrami operator 4 on (M, g). Its positive square root, √ 4, is a selfadjoint pseudodifferential operator of order one, of principal symbolσP(√
4) =g?, the cometric ofg(defined onT∗M). The spectrum of√
4is discrete and is denoted by Spec(√
4). The set of normalized (i.e., of norm one inL2(M, dxg)) real-valued eigenfunctionsφis denoted byE.
We say that the manifold M is Zoll whenever all its geodesics are periodic (see [3]). Zoll manifolds have been characterized within a spectral viewpoint in [9,11], where it has been shown that, in some sense, periodicity of geodesics is equivalent to periodicity in the spectrum of √
4 (see Section1.3for details).
∗Laboratoire de Math´ematiques et de Physique Th´eorique, UFR Sciences et Technologie, Facult´e Fran¸cois Ra- belais, Parc de Grandmont, 37200 Tours, France (emmanuel.humbert@lmpt.univ-tours.fr).
†IRMA, Universit´e de Strasbourg, CNRS UMR 7501, 7 rue Ren´e Descartes, 67084 Strasbourg, France (yannick.privat@unistra.fr).
‡Sorbonne Universit´e, Universit´e Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, ´equipe CAGE, F-75005 Paris (emmanuel.trelat@sorbonne-universite.fr).
Given anyT >0 and any Lebesgue measurable subsetωofM, denoting byχωthe characteristic function ofω, we define the geometric quantities
g2T(ω) = inf
γ∈Γ
1
Tχω(γ(t))dt and g2(ω) = lim inf
T→+∞g2T(ω) and, denoting byE the set of eigenfunctions φof √
4 of norm one in L2(M, dxg), we define the spectral quantities
g1(ω) = inf
φ∈E
Z
ω
φ2dxg
and, forωBorel measurable,
g100(ω) = inf
µ∈I(S∗M)µ(π−1(ω))
whereI(S∗M) is the set of probability Radon measures µon S∗M that are invariant under the geodesic flow. It is always true that g2T(ω) 6 g2(ω) 6 g100(ω). These geometric and spectral functionals are defined in a more general setting in Sections1.1and 1.2, as well as several others which are of interest.
Our main result (Theorem2), formulated in Section1.4, provides new characterizations of Zoll manifolds and relations with quantum limits, among which we quote the following:
• M is Zoll ⇐⇒ g2(ω) =g100(ω) for everyω⊂M Borel measurable.
• M is Zoll and the Dirac measure δγ along any periodic ray γ ∈ Γ is a quantum limit on M
⇐⇒ there existsT >0 such thatg1(ω)6gT2(ω) for any closed subsetω⊂M.
• Ifg1(ω)6g100(ω) for any closed subsetω⊂M then the Dirac measureδγ along any periodic rayγ∈Γ is a quantum limit on M.
• Assume that the spectrum of√
4 is uniformly locally finitea. ThenM is Zoll and for every geodesic rayγthere exists a quantum limitµonMsuch thatµ(γ(R))>0.
aThis means that there exists` >0 andm∈IN∗such that the intersection of the spectrum with any interval of length`has at mostmdistinctelements (allowing multiplicity with arbitrarily large order).
Here, a quantum limit onM is defined as a probability Radon measure onM that is a weak limit of a sequence of probability measuresφ2λdxg. The last item above slightly generalizes some results of [9,11,17].
The study of g1(ω) and g2(ω) done in this paper is in particular motivated by the following inequality on the observability constant for the wave equation onM:
1
2min(g1(ω), g2(˚ω))6 lim
T→+∞
CT(ω) T 6 1
2min(g1(ω), g2(ω))
which is valid for any Lebesgue measurable subsetω of M (see [13, Theorems 2 and 3]). Here, given anyT >0, the observability constant CT(ω) is defined as the largest possible nonnegative constantC such that the observability inequality
Ck(y(0,·), ∂ty(0,·))k2L2(M)×H−1(M)6 Z T
0
Z
ω
|y(t, x)|2dxgdt is satisfied for all possible solutions of the wave equation∂tty− 4y= 0.
The article is organized as follows. In Sections1.1and1.2, we define with full details the various geometric and spectral quantities that are of interest for the forthcoming results. In Section1.3, we recall several known results about new characterizations of Zoll manifolds. In Section 1.4, we gather all the new results and estimates about the characterization of Zoll manifolds and the relations with quantum limits. Finally, the proofs of these results are all postponed to Section2.
1.1 Geometric quantities
Given any bounded measurable functionaon (S∗M, µL) and given anyT >0, we define gT2(a) = inf
z∈S∗M
1 T
Z T 0
a◦ϕt(z)dt= inf
z∈S∗M¯aT(z) where ¯aT(z) = T1RT
0 a◦ϕt(z)dt. Note that gT2(a) = g2T(a◦ϕt), i.e., g2T is invariant under the geodesic flow. We set
g2(a) = lim inf
T→+∞ inf
z∈S∗M
1 T
Z T 0
a◦ϕt(z)dt
| {z }
gT2(a)
= lim inf
T→+∞ inf
z∈S∗M¯aT(z)
and
g20(a) = inf
z∈S∗Mlim inf
T→+∞
1 T
Z T 0
a◦ϕt(z)dt= inf
z∈S∗Mlim inf
T→+∞¯aT(z) Note that we always haveg2(a)6g02(a).
