ENTROPY OF EIGENFUNCTIONS
NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER
Abstra
t. We study the high�energy limit for eigenfun
tions of the lapla
ian, on a

ompa
t negatively 
urved manifold. We review the re
ent result of Anantharaman�
Nonnenma
her [4℄ giving a lower bound on the Kolmogorov�Sinai entropy of semi
lassi
al
measures. The bound proved here improves the result of [4℄ in the 
ase of variable negative

urvature.
1. Motivations
The theory of quantum 
haos tries to understand how the 
haoti
 behaviour of a 
lassi-

al Hamiltonian system is re�e
ted in its quantum 
ounterpart. For instan
e, let M be a

ompa
t Riemannian C∞ manifold, with negative se
tional 
urvatures. The geodesi
 �ow
has the Anosov property, whi
h is 
onsidered as the ideal 
haoti
 behaviour in the theory
of dynami
al systems. The 
orresponding quantum dynami
s is the unitary �ow gener-
ated by the Lapla
e-Beltrami operator on L2(M). One expe
ts that the 
haoti
 properties
of the geodesi
 �ow in�uen
e the spe
tral theory of the Lapla
ian. The Random Matrix

onje
ture [7℄ asserts that the large eigenvalues should, after proper unfolding, statisti-

ally resemble those of a large random matrix, at least for a generi
 Anosov metri
. The
Quantum Unique Ergodi
ity 
onje
ture [26℄ (see also [6, 30℄) des
ribes the 
orresponding
eigenfun
tions ψk: it 
laims that the probability measure |ψk(x)|
2dx should approa
h (in
the weak topology) the Riemannian volume, when the eigenvalue tends to in�nity. In fa
t
a stronger property should hold for the Wigner transform Wψ, a fun
tion on the 
otangent
bundle T ∗M , (the 
lassi
al phase spa
e) whi
h simultaneously des
ribes the lo
alization of
the wave fun
tion ψ in position and momentum.
We will adopt a semi
lassi
al point of view, that is 
onsider the eigenstates of eigenvalue
unity of the semi
lassi
al Lapla
ian −~2△, thereby repla
ing the high-energy limit by the
semi
lassi
al limit ~ → 0. We denote by (ψk)k∈N an orthonormal basis of L
2(M) made of
eigenfun
tions of the Lapla
ian, and by (− 1
)k∈N the 
orresponding eigenvalues:
(1.1) − ~2k△ψk = ψk, with ~k+1 ≤ ~k .
We are interested in the high-energy eigenfun
tions of −△, in other words the semi
lassi
al
limit ~k → 0.
The Wigner distribution asso
iated to an eigenfun
tion ψk is de�ned by
Wk(a) = 〈Op~k(a)ψk, ψk〉L2(M), a ∈ C
∗M) .
Here Op
is a quantization pro
edure, set at the s
ale (wavelength) ~k, whi
h asso
iates to
any smooth phase spa
e fun
tion a (with ni
e behaviour at in�nity) a bounded operator on
http://arxiv.org/abs/0704.1564v1
2 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
L2(M). See for instan
e [13℄ or [14℄ for various quantizations Op
. On a manifold,
one 
an use lo
al 
oordinates to de�ne Op in a �nite system of 
harts, then glue the obje
ts
de�ned lo
ally thanks to a smooth partition of unity [11℄. For standard quantizations Op
the Wigner distribution is of the form Wk(x, ξ) dx dξ, where Wk(x, ξ) is a smooth fun
tion
on T ∗M , 
alled the Wigner transform of ψ. If a is a fun
tion on the manifold M , Op

an be taken as the multipli
ation by a, and thus we have Wk(a) =
a(x)|ψk(x)|
2dx: the
Wigner transform is thus a mi
rolo
al lift of the density |ψk(x)|
. Although the de�nition
ofWk depends on a 
ertain number of 
hoi
es, like the 
hoi
e of lo
al 
oordinates, or of the
quantization pro
edure (Weyl, anti-Wi
k, �right� or �left� quantization...), its asymptoti
behaviour when ~k −→ 0 does not. A