Given two functions onS∗M such thata= ˜a µL-almost everywhere, we may haveg2(a)6=g2(˜a) andg20(a)6=g20(˜a). Indeed, takinga= 1 everywhere onS∗M and ˜a= 1 as well except along one geodesic, one hasg2(a) = 1 andg2(˜a) = 0, althougha= ˜aalmost everywhere.
Note that gT2, g2 and g20 are inner measures on S∗M, which are invariant under the geodesic flow. They are superadditive but not subadditive in general (and thus, they are not measures).
We can pushforward them to M under the canonical projection π : S∗M → M: given any bounded measurable functionf on (M, dxg), we set
(π∗gT2)(f) =gT2(π∗f) =g2T(f◦π) = inf
z∈S∗M
1 T
Z T 0
f◦π◦ϕt(z)dt= inf
γ∈Γ
1 T
Z T 0
f(γ(t))dt and accordingly,
(π∗g2)(f) = lim inf
T→+∞inf
γ∈Γ
1 T
Z T 0
f(γ(t))dt
| {z }
g2T(f)
, (π∗g02)(f) = inf
γ∈Γlim inf
T→+∞
1 T
Z T 0
f(γ(t))dt,
that we simply denote bygT2(f), g2(f) and g20(f) respectively when the context is clear. Also, given any Lebesgue measurable1 subsetω of M, denoting byχω the characteristic function ofω, defined by χω(x) = 1 if x∈ ω and χω(x) = 0 otherwise, we often denote by gT2(ω), g2(ω) and g02(ω) instead ofg2T(χω),g2(χω) andg02(χω) respectively. Note that the real number
lim inf
T→+∞
1 T
Z T 0
χω(γ(t))dt
1Here, measurability is considered in the Lebesgue sense, that is, for instance, for the measureπ∗µLonM.
represents the average time spent by the rayγ inω.
It is interesting to notice that, forω⊂M open, ifg2(ω) = 0 theng20(ω) = 0 (see Lemma6).
Remark 1. Given any bounded measurable functionaon (S∗M, µL), we have g2T(a)6g2(a) ∀T >0 and g2(a) = lim
T→+∞gT2(a) = sup
T >0
gT2(a) = sup
T >0
inf ¯aT. Indeed, let Tm converging to +∞ such that limmg2Tm(a) = g2(a). In the following, bxcdenotes the integer part of the real numberx. WritingTm=bTTmcT+δmfor someδm∈[0, T], and setting nm=bTTmc, we have
gT2m(a) = inf 1 Tm
Z nmT 0
a◦ϕtdt+ 1 Tm
Z nMT+δm nmT
a◦ϕtdt
!
>inf 1 Tm
nm−1
X
k=0
Z (k+1)T kT
a◦ϕtdt
! .
Noting that T1 R(k+1)T
kT a◦ϕtdt> gT2(a) for everyk, we obtaing2Tm(a) > nTmmTgT2(a). The claim follows by lettingTm tend to +∞. Note that this argument is exactly the one used to establish Fekete’s Lemma: indeed, forafixed the functionT 7→T g2T(a) is superadditive.
Remark 2. We have g2T(a) 6 µ(a) for every T > 0, for every Borel measurable functiona on S∗M, and for every probability measureµonS∗M that is invariant under the geodesic flow. We will actually establish in Lemma4a more general result.
Remark 3. Setting at =a◦ϕt, and assuming that a ∈C0(S∗M) is the principal symbol of a pseudo-differential operatorA∈Ψ0(M) (of order 0), that is,a=σP(A), we have, by the Egorov theorem (see [12,21]),
at=a◦ϕt=σP(At) with At=e−it
√4Aeit
√4
whereσP(·) is the principal symbol. Accordingly, we have ¯aT =σP( ¯AT) with A¯T = 1
T Z T
0
Atdt= 1 T
Z T 0
e−it
√4Aeit
√4dt.
We provide hereafter a microlocal interpretation of the functionalsgT2,g2,g20 and give a relationship with the wave observability constant.
Microlocal interpretation of g2T, g2, g02, and of the wave observability constant. Let fT be such that ˆfT(t) = T1χ[0,T](t), i.e., fT(t) = 2π1ieiT t/2sinc(T t/2)). Note that R
IRfˆT = 1, i.e., equivalently,fT(0) = 1. Using thata◦etX = (etX)∗a=etLXa=eitSa, we get
gT2(a) = inf
z∈S∗M
1 T
Z T 0
a◦etX(z)dt= inf
z∈S∗M
Z
IR
fˆT(t)eitSa dt(z) = inffT(S)a.
Besides, settingA= Op(a), we have A¯T(a) = 1
T Z T
0
e−it
√4aeit
√4dt= Z
IR
fˆT(t)e−it
√4aeit
√4dt=AfT =X
λ,µ
fT(λ−µ)PλAPµ. Restricting to half-waves, the wave observability constant is therefore given (see [13]) by
CT(a) = inf
kyk=1hA¯T(a)y, yi= inf
kyk=1hAfTy, yi.
Note that
hAfTy, yi=X
λ,µ
fT(λ−µ)hAPλy, Pµyi=X
λ,µ
fT(λ−µ) Z
M
a Pλy Pµy
=X
λ,µ
fT(λ−µ)aλa¯µ
Z
M
aφλφµ
and we thus recover the expression ofCT(a) by series expansion.
Note also that, as said before, the principal symbol ofAfT = ¯AT(a) is σP(AfT) =σP( ¯AT(a)) =afT =
Z
IR
fˆT(t)a◦etXdt=fT(S)a and thatg2T(a) = infσP( ¯AT(a)).
Note as well that
g20(a) = inf
S∗Mlim inf
T→+∞fT(S)a
and that fT converges pointwise to χ{0} as T → +∞, and uniformly to 0 outside of 0. Since S = 1iLX is selfadjoint with compact inverse, it has a discrete spectrum 0 = µ0 < µ1 < · · · associated with eigenfunctions ψj. If a = P
ajψj, then fT(S)a = P
fT(µj)ajψj → a0ψ0 as T→+∞. In other words, we have
g20(a) = infQ0a
whereQ0is the projection onto the eigenspace ofSassociated with the eigenvalue 0, which is also the set of functions that are invariant under the geodesic flow.
1.2 Spectral quantities
Recall thatE is the set of normalized (i.e., of norm one inL2(M, dxg)) real-valued eigenfunctions φof√
4. Choosing a quantization2 Op onM, given any symbola∈ S0(M) of order 0, we define g1(a) = inf
φ∈EhOp(a)φ, φi
Note that this definition depends on the chosen quantization. In order to get rid of the quantization, one could defineg1(A) = infφ∈EhAφ, φifor everyA∈Ψ0(M) that is nonnegative and selfadjoint.
Note thatg1(A) is then the infimum of eigenvalues of the operatorA onL2(M, dxg).
As we have done forg2, we pushforward the functionalg1toM under the canonical projection π, and we set (noting that Op(f◦π)φ=f φ)
(π∗g1)(f) =g1(f ◦π) = inf
φ∈E
Z
M
f φ2dxg
for everyf ∈C0(M), which we also denote byg1(f). Note that the definition ofg1(f) still makes sense for essentially bounded Lebesgue measurable functions f on (M, dxg) which need not be continuous.
Accordingly, given any Lebesgue measurable subsetω ofM, we will often denote by g1(ω) = inf
φ∈E
Z
ω
φ2dxg
2A quantization is constructed by covering the closed manifoldMwith a finite number of coordinate charts; once this covering is fixed, by using a smooth partition of unity, we define the quantization of symbols that are supported in some coordinate charts, and there we choose a quantization, for instance the Weyl quantization.
to designate the quantityg1(χω). Note that, likeg2, the functional g1is an inner measure (which is not sub-additive in general).
Remark 4. It is interesting to note that, given any Lebesgue measurable subsetωofM such that
∂ω=ω\˚ω has zero Lebesgue measure (i.e.,ω is a Jordan measurable set), we have g1(˚ω) =g1(ω) =g1(ω).
Indeed, in this case we haveR
˚ωφ2dxg=R
ωφ2dxg=R
ωφ2dxgfor every φ∈ E.
More generally, we have g1(f) =g1( ˜f) for all essentially bounded Lebesgue measurable func- tionsf and ˜f coinciding Lebesgue almost everywhere on (M, dxg).
Quantum limits. We recall that a quantum limit (QL in short) µ, also called semi-classical measure, is a probability Radon (i.e., probability Borel regular) measure onS∗M that is a closure point (weak limit), asλ→+∞, of the family of Radon measuresµλ(a) =hOp(a)φλ, φλi (which are asymptotically positive by the G˚arding inequality), whereφλdenotes an eigenfunction of norm 1 associated with the eigenvalueλof√
4. We speak of aQL onM to refer to a closure point (for the weak topology) of the sequence of probability Radon measuresφ2λdxgonM asλ→+∞. Note that QLs do not depend on the choice of a quantization. We denote byQ(S∗M) (resp., Q(M)) the set of QLs (resp., the set of QLs onM). Both are compact sets.
Given any µ∈ Q(S∗M), the Radon measure π∗µ, image of µunder the canonical projection π:S∗M →M, is a probability Radon measure onM. It is defined, equivalently, by (π∗µ)(f) = µ(π∗f) =µ(f◦π) for everyf ∈C0(M) (note that, in local coordinates (x, ξ) inS∗M, the function f ◦π is a function depending only on x), or by (π∗µ)(ω) = µ(π−1(ω)) for every ω ⊂ M Borel measurable (or Lebesgue measurable, by regularity). It is easy to see that3
π∗Q(S∗M) =Q(M). (1)
In other words, QLs onM are exactly the image measures underπof QLs.
Given any bounded Borel measurable functionaonS∗M, we define g10(a) = inf
µ∈Q(S∗M)
Z
S∗M
a dµ
As before, we pushforward the functional g01 to M, by setting (π∗g01)(f) = g10(f ◦π) for every f ∈C0(M), which we often denote byg01(f). Thanks to (1), we have
g10(f) = inf
ν∈Q(M)ν(f).
It makes also sense to defineg01(ω) for any measurable subsetω ofM, by setting g01(ω) = inf
ν∈Q(M)ν(ω)
Remark 5. In contrast to Remark4, we may haveg01(ω)6=g10(ω) even for a Jordan setω. This is the case if one takesM =S2, the unit sphere in IR3, andωthe open northern hemisphere. Indeed, the Dirac along the equator is a QL (see, e.g., [14]) and thusg10(ω) = 0. But we haveg01(ω) = 1/2 (the infimum is reached for any QL that is the Dirac along a great circle transverse to the equator).