ordingly, we 
all semi
lassi
al measures the limit
points of the sequen
e (Wk)k∈N, in the distribution topology.
In the semi
lassi
al limit, �quantum me
hani
s 
onverges to 
lassi
al me
hani
s�. We
will denote |·|x the norm on T
xM given by the metri
. The geodesi
 �ow (g
t)t∈R is the
Hamiltonian �ow on T ∗M generated by the Hamiltonian H(x, ξ) =
|ξ|2x
. A quantization
of this Hamiltonian is given by the res
aled Lapla
ian −~
, whi
h generates the unitary
�ow (U t
) = (exp(it~△
)) a
ting on L2(M). The semi
lassi
al 
orresponden
e of the �ows
) and (gt) is expressed through the Egorov Theorem :
Theorem 1.1. Let a ∈ C∞c (T
∗M). Then, for any given t in R,
(1.2) ‖U−t
(a)U t
(a ◦ gt)‖L2(M) = O(~) , ~ → 0 .
The 
onstant implied in the remainder grows (often exponentially) with t, whi
h rep-
resents a notorious problem when one wants to study the large time behaviour of (U t
Typi
ally, the quantum-
lassi
al 
orresponden
e will break down for times t of the order
of the Ehrenfest time (3.25).
Using (1.2) and other standard semi
lassi
al arguments, one shows the following :
Proposition 1.2. Any semi
lassi
al measure is a probability measure 
arried on the energy
layer E = H−1(1
) (whi
h 
oin
ides with the unit 
otangent bundle S∗M). This measure is
invariant under the geodesi
 �ow.
Let us 
all M the set of gt-invariant probability measures on E . This set is 
onvex
and 
ompa
t for the weak topology. If the geodesi
 �ow has the Anosov property � for
instan
e if M has negative se
tional 
urvature � that set is very large. The geodesi
�ow has 
ountably many periodi
 orbits, ea
h of them 
arrying an invariant probability
measure. There are many other invariant measures, like the equilibrium states obtained
by variational prin
iples [19℄, among them the Liouville measure µLiouv, and the measure
of maximal entropy. Note that, for all these examples of measures, the geodesi
 �ow
a
ts ergodi
ally, meaning that these examples are extremal points in M. Our aim is to
determine, at least partially, the set Msc formed by all possible semi
lassi
al measures. By
its de�nition, Msc is a 
losed subset of M, in the weak topology.
For manifolds su
h that the geodesi
 �ow is ergodi
 with respe
t to the Liouville measure,
it has been known for some time that almost all eigenfun
tions be
ome equidistributed over
E , in the semi
lassi
al limit. This property is dubbed as Quantum Ergodi
ity :
ENTROPY OF EIGENFUNCTIONS 3
Theorem 1.3. [27, 32, 11℄ Let M be a 
ompa
t Riemannian manifold, assume that the
a
tion of the geodesi
 �ow on E = S∗M is ergodi
 with respe
t to the Liouville measure.
Let (ψk)k∈N be an orthonormal basis of L
2(M) 
onsisting of eigenfun
tions of the Lapla
ian
(1.1), and let (Wk) be the asso
iated Wigner distributions on T
Then, there exists a subset S ⊂ N of density 1, su
h that
(1.3) Wk −→µLiouv, k → ∞, k ∈ S.
The question of existen
e of �ex
eptional� subsequen
es of eigenstates with a di�erent
behaviour is still open. On a negatively 
urved manifold, the geodesi
 �ow satis�es the
ergodi
ity assumption, and in fa
t mu
h stronger properties : mixing, K�property, et
.
For su
h manifolds, it has been postulated in the Quantum Unique Ergodi
ity 
onje
ture
[26℄ that the full sequen
e of eigenstates be
omes semi
lassi
ally equidistributed over E :
one 
an take S = N in the limit (1.3). In other words, this 
onje
ture states that there
exists a unique semi
lassi
al measure, and Msc = {µLiouv}.
So far the most pre
ise results on this question were obtained for manifolds M with

onstant negative 
urvature and arithmeti
 properties: see Rudni
k�Sarnak [26℄, Wolpert
[31℄. In that very parti
ular situation, there exists a 
ountable 
ommutative family of
self�adjoint operators 
ommuting with the Lapla
ian : the He
ke operators. One may
thus de
ide to restri
t the attention to 
ommon bases of eigenfun
tions, often 
alled �arith-
meti
� eigenstates, or He
ke eigenstates. A few years ago, Lindenstrauss [24℄ proved that
any sequen
e of arithmeti
 eigenstates be
ome asymptoti
ally equidistributed. If there is
some degenera
y in the spe
trum of the Lapla
ian, note that it 
ould be possible that the
Quantum Unique Ergodi
ity 
onje
tured by Rudni
k and Sarnak holds for one orthonormal
basis but not for another. On su
h arithmeti
 manifolds, it is believed that the spe
trum
of the Lapla
ian has bounded multipli
ity: if this is really the 
ase, then the semi
lassi
al
equidistribution easily extends to any sequen
e of eigenstates.
Nevertheless, one may be less optimisti
 when extending the Quantum Unique Ergod-
i
ity 
onje
ture to more general systems. One of the simplest example of a symple
ti
Anosov dynami
al system is given by linear hyperboli
 automorphisms of the 2-torus, e.g.
Arnold's �
at map�
. This system 
an be quantized into a sequen
e ofN×N unitary
matri
es � the propagators, where N ∼ ~−1 [18℄. The eigenstates of these matri
es satisfy
a Quantum Ergodi
ity theorem similar with Theorem 1.3, meaning that almost all eigen-
states be
ome equidistributed on the torus in the semi
lassi
al limit [9℄. Besides, one 
an