3Indeed, given anyf∈C0(M) and anyλ∈Spec(√
4), we have (π∗µλ)(f) =µλ(π∗f) =hOp(π∗f)φλ, φλi=
Z
M
f φ2λdxg,
because Op(π∗f)φλ=f φλ. The equality then easily follows by weak compactness of probability Radon measures.
Remark 6. Given anyν ∈ Q(M), there exists a sequence ofλ→+∞such thatφ2λdxg* ν, and it follows from the Portmanteau theorem (see AppendixA.1) thatν(ω)>limλ→+∞R
ωφ2λdxg>g1(ω) for any closed subsetωofM. Hence
g1(ω)6g10(ω) ∀ω⊂M closed,
or, more generally, for every Borel subsetωofM not charging any QL onM. Even more generally, we have
g1(a)6g01(a)
for every bounded Borel functionaonS∗M for which theµ-measure of the set of discontinuities ofais zero for everyµ∈ Q(S∗M). In particular the inequality holds true for anya∈ S0(M). We refer to Lemma5for this result.
The above inequality may be wrong without the specific assumption on ω or on a. Indeed, consider again the example given in Remark5: M =S2,ω is the open northern hemisphere, then g1(ω) =g1(ω) = 1/2 (by symmetry arguments as in [15]), whereasg10(ω) = 0.
Finally, it is interesting to notice that, for ω ⊂ M open, if g1(ω) = 0 then g10(ω) = 0 (see Lemma5).
Invariant measures. LetI(S∗M) be the set of probability Radon measures onS∗M that are invariant under the geodesic flow. It is a compact set. It is well known that, by the Egorov theorem, we haveQ(S∗M)⊂ I(S∗M).4 The converse inclusion is not true. However, it is known that, ifM has the spectral gap property5, thenM is Zoll (i.e., all its geodesics are periodic) and Q(S∗M) =I(S∗M) (see [17, Theorem 2 and Remark 3]).
Given any bounded Borel measurable functionaonS∗M, we define g001(a) = inf
µ∈I(S∗M)
Z
S∗M
a dµ SinceQ(S∗M)⊂ I(S∗M), we have
g100(a)6g10(a)
for every bounded Borel measurable function a on S∗M. As before, given any bounded Borel measurable functionf onM, the notationg001(f) stands forg100(f◦π) without any ambiguity.
Remark 7. Since the set of extremal points of I(S∗M) is the set of ergodic measures, in the definition ofg100we can replaceI(S∗M) by the set of ergodic measures in the infimum.
Remark 8. It follows from [20] that, if the manifold M (which is connected and compact) is of negative curvature then the set of Dirac measuresδγ along periodic geodesic raysγ∈Γ is dense inI(S∗M) for the vague topology, and therefore
g100(a) = inf
γ∈Γ periodicδγ(a) = inf (1
T Z T
0
a◦ϕt(z)dt | z∈S∗M, T >0, ϕT(z) =z )
for every continuous functionaonS∗M.
4Indeed, by the Egorov theorem (see also Remark 3), at = a◦ ϕt is the principal symbol of At = e−it
√4Op(a)eit
√4. Let µ∈ Q(S∗M). By definition,µ(a) is the limit of (some subsequence of)hOp(a)φj, φji, henceµ(a◦ϕt) = limhOp(a)eit
√4φj, eit
√4φji=µ(a) becauseeit
√4φj=eitλjφj.
5We say that M has the spectral gap property if there existsc >0 such that|λ−µ|>cfor any twodistinct eigenvaluesλandµof√
4. This property allows multiplicity.
1.3 Known results on Zoll manifolds
Recall that a Zoll manifold is a smooth connected closed Riemannian manifold without boundary, of which all geodesics are periodic. Thanks to a theorem by Wadsley (see [3]), they have a least common periodT > 0. This does not mean that all geodesics are T-periodic: there may exist exceptional geodesics with period less thanT, like in the lens-spaces, that are quotients ofS2m−1 by certain finite cyclic groups of isometries.
Note that, in some of the existing literature, “Zoll manifold” means that not only all geodesics are periodic, but also, have the same period. Here, we relax the latter statement (we could name this kind of manifold a “weak Zoll manifold”).
We consider the eigenvaluesλof the operator√
4considered on the compact manifoldM. Let X be the Hamiltonian vector field onS∗M of√
4. Note thatetX =ϕtfor everyt∈IR. Denoting byLX the Lie derivative with respect toX, we define the self-adjoint operatorS= 1iLX. We also define Σ as the set of closure points of allλ−µ.
LetA∈Ψ0(M), of principal symbol a. For every functionf on IR, we set Af =
Z
IR
fˆ(t)e−it
√4Aeit
√4dt.
Following [9], its principal symbol is computed on a finite time interval (by the Egorov theorem) and by passing to the limit (by using the Lebesgue dominated convergence theorem), and we get
af =σP(Af) = Z
IR
fˆ(t)a◦etXdt= Z
IR
fˆ(t)(etX)∗a dt.
We will also denoteϕt=etX. Besides, we have
(etX)∗a=a◦etX =etLXa=eitSa, and hence
af =σP(Af) = Z
IR
f(t)eˆ itSa dt=f(S)a.