hoose orthonormal eigenbases of the propagators, su
h that the whole sequen
e of eigen-
states is semi
lassi
ally equidistributed [22℄. Still, be
ause the spe
tra of the propagators
are highly degenerate, one 
an also 
onstru
t sequen
es of eigenstates with a di�erent limit
measure [15℄, for instan
e, a semi
lassi
al measure 
onsisting in two ergodi
 
omponents:
half of it is the Liouville measure, while the other half is a Dira
 peak on a single (unsta-
ble) periodi
 orbit. It was also shown that this half-lo
alization is maximal for this model
[16℄ : a semi
lassi
al measure 
annot have more than half its mass 
arried by a 
ountable
union of periodi
 orbits. The same type of half-lo
alized eigenstates were 
onstru
ted by
two of the authors for another solvable model, namely the �Walsh quantization� of the
4 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
baker's map on the torus [3℄; for that model, there exist ergodi
 semi
lassi
al measures of
purely fra
tal type (that is, without any Liouville 
omponent). Another type of semi
las-
si
al measure was re
ently obtained by Kelmer for quantized hyperboli
 automorphisms
on higher-dimensional tori [20℄: it 
onsists in the Lebesgue measure on some invariant

o-isotropi
 subspa
e of the torus.
For these Anosov models on tori, the 
onstru
tion of ex
eptional eigenstates strongly
uses nongeneri
 algebrai
 properties of the 
lassi
al and quantized systems, and 
annot be
generalized to nonlinear systems.
2. Main result.
In order to understand the set Msc, we will attempt to 
ompute the Kolmogorov�Sinai
entropies of semi
lassi
al measures. We work on a 
ompa
t Riemannian manifold M of
arbitrary dimension, and assume that the geodesi
 �ow has the Anosov property. A
tually,
our method 
an without doubt be adapted to more general Anosov Hamiltonian systems.
The Kolmogorov�Sinai entropy, also 
alled metri
 entropy, of a (gt)-invariant probability
measure µ is a nonnegative number hKS(µ) that des
ribes, in some sense, the 
omplexity of
a µ-typi
al orbit of the �ow. The pre
ise de�nition will be given later, but for the moment
let us just give a few fa
ts. A measure 
arried on a 
losed geodesi
 has vanishing entropy.
In 
onstant 
urvature, the entropy is maximal for the Liouville measure. More generally,
for any Anosov �ow, the energy layer E is foliated into unstable manifolds of the �ow. An
upper bound on the entropy of an invariant probability measure is then provided by the
Ruelle inequality:
(2.1) hKS(µ) ≤
log Ju(ρ)dµ(ρ)
In this inequality, Ju(ρ) is the unstable Ja
obian of the �ow at the point ρ ∈ E , de�ned
as the Ja
obian of the map g−1 restri
ted to the unstable manifold at the point g1ρ (note
that the average of log Ju over any invariant measure is negative). The equality holds in
(2.1) if and only if µ is the Liouville measure on E [23℄. If M has dimension d and has

onstant se
tional 
urvature −1, the above inequality just reads hKS(µ) ≤ d− 1.
Finally, an important property of the metri
 entropy is that it is an a�ne fun
tional on
M. A

ording to the Birkho� ergodi
 theorem, for any µ ∈ M and for µ�almost every
ρ ∈ E , the weak limit
µρ = lim
|t|−→∞
δgsρds
exists, and is an ergodi
 probability measure. We 
an then write
µρdµ(ρ),
whi
h realizes the ergodi
 de
omposition of µ. The a�neness of the KS entropy means
hKS(µ) =
hKS(µ
ρ)dµ(ρ).
ENTROPY OF EIGENFUNCTIONS 5
An obvious 
onsequen
e is the fa
t that the range of hKS on M is an interval [0, hmax].
In the whole arti
le, we 
onsider a 
ertain subsequen
e of eigenstates (ψkj )j∈N of the
Lapla
ian, su
h that the 
orresponding sequen
e of Wigner distributions (Wkj) 
onverges
to a semi
lassi
al measure µ. In the following, the subsequen
e (ψkj )j∈N will simply be
denoted by (ψ~)~→0, using the slightly abusive notation ψ~ = ψ~kj for the eigenstate ψkj .
Ea
h eigenstate ψ~ thus satis�es
(2.2) (−~2 △−1)ψ~ = 0 .
In [2℄ the �rst author proved that the entropy of any µ ∈ Msc is stri
tly positive. In [4℄,
more expli
it lower bounds were obtained. The aim of this paper is to improve the lower
bounds of [4℄ into the following
Theorem 2.1. Let µ be a semi
lassi
al measure asso
iated to the eigenfun
tions of the
Lapla
ian on M . Then its metri
 entropy satis�es
(2.3) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
(d− 1)
λmax ,
where d = dimM and λmax = limt→±∞
log supρ∈E |dg
ρ| is the maximal expansion rate of
the geodesi
 �ow on E .
In parti
ular, if M has 
onstant se
tional 
urvature −1, we have
(2.4) hKS(µ) ≥
In dimension d, we always have
log Ju(ρ)dµ(ρ)
≤ (d− 1)λmax ,
so the above bound is an improvement over the one obtained in [4℄,
(2.5) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
− (d− 1)λmax .
In the 
ase of 
onstant or little-varying 
urvature, the bound (2.4) is mu
h sharper than
the one proved in [2℄. On the other hand, if the 
urvature varies a lot (still being negative
everywhere), the right hand side of (2.3) may a
tually be negative, in whi
h 
ase the
bound is trivial. We believe this �problem� to be a te
hni
al short
oming of our method,
and a
tually 
onje
ture the following bound:
(2.6) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
Extended to the 
ase of the quantized torus automorphisms or the Walsh-quantized baker's
map, this bound is saturated for the half-lo
alized semi
lassi
al measures 
onstru
ted in
[15℄, as well as those obtained in [20, 3℄. This bound allows 
ertain ergodi
 
omponents
to be 
arried by 
losed geodesi
s, as long as other 
omponents have positive entropy. This
may be 
ompared with the following result obtained by Bourgain and Lindenstrauss in the

ase of arithmeti
 surfa
es :
6 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
Theorem 2.2. [8℄ Let M be a 
ongruen
e arithmeti
 surfa
e, and (ψj) an orthonormal
basis of eigenfun
tions for the Lapla
ian and the He
ke operators.
Let µ be a 
orresponding semi
lassi
al measure, with ergodi
 de
omposition µ =
µρdµ(ρ).
Then, for µ-almost all ergodi
 
omponents we have hKS(µ
ρ) ≥ 1
As dis
ussed above, the Liouville measure is the only one satisfying hKS(µ) =
log Ju(ρ) dµ(ρ)
[23℄, so the Quantum Unique Ergodi
ity would be proven in one 
ould repla
e 1/2 by 1 on
the right hand side of (2.6). However, we believe that (2.6) is the optimal result that 
an
be obtained without using mu
h more pre
ise information, like for instan
e a sharp 
ontrol
on the spe
tral degenera
ies, or �ne information on the lengths of 
losed geodesi
s.
Indeed, in the above mentioned examples of Anosov systems where the Quantum Unique
Ergodi
ity 
onje
ture is wrong and the bound (2.6) sharp, the quantum spe
trum has very
high degenera
ies, whi
h 
ould be responsible for the possibility to 
onstru
t ex
eptional
eigenstates. Su
h high degenera
ies are not expe
ted in the 
ase of the Lapla
ian on a neg-
atively 
urved manifold. For the moment, however, there is no 
lear understanding of the
pre
ise relation between spe
tral degenera
ies and failure of Quantum Unique Ergodi
ity.
A
knowledgements. N.A and S.N. were partially supported by the Agen
e Nationale
de la Re
her
he, under the grant ANR-05-JCJC-0107-01. They bene�ted from numerous
dis
ussions with Y. Colin de Verdière and M. Zworski. S.N. is grateful to the Mathemati
al
Department in Bonn for its hospitality in De
ember 2006.
3. Outline of the proof
We start by re
alling the de�nition and some properties of the metri
 entropy asso
iated
with a probability measure on T ∗M , invariant through the geodesi
 �ow. In �3.2 we extend
the notion of entropy to the quantum framework. Our approa
h is semi
lassi
al, so we want
the 
lassi
al and quantum entropies to be 
onne
ted in some way when ~ → 0. The weights
appearing in our quantum entropy are estimated in Thm. 3.1, whi
h was proven and used
in [2℄. In �3.2.1 we also 
ompare our quantum entropy with several �quantum dynami
al
entropies� previously de�ned in the literature. The proof of Thm. 2.1 a
tually starts in
�3.3, where we present the algebrai
 tool allowing us to take advantage of our estimates
(3.9) (or their optimized version given in Thm. 3.5), namely an �entropi
 un
ertainty
prin
iple� spe
i�
 of the quantum framework. From �3.4 on, we apply this �prin
iple� to
the quantum entropies appearing in our problem, and pro
eed to prove Thm. 2.1. Although
the method is basi
ally the same as in [4℄, several small modi�
ations allow to �nally obtain
the improved lower bound (2.3), and also simplify some intermediate proofs, as explained
in Remark 3.6.
3.1. De�nition of the metri
 entropy. In this paper we will meet several types of
entropies, all of whi
h are de�ned using the fun
tion η(s) = −s log s, for s ∈ [0, 1]. We
start with the Kolmogorov-Sinai entropy of the geodesi
 �ow with respe
t to an invariant
probability measure.
Let µ be a probability measure on the 
otangent bundle T ∗M . Let P = (E1, . . . , EK) be
a �nite measurable partition of T ∗M : T ∗M =
i=1Ei. We will denote the set of indi
es
ENTROPY OF EIGENFUNCTIONS 7
{1, . . . , K} = [[1, K]]. The Shannon entropy of µ with respe
t to the partition P is de�ned
hP(µ) = −
µ(Ek) logµ(Ek) =
µ(Ek)
For any integer n ≥ 1, we denote by P∨n the partition formed by the sets
(3.1) E
= Eα0 ∩ g
−1Eα1 . . . ∩ g
−n+1Eαn−1 ,
where α = (α0, . . . , αn−1) 
an be any sequen
e in [[1, K]]
(su
h a sequen
e is said to be
of length |α| = n). The partition P∨n is 
alled the n-th re�nement of the initial partition
P = P∨1. The entropy of µ with respe
t to P∨n is denoted by
(3.2) hn(µ,P) = hP∨n(µ) =
α∈[[1,K]]n
If µ is (gt)�invariant, it follows from the 
onvexity of the logarithm that
(3.3) ∀n,m ≥ 1, hn+m(µ,P) ≤ hn(µ,P) + hm(µ,P),
in other words the sequen
e (hn(µ,P))n∈N is subadditive. The entropy of µ with respe
t
to the a
tion of the geodesi
 �ow and to the partition P is de�ned by
(3.4) hKS(µ,P) = lim
hn(µ,P)
= inf
hn(µ,P)
Ea
h weight µ(E
) measures the µ�probability to visit su