Denoting byPλ the projection onto the eigenspace corresponding to the eigenvalueλ, we have p4= X
λ∈Spec(√ 4)
λPλ, eit
√4= X
λ∈Spec(√ 4)
eitλPλ, (2) and we obtain
Af =X
λ,µ
f(λ−µ)PλAPµ.
Note that, by definition ofS, using that LXa ={H, a} where H = σP(√
4) (Hamiltonian), we haveSa= 1i{a, H}. As a consequence, the eigenfunctions ofS corresponding to the eigenvalue 0 are exactly the functions that are invariant under the geodesic flow.
It is remarkable that periodicity of geodesics and periodicity of the spectrum are closely related (see [5,7,9,11]). We gather these classical results in the following theorem.
Theorem 1([5,7,9, 11]). We have the following results:
• Spec(S)⊂Σ.
• If there exists a non-periodic geodesic, then Spec(S) =IR, and thus Σ =IR.
• M is Zoll if and only if Σ 6=IR. In this case, we have Σ = 2πT Z, where T is the smallest common period.
Note that, if Σ = IR, then there exists a non-periodic geodesic. Indeed, otherwise any geodesic would be periodic, and by the Wadsley theorem, M would be Zoll and then this implies that Σ =2πT Z.
Actually, if M is Zoll, then, denoting by T the (common) period, we have λj ' 2πT (σ+ni) for someni ∈ Z(asymptotic spectrum of √
4). In other words, this means that the eigenvalues cluster along the net 2πσT +2πT Z.
1.4 Main results: new characterizations of Zoll manifolds
Given a periodic rayγ on M, the Dirac measureδγ on M is defined byδγ(f) = T1 RT
0 f(γ(t))dt for everyf ∈ C0(M), where T is the period of γ. Accordingly, given a periodic geodesic ˜γ on S∗M, the Dirac measureδ˜γ on S∗M is defined by δγ(a) = T1 RT
0 a◦ϕt(z)dt = ¯aT(z) for every a∈C0(S∗M), where ˜γ(t) =ϕt(z) for somez∈S∗M.
Note that, for a periodic geodesic ˜γ on S∗M, settingγ =π◦˜γ, we have δγ ∈ Q(M) if and only if δγ˜ ∈ Q(S∗M) (this follows from Proposition 1 in Appendix A.2 and from the fact that Q(S∗M)⊂ I(S∗M)). Hence, in the theorem below, saying that “the Dirac along any periodic ray is a QL onM” is equivalent to saying that “the Dirac along any geodesic is a QL”.
Before stating the result hereafter, we give two definitions concerning the spectrum:
• We say thatM has thespectral gap property if there existsc >0 such that|λ−µ|>cfor any twodistincteigenvalues λandµof√
4. This property allows multiplicity.
• We say that the spectrum is uniformly locally finite if there exist ` >0 and m ∈IN∗ such that the intersection of the spectrum with any interval of length ` has at mostm distinct elements. This does not preclude multiplicity with arbitrarily large order.
Also, in order to appreciate the contents of the following result, it is useful to note that gT2(a)6g2(a)6g02(a)6g100(a)6g01(a)
for everyT >0 and for every bounded Borel measurable functionaonS∗M, and that g1(a)6g10(a) ∀a∈C0(S∗M)
g1(ω)6g10(ω) ∀ω⊂M closed These facts will be proved in Lemmas4and5.
The next theorem, which is the main result of this paper, gives new characterizations of Zoll manifolds and relations with quantum limits.
Theorem 2.
1. The following statements are equivalent:
• M is Zoll and δγ ∈ Q(M)for every periodic rayγ∈Γ;
• g2(a) =g20(a) =g100(a) =g10(a)for every bounded Borel measurable functionaonS∗M;
• g02(ω) =g100(ω) =g01(ω)for every ω⊂M Borel measurable;
• there existsT >0such that g1(ω)6g2T(ω)for any closed subsetω⊂M.
Moreover, the smallestT >0 such thatg1(ω)6gT2(ω) for any closed subsetω ⊂M is the smallest period of geodesics of the Zoll manifoldM.
2. The following statements are equivalent:
• M is Zoll;
• g2(a) =g20(a)for every bounded Borel measurable functionaonS∗M;
• g2(ω) =g20(ω) for everyω⊂M Borel measurable;
• g02(a) =g100(a)for every bounded Borel measurable functionaon S∗M;
• g02(ω) =g100(ω)for everyω⊂M Borel measurable;
• there existsT >0such that g2T(ω) =g2(ω)for every open subset ω⊂M;
• there existsT >0such that g2T(ω) =g2(ω)for every closed subsetω⊂M;
• there existsT >0such that g2T(ω) =g02(ω)for any closed subsetω⊂M;
• g2(ω) =g20(ω) for every closed subsetω⊂M.
Moreover, the smallestT >0such that the items above are satisfied is the smallest period of geodesics of the Zoll manifoldM.
3. If g1(ω) 6 g100(ω) for any ω ⊂ M closed then δγ ∈ Q(M) for every periodic ray γ ∈ Γ.
Moreover, for every minimally invariant6compact setK, there exists a quantum limit whose support isK.
4. If M is Zoll and is a two-point homogeneous space7 and if the Dirac along any periodic geodesic is a QL then Q(S∗M) =I(S∗M).