essively Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 through the geodesi
 �ow. Roughly speaking, the entropy measures
the exponential de
ay of these probabilities when n gets large. It is easy to see that
hKS(µ,P) ≥ β if there exists C su
h that µ(Eα) ≤ C e
, for all n and all α ∈ [[1, K]]
Finally, the Kolmogorov-Sinai entropy of µ with respe
t to the a
tion of the geodesi
�ow is de�ned as
(3.5) hKS(µ) = sup
hKS(µ,P),
the supremum running over all �nite measurable partitions P. The 
hoi
e to 
onsider the
time 1 of the geodesi
 �ow in the de�nition (3.1) may seem arbitrary, but the entropy has
a natural s
aling property : the entropy of µ with respe
t to the �ow (gat) is |a|�times its
entropy with respe
t to (gt).
Assume µ is 
arried on the energy layer E . Due to the Anosov property of the geodesi
�ow on E , it is known that the supremum (3.5) is rea
hed as soon as the diameter of the
partition P ∩ E (that is, the maximum diameter of its elements Ek ∩ E) is small enough.
Furthermore, let us assume (without loss of generality) that the inje
tivity radius of M is
larger than 1. Then, we may restri
t our attention to partitions P obtained by lifting on
E a partition of the manifoldM , that is take M =
k=1Mk and then Ek = T
∗Mk. In fa
t,
if the diameter of Mk in M is of order ε, then the diameter of the partition P
∨2 ∩E in E is
also of order ε. This spe
ial 
hoi
e of our partition is not 
ru
ial, but it simpli�es 
ertain
aspe
ts of the analysis.
8 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The existen
e of the limit in (3.4), and the fa
t that it 
oin
ides with the in�mum, follow
from a standard subadditivity argument. It has a 
ru
ial 
onsequen
e : if (µi) is a sequen
e
of (gt)�invariant probability measures on T ∗M , weakly 
onverging to a probability µ, and
if µ does not 
harge the boundary of the partition P, we have
hKS(µ,P) ≥ lim sup
hKS(µi,P) .
In parti
ular, assume that for i large enough, the following estimates hold :
(3.6) ∀n ≥ 1, ∀α ∈ [[1, K]]
, µi(Eα) ≤ Ci e
−βn ,
with β independent of i. This implies for i large enough hKS(µi,P) ≥ β, and this estimate
goes to the limit to yield hKS(µ) ≥ β.
3.2. From 
lassi
al to quantum dynami
al entropy. Sin
e our semi
lassi
al measure
µ is de�ned as a limit of Wigner distributions W~, a naive idea would be to estimate
from below the KS entropy of W~ and then take the limit ~ → 0. This idea 
annot work
dire
tly, be
ause the Wigner transformsW~ are neither positive, nor are they (g
t)�invariant.
Therefore, one 
annot dire
tly use the (formal) integrals W~(Eα) =
W~(x, ξ) dx dξ to

ompute the entropy of the semi
lassi
al measure.
Instead, the method initiated by the �rst author in [2℄ is based on the following remarks.
Ea
h integral W~(Eα) 
an also be written as W~(1lEα) =
W~ 1lEα , where 1lEα is the