5. Under the spectral gap assumption,M is Zoll andδγ ∈ Q(M)for every periodic rayγ∈Γ.
6. If the spectrum is uniformly locally finite thenM is Zoll and for every rayγ∈Γ there exists ν∈ Q(M)such thatν(γ(IR))>0.
Remark 9. The statements 3 and 4 of the theorem above are already known. Statement 3 is exactly the contents of [17, Theorems 4 and 8]. Concerning the statement 4: under the spectral gap assumption, we have Σ6= IR and thus M is Zoll by Theorem 1, and the fact that the Dirac along any geodesic is a QL has been established in [17, Theorem 2 and Remark 3].
Remark 10. The assumption of uniform locally finite spectrum means that clustering is possible but only with a uniformly bounded number of distinct eigenvalues, but it allows arbitrary large multiplicities of eigenvalues.
Note that, by Theorem1,M is Zoll if and only if Σ6= IR. Therefore, in turn we have obtained that if the spectrum is uniformly locally finite then we must have Σ 6= IR. This rules out the possibility of having a spectrum consisting, for instance, of allj2/n, j2/n+qj, forj ∈IN∗, where (qj)j∈IN∗ is a countable description ofQ(because then we haveQ⊂Σ, and hence Σ = IR).
Remark 11. By the Egorov theorem, we have the inclusionQ(S∗M)⊂ I(S∗M), and in general this inclusion is strict. Remarks are in order:
• We have Q(S∗M) = I(S∗M) when M is the sphere in any dimension endowed with its canonical metric (see [14]), or more generally, when M is a compact rank-one symmetric space, which is a special case of a Zoll manifold (see [17]).
• We do not know if, for a given Riemannian manifold M, the equality Q(S∗M) =I(S∗M) implies thatM is Zoll.
6A minimally invariant set is a nonempty closed invariant set containing no proper closed invariant subset.
7By definition, this means that, given pointsx0, x1, y0, y1 inM such thatd(x0, y0) =d(x1, y1), there exists an isometryϕofMsuch thatϕ(x0) =x1andϕ(y0) =y1. This is equivalent to say that the group Iso(M) of isometries ofM acts transitively onM×M.
• Conversely,M Zoll does not implyQ(S∗M) =I(S∗M). Indeed, by [18, Theorem 1.4], there exist two-dimensional Zoll manifolds M (Tannery surfaces) for which there exists a ray γ (and even, an open set of rays) such thatδγ ∈ Q(M/ ). In particular, for such Zoll manifolds we have Q(S∗M) ( I(S∗M). Also, by Theorem 2, there must exist a Borel measurable subsetω⊂M (and even, an open subset) such thatg100(ω)< g10(ω).
• We do not know any example of a Zoll manifold M for which the spectrum is uniformly locally finite andQ(S∗M)(I(S∗M).
• It is interesting to note that ifM is of negative curvature then the Diracδγ along a periodic rayγ∈Γ can never be a QL (see [1, 2]).
2 Proofs
2.1 General results
Note the obvious fact that if aand b are functions such that a6 b and for which the following quantities make sense, then gT2(a) 6 gT2(b), g2(a) 6 g2(b), g1(a) 6 g1(b), g01(a) 6 g10(b) and g001(a)6g100(b). In other words, the functionals that we have defined are nondecreasing.
2.1.1 Semi-continuity properties
Lemma 1. Letω be a subset ofM, letT >0be arbitrary and let(hk)k∈IN∗ be a uniformly bounded sequence of Borel measurable functions onM.
• If hk converges pointwise toχω, then lim sup
k→+∞
g2T(hk)6gT2(ω), lim sup
k→+∞
g01(hk)6g10(ω), lim sup
k→+∞
g001(hk)6g001(ω), and if moreoverχω6hk for every k∈IN∗, then
gT2(ω) = lim
k→+∞gT2(hk) = inf
k∈IN∗gT2(hk), g01(ω) = lim
k→+∞g10(hk) = inf
k∈IN∗g01(hk), g001(ω) = lim
k→+∞g100(hk) = inf
k∈IN∗g001(hk).
• If hk converges Lebesgue almost everywhere toχω, then lim sup
k→+∞
g1(hk)6g1(ω), and if moreoverχω6hk for every k∈IN∗, then
g1(ω) = lim
k→+∞g1(hk) = inf
k∈IN∗g1(hk).
Proof.
• Let γ ∈ Γ be arbitrary. By pointwise convergence, we have hk(γ(t)) → χω(γ(t)) for ev- ery t ∈ [0, T], and it follows from the dominated convergence theorem that g2T(hk) 6
1 T
RT
0 hk(γ(t))dt → T1 RT
0 χω(γ(t))dt, and thus lim supk→+∞gT2(hk) 6 T1
RT
0 χω(γ(t))dt.
Since this inequality is valid for anyγ∈Γ, the first inequality follows.
By compactness of QLs, there exists ν ∈ Q(M) such that g10(ω) = ν(ω). By dominated convergence, we haveg01(hk) 6R
Mhkdν →R
Mχωdν =ν(ω) = g10(ω), whence the second inequality. The proof forg001 is similar.
If moreoverχω6hkthen lim supk→+∞gT2(hk)6g2T(ω)6gT2(hk) and lim supk→+∞g10(hk)6 g10(ω)6g10(hk) and the result follows.
• We have g1(ω) = infφ∈Eνφ(ω) with νφ = φ2dxg. Let φ ∈ E be arbitrary. For every k ∈ IN∗ we have g1(hk) 6 R
Mhkφ2dxg, and besides, by dominated convergence we have R
Mhkφ2dxg → R
ωφ2dxg = νφ(ω), hence lim supk→+∞g1(hk) 6 νφ(ω). Since φ ∈ E is arbitrary, we get lim supk→+∞g1(hk)6infφ∈Eνφ(ω) =g1(ω).
If moreoverχω6hk then lim supk→+∞g1(hk)6g1(ω)6g1(hk) and the result follows.
Note that, in the proof forg2T andg10, we use the fact thathk(x)→χω(x) forevery x. Almost everywhere convergence (in the Lebesgue sense) would not be enough.
Remark 12. We denote bydthe geodesic distance on (M, g). It is interesting to note that, given any subsetωofM:
• ω is open if and only if there exists a sequence of continuous functions hk onM satisfying 06hk6hk+16χωfor every k∈IN∗ and converging pointwise toχω.
Indeed, ifω is open, then one can take for instance hk(x) = min(1, k d(x, ωc)). Conversely, since hk(x) = 0 for every x ∈ ωc, by continuity of hk it follows that hk(x) = 0 for every x∈ωc= (˚ω)c. Now takex∈ω\˚ω. We havehk(x) = 0, andhk(x)→χω(x), henceχω(x) = 0 and thereforex∈ωc. Hence ωis open.
• ω is closed if and only if there exists a sequence of continuous functionshk onM satisfying 06χω6hk+16hk61 for everyk∈IN∗ and converging pointwise toχω.
Indeed, if ω is closed, then one can take hk(x) = max(0,1−k d(x, ω)). Conversely, since hk(x) = 1 for every x∈ ω, by continuity ofhk it follows that hk(x) = 1 for everyx ∈ω, and thusχω6hk61. Now takex∈ω\ω. We havehk(x) = 1 andhk(x)→χω(x), hence χω(x) = 1 and thereforex∈ω. Henceω is closed.
Lemma 2. Let ω be an open subset of M and let T > 0 be arbitrary. For every sequence of continuous functionshkonM converging pointwise toχω, satisfying moreover06hk 6hk+16χω
for everyk∈IN∗, we have gT2(ω) = lim
k→+∞g2T(hk) = sup
k∈IN∗
g2T(hk), g2(ω) = lim
k→+∞g2(hk) = sup
k∈IN∗
g2(hk), g10(ω) = lim
k→+∞g10(hk) = sup
k∈IN∗
g10(hk), g100(ω) = lim
k→+∞g001(hk) = sup
k∈IN∗
g100(hk).
Note that, for g1, the property g1(ω) = limk→+∞g1(hk) = supk∈IN∗g1(hk) may fail. Indeed, takeM =S2,ω the open Northern hemisphere, theng1(ω) = 1/2 andg1(hk) = 0 for everyk.
Proof. Since hk 6 χω, we have gT2(hk) 6 g2T(ω). By continuity of hk and by compactness of geodesics, there exists a rayγk such that g2T(hk) = T1RT
0 hk(γk(t))dt. Again by compactness of geodesics, up to some subsequenceγk converges to a ray ¯γ inC0([0, T], M). We claim that
lim inf
k→+∞hk(γk(t))>χω(¯γ(t)) ∀t∈[0, T].
Indeed, either ¯γ(t)∈/ ωand thenχω(¯γ(t)) = 0 and the inequality is obviously satisfied, or ¯γ(t)∈ω and then, using thatω is open, fork large enough we haveγk(t)∈U whereU ⊂ω is a compact neighborhood of ¯γ(t). Since hk is monotonically nondecreasing and χω is continuous on U, it follows from the Dini theorem thathk converges uniformly to χω on U, and then we infer that hk(γk(t))→1 =χω(¯γ(t)). The claim is proved. Now, we infer from the Fatou lemma that
gT2(ω)6 1 T
Z T 0
χω(¯γ(t))dt6 1 T
Z T 0
lim inf
k→+∞hk(γk(t))dt 6lim inf
k→+∞
1 T
Z T 0
hk(γk(t))dt= lim inf
k→+∞g2T(hk)6gT2(ω) and we get the equality.
Sinceg2(ω) = supT >0g2T(ω) by Remark1, interverting the sup yields g2(ω) = sup
T >0
sup
k∈IN∗
gT2(hk) = sup
k∈IN∗
sup
T >0
gT2(hk) = sup
k∈IN∗
g2(hk).
By compactness of QLs, there exists νk ∈ Q(M) such that g01(hk) = νk(hk). Again by com- pactness of QLs, up to some subsequence we have νk * ν¯ ∈ Q(M). Since ω is open, we can write
ω=∪ε>0Vε, Vε={x∈ω | d(x, ωc)> ε},
Vε being open. Let ε > 0 be arbitrarily fixed. By construction, Vε = {x ∈ ω | d(x, ωc)> ε}
is a compact set contained in the open set ω. Since hk converges monotonically pointwise to χω, by the Dini theorem, hk converges uniformly to 1 on Vε, and thus without loss of gen- erality we write that χVε 6 hk. By the Portmanteau theorem (see Appendix A.1), we have
¯
ν(Vε)6lim infk→+∞νk(Vε), and we haveνk(Vε) =R
MχVεdνk 6R
Mhkdνk =g10(hk). It follows that ¯ν(Vε) 6lim infk→+∞g01(hk). Now we let ε converge to 0 and we get that g01(ω) 6ν(ω)¯ 6 lim infk→+∞g01(hk). The result forg10 follows. The proof forg001 is similar.