hara
teristi
 fun
tion on the set E
, that is
(3.7) 1lEα = (1lEαn−1 ◦ g
n−1)× . . .× (1lEα1 ◦ g)× 1lEα0 .
Remember we took Ek = T
∗Mk, where the Mk form a partition of M .
From the de�nition of the Wigner distribution, this integral 
orresponds formally to the
overlap 〈ψ~,Op~(1lEα )ψ~〉. Yet, the 
hara
teristi
 fun
tions 1lEα have sharp dis
ontinuities,
so their quantizations 
annot be in
orporated in a ni
e pseudodi�erential 
al
ulus. Besides,
the set E
is not 
ompa
tly supported, and shrinks in the unstable dire
tion when n =
|α| −→ +∞, so that the operator Op
(1lEα ) is very problemati
.
We also note that an overlap of the form 〈ψ~,Op~(1lEα)ψ~〉 is a hybrid expression: this is
a quantum matrix element of an operator de�ned in terms of the 
lassi
al evolution (3.7).
From the point of view of quantum me
hani
s, it is more natural to 
onsider, instead, the
operator obtained as the produ
t of Heisenberg-evolved quantized fun
tions, namely
(3.8) (U−n+1
Pαn−1U
) (U−n+2
Pαn−2U
) · · · (U−1
Pα1U~)Pα0 .
Here we used the shorthand notation Pk = 1lMk , k ∈ [[1, K]] (multipli
ation operators). To
remedy the fa
t that the fun
tions 1lMk are not smooth, whi
h would prevent us from using
a semi
lassi
al 
al
ulus, we apply a 
onvolution kernel to smooth them, obtain fun
tions
1lsmMk ∈ C
∞(M), and 
onsider Pk
= 1lsmMk (we 
an do this keeping the property
k=1 1l
In the following, we will use the notation A(t)
= U−t
for the Heisenberg evolution
of the operator A though the S
hrödinger �ow U t
= exp(−it~△
). The norm ‖•‖ will denote
ENTROPY OF EIGENFUNCTIONS 9
either the Hilbert norm on L2(M), or the 
orresponding operator norm. The subsequent
�purely quantum� norms were estimated in [2, Thm. 1.3.3℄:
Theorem 3.1. (The main estimate [2℄) Set as above Pk
= 1lsmMk . For every K > 0,
there exists ~K > 0 su
h that, uniformly for all ~ < ~K, for all n ≤ K| log ~|, for all
(α0, . . . , αn−1) ∈ [[1, K]]
(3.9) ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 ψ~‖ ≤ 2(2π~)
−d/2 e−
n(1 +O(ε))n.
The exponent Λ is given by the �smallest expansion rate�:
Λ = − sup
log Ju(ρ)dν(ρ) = inf
λ+i (γ).
The in�mum on the right hand side runs over the set of 
losed orbits on E , and the λ+i denote
the positive Lyapunov exponents along the orbit, that is the logarithms of the expanding
eigenvalues of the Poin
aré map, divided by the period of the orbit. The parameter ε > 0
is an upper bound on the diameters of the supports of the fun
tions 1lsmMk in M .
From now on we will 
all the produ
t operator
(3.10) P
= Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 , α ∈ [[1, K]]
To prove the above estimate, one a
tually 
ontrols the operator norm
(3.11) ‖P
(χ)‖ ≤ 2(2π~)−d/2 e−
n(1 +O(ε))n ,
where χ ∈ C∞c (E
ε) is an energy 
uto� su
h that χ = 1 near E , supported inside a neigh-
bourhood Eε = H−1([1
− ε, 1
+ ε]) of E .
In quantum me
hani
s, the matrix element 〈ψ~, Pαψ~〉 looks like the �probability�, for a
parti
le in the state ψ~, to visit su

essively the phase spa
e regions Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 of the S
hrödinger �ow. Theorem 3.1 implies that this �probability�
de
ays exponentially fast with n, with rate Λ
, but this de
ay only starts around the time
(3.12) n1
d| log ~|
whi
h is a kind of �Ehrenfest time� (see (3.25) for another de�nition of Ehrenfest time).
Yet, be
ause the matrix elements 〈ψ~, Pαψ~〉 are not real in general, they 
an hardly be
used to de�ne a �quantum measure�. Another possibility to de�ne the probability for the
parti
le to visit the sets Eαk at times k, is to take the squares of the norms appearing in
(3.9):
(3.13) ‖P
2 = ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0ψ~‖
Now we require the smoothed 
hara
teristi
 fun
tions 1lsmMi to satisfy the identity
(3.14)
1lsmMk(x)
= 1 for any point x ∈M .
10 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
We denote by Psm the smooth partition of M made by the fun
tions
(1lsmMk)
. The

orresponding set of multipli
ation operators (Pk)
= Pq forms a �quantum partition of
unity� :
(3.15)
P 2k = IdL2 .
For any n ≥ 1, we re�ne the quantum partition Pq into (Pα)|α|, as in (3.10). The weights
(3.13) exa
tly add up to unity, so it makes sense to 
onsider the entropy
(3.16) hn(ψ~,Pq)
α∈[[1,K]]n
3.2.1. Conne
tion with other quantum entropies. This entropy appears to be a parti
ular

ase of the �general quantum entropies� des
ribed by Sªom
zy«ski and �y
zkowski [28℄,
who already had in mind appli
ations to quantum 
haos. In their terminology, a family of
bounded operators π = (πk)
k=1 on a Hilbert spa
e H satisfying
(3.17)
π∗k πk = IdH
provides an �instrument� whi
h, to ea
h index k ∈ [[1,N ]], asso
iates the following map on
density matri
es:
ρ 7→ I(k)ρ = πk ρ π
k , a nonnegative operator with tr(I(k)ρ) ≤ 1 .
From a unitary propagator U and its adjoint a
tion Uρ = UρU−1, they propose to 
onstru
t
the re�ned instrument
I(α)ρ
= I(αn−1) ◦ · · · U ◦ I(α1) ◦ U ◦ I(α0)ρ = U
−n+1 π
Un−1 , α ∈ [[1,N ]]
where we used (3.10) to re�ne the operators πk into πα. We obtain the probability weights
(3.18) tr(I(α)ρ) = tr(π
) , α ∈ [[1,N ]]
For any U-invariant density ρ, these weights provide an entropy
(3.19) hn(ρ, I) =
α∈[[1,N ]]n
tr(I(α)ρ)
One easily 
he
ks that our quantum partition Pq = (Pk)
k=1 satis�es (3.17), and that if
one takes ρ = |ψ~〉〈ψ~| the weights tr(I(α)ρ) exa
tly 
orrespond to our weights ‖Pαψ‖
Hen
e, the entropy (3.19) 
oin
ides with (3.16).
Around the same time, Ali
ki and Fannes [1℄ used the same quantum partition (3.17)
(whi
h they 
alled ��nite operational partitions of unity�) to de�ne a di�erent type of
entropy, now 
alled the �Ali
ki-Fannes entropy� (the de�nition extends to general C∗-
dynami
al systems). For ea
h n ≥ 1 they extend the weights (3.18) to �o�-diagonal entries�
to form a N n ×N n density matrix ρn:
(3.20) [ρn]α′,α = tr(πα′ ρ π
), α,α′ ∈ [[1,N ]]
ENTROPY OF EIGENFUNCTIONS 11
The AF entropy of the system (U , ρ) is then de�ned as follows: take the Von Neumann
entropy of these density matri
es, hAFn (ρ, π) = tr η(ρn), then take lim supn→∞
hAFn (ρ, π)
and �nally take the supremum over all possible �nite operational partitions of unity π.
We mention that tra
es of the form (3.20) also appear in the �quantum histories� ap-
proa
h to quantum me
hani
s (see e.g. [17℄, and [28, Appendix D℄ for referen
es).
3.2.2. Naive treatment of the entropy hn(ψ~,Pq). For �xed |α| > 0, the Egorov theorem
shows that ‖P

onverges to the 
lassi
al weight µ
(1lsmMα )
when ~ → 0, so for �xed
n > 0 the entropy hn(ψ~,Pq) 
onverges to hn(µ,Psm), de�ned as in (3.2), the 
hara
teristi
fun
tions 1lMk being repla
ed by their smoothed versions (1l
)2. On the other hand, from
the estimate (3.11) the entropies hn(ψ~,Pq) satisfy, for ~ small enough,
(3.21) hn(ψ~,Pq) ≥ n
Λ+O(ε)
− d| log ~|+O(1) ,
for any time n ≤ K| log ~|. For large times n ≈ K| log ~|, this provides a lower bound
hn(ψ~,Pq) ≥
Λ +O(ε)
+O(1/| log ~|) ,
whi
h looks very promising sin
e K 
an be taken arbitrary large: we 
ould be tempted to
take the semi
lassi
al limit, and dedu
e a lower bound hKS(µ) ≥ Λ.
Unfortunately, this does not work, be
ause in the range {n > n1} where the estimate
(3.21) is useful, the Egorov theorem breaks down, the weights (3.13) do not approximate
the 
lassi
al weights µ
(1lsmMα )
, and there is no relationship between hn(ψ,Pq) and the