Remark 13. The results of Lemmas 1 and 2 are valid as well for subsets of S∗M (which is a metric space).
Lemma 3. Let ωbe an open subset ofM and letT >0be arbitrary. There existsγ∈Γ such that gT2(ω) = T1 RT
0 χω(γ(t))dt, i.e., the infimum in the definition of gT2(ω)is reached.
Proof. The argument is almost contained in the proof of Lemma2, but for completeness we give the detail. Let (γk)k∈IN∗ be a sequence of rays such that T1 RT
0 χω(γk(t))dt→g2T(ω). By compactness of geodesics,γk(·) converges uniformly to some ray γ(·) on [0, T].
Let t ∈[0, T] be arbitrary. If γ(t)∈ ω then for k large enough we have γk(t) ∈ω, and thus 1 =χω(γ(t))6χω(γk(t)) = 1. If γ(t)∈ M \ω then 0 = χω(γ(t))6χω(γk(t)) for any k. In all cases, we have obtained the inequalityχω(γ(t))6lim infk→+∞χω(γk(t)), for everyt∈[0, T].
By the Fatou lemma, we infer that gT2(ω)6 1
T Z T
0
χω(γ(t))dt6 1 T
Z T 0
lim inf
k→+∞χω(γk(t))dt6lim inf
k→+∞
1 T
Z T 0
χω(γk(t))dt=gT2(ω) and the equality follows.
2.1.2 General inequalities Lemma 4. We have
gT2(a)6g2(a)6g02(a)6g100(a)6g01(a)
for everyT >0and for every bounded Borel measurable function aonS∗M. Proof. Given anyµ∈ I(S∗M), we haveR
S∗Ma dµ=R
S∗Ma◦φtdµfor any bounded Borel measur- able functionaonS∗M and for anyt∈IR, and thusR
S∗Ma dµ=R
S∗M 1 T
RT
0 a◦φtdµ=R
S∗M¯aTdµ for anyT >0. Passing to the limit we getR
S∗Ma dµ=R
S∗Mlim infT→+∞¯aTdµ>g02(a), because g02(a) = inf lim infT→+∞¯aT by definition. The result follows.
Lemma 5. • We have g1(a) 6g10(a) for every bounded Borel function a on S∗M for which the µ-measure of the set of discontinuities ofais zero for every µ∈ Q(S∗M). In particular the inequality is valid for every continuous function aonS∗M.
• Given any closed subsetω of M (or ofS∗M), we have g1(ω)6g10(ω).
• Let ω be an open subset of M. Ifg1(ω) = 0 theng10(ω) = 0.
Proof. The first claim follows by the Portmanteau theorem (see Appendix A.1) and by definition of QLs.
The second claim follows by using Lemma1and Remark 12.
Let us prove the third item. If g1(ω) = 0 for some measurable subset ω of positive Lebesgue measure, then either R
ωφ2dxg = 0 for some φ ∈ E or lim infλ→+∞R
ωφ2λdxg = 0. The first possibility cannot occur: it cannot happen thatR
ωφ2dxg= 0 because this would imply thatφ= 0 on a subsetωof positive Lebesgue measure and thus of Hausdorff dimensionn, which is impossible by results of [6, 10, 16] (this impossibility is obvious when the manifold M is analytic because thenφ is analytic). Therefore lim infλ→+∞R
ωφ2λdxg = 0. Up to some subsequence, there exists ν∈ Q(M) that is the weak limit ofφ2λdxg. Now, by the Portmanteau theorem (see AppendixA.1), sinceω is open we haveν(ω)6lim infλ→+∞R
ωφ2λdxg, and thusν(ω) = 0. The claim follows.
Lemma 6. Let ω be an open subset of M. Ifg2(ω) = 0then g02(ω) =g001(ω) = 0.
Proof. By Remark 1, we have gT2(ω) 6 g2(ω) and thus gT2(ω) = 0 for every T > 0. Now, by Lemma 3, for every k ∈ IN∗ there exists γk ∈Γ such that g2k(ω) = k1Rk
0 χω(γk(t))dt = 0, hence χω(γk(t)) = 0 (i.e.,γk(t)∈M\ω) for almost everyt∈[0, k]. By compactness of geodesics, up to some subsequenceγk converges to a rayγ∈Γ uniformly on any compact interval. Using the fact thatM\ωis closed, we infer thatγ(IR)⊂M\ω. Not only it follows thatg20(ω) = 0, but also that g001(ω) = 0. To obtain the last statement, take an invariant probability measureν on the compact set γ(IR) (there always exists at least one such measure) and consider the invariant probability measureνγ onM defined byνγ(E) =ν(E∩γ(IR)) for every Borel setE⊂M.
2.2 Proof of Theorem 2
Lemma 7. If M is Zoll theng2(a) =g02(a) =g100(a)for any bounded Borel measurable functiona onS∗M.
If moreover δγ ∈ Q(M)for every periodic ray γ ∈ Γ then g2(a) = g02(a) =g100(a) =g01(a) for any bounded Borel measurable functionaonS∗M.