lassi
al entropies hn(µ,Psm).
This breakdown of the quantum-
lassi
al 
orresponden
e around the Ehrenfest time is
ubiquitous for 
haoti
 dynami
s. It has been observed before when studying the 
onne
tion
between the Ali
ki-Fannes entropy for the quantized torus automorphisms and the KS
entropy of the 
lassi
al dynami
s [5℄: the quantum entropies hAFn (ψ~,Pq) follow the 
lassi
al
hn(µ,Psm) until the Ehrenfest time (and therefore grow linearly with n), after whi
h they
�saturate�, to produ
e a vanishing entropy lim supn→∞
hAFn (ψ~,Pq).
To prove the Theorem 2.1, we will still use the estimates (3.11), but in a more subtle
way, namely by referring to an entropi
 un
ertainty prin
iple.
3.3. Entropi
 un
ertainty prin
iple. The theorem below is an adaptation of the en-
tropi
 un
ertainty prin
iple 
onje
tured by Deuts
h and Kraus [12, 21℄ and proved by
Massen and U�nk [25℄. These authors were investigating the theory of measurement in
quantum me
hani
s. Roughly speaking, this result states that if a unitary matrix has
�small� entries, then any of its eigenve
tors must have a �large� Shannon entropy.
Let (H, 〈., .〉) be a 
omplex Hilbert spa
e, and denote ‖ψ‖ =
〈ψ, ψ〉 the asso
iated
norm. Consider a quantum partition of unity (πk)
k=1 on H as in (3.17). If ‖ψ‖ = 1,
we de�ne the entropy of ψ with respe
t to the partition π as in (3.16), namely hπ(ψ) =
12 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
‖πk ψ‖
. We extend this de�nition by introdu
ing the notion of pressure, asso
i-
ated to a family v = (vk)k=1,...,N of positive real numbers: the pressure is de�ned by
pπ,v(ψ)
‖πk ψ‖
‖πk ψ‖
2 log v2k.
In Theorem 3.2, we a
tually need two partitions of unity (πk)
k=1 and (τj)
j=1, and two
families of weights v = (vk)
k=1, w = (wj)
j=1, and 
onsider the 
orresponding pressures
pπ,v(ψ), pτ,w(ψ). Besides the appearan
e of the weights v, w, we bring another modi�
ation
to the statement in [25℄ by introdu
ing an auxiliary operator O.
Theorem 3.2. [4, Thm. 6.5℄ Let O be a bounded operator and U be an isometry on H.
De�ne c
(v,w)
O (U)
= supj,k wj vk ‖τj U π
kO‖, and V = maxk vk, W = maxj wj.
Then, for any ǫ ≥ 0, for any normalized ψ ∈ H satisfying
(3.22) ∀k = 1, . . . ,N , ‖(Id−O) πk ψ‖ ≤ ǫ ,
the pressures pτ,w
, pπ,v
satisfy
+ pπ,v
≥ −2 log
(v,w)
O (U) +N V W ǫ
Example 1. The original result of [25℄ 
orresponds to the 
ase where H = CN , O = Id,
ǫ = 0, N = M, vk = wj = 1, and the operators πk = τk are the orthogonal proje
tors on
some orthonormal basis (ek)
k=1 of H. In this 
ase, the theorem asserts that
hπ(U ψ) + hπ(ψ) ≥ −2 log c(U)
where c(U) = supj,k |〈ek,Uej〉| is the supremum of all matrix elements of U in the orthonor-
mal basis (ek). As a spe
ial 
ase, one gets hπ(ψ) ≥ − log c(U) if ψ is an eigenfun
tion of
3.4. Applying the entropi
 un
ertainty prin
iple to the Lapla
ian eigenstates.
In this se
tion we explain how to use Theorem 3.2 in order to obtain nontrivial information
on the quantum entropies (3.16) and then hKS(µ). For this we need to de�ne the data
to input in the theorem. Ex
ept the Hilbert spa
e H = L2(M), all other data depend on
the semi
lassi
al parameter ~: the quantum partition π, the operator O, the positive real
number ǫ, the weights (vj), (wk) and the unitary operator U .
As explained in se
tion 3.2, we partition M into M = ⊔Kk=1Mk, 
onsider open sets
Ωk ⊃⊃Mk (whi
h we assume to have diameters ≤ ε), and 
onsider smoothed 
hara
teristi
fun
tions 1lsmMk supported respe
tively inside Ωk, and satisfying the identity (3.14). The
asso
iated multipli
ation operators on H are form a quantum partition (Pk)
k=1, whi
h we
had 
alled Pq. To alleviate notations, we will drop the subs
ript q.
From (3.15), and using the unitarity of U~, one realizes that for any n ≥ 1, the families
of operators P∨n = (P ∗
)|α|=n and T
∨n = (P
)|α|=n (see (3.10)) make up two quantum
partitions of unity as in (3.17), of 
ardinal Kn.
ENTROPY OF EIGENFUNCTIONS 13
3.4.1. Sharp energy lo
alization. In the estimate (3.11), we introdu
ed an energy 
uto� χ
on a �nite energy strip Eε, with χ ≡ 1 near E . This 
uto� does not appear in the estimate
(3.9), be
ause, when applied to the eigenstate ψ~, the operator Op~(χ) essentially a
ts like
the identity.
The estimate (3.11) will a
tually not su�
e to prove Theorem 2.1. We will need to
optimize it by repla
ing χ in (3.11) with a �sharp� energy 
uto�. For some �xed (small)
δ ∈ (0, 1), we 
onsider a smooth fun
tion χδ ∈ C
∞(R; [0, 1]), with χδ(t) = 1 for |t| ≤ e
and χδ(t) = 0 for |t| ≥ 1. Then, we res
ale that fun
tion to obtain the following family of
~-dependent 
uto�s near E :
(3.23) ∀~ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗M, χ(n)(ρ; ~)
e−nδ ~−1+δ(H(ρ)− 1/2)
The 
uto� χ(n) is supported in a tubular neighbourhood of E of width 2~1−δ enδ. We will
always assume that this width is << ~1/2 in the semi
lassi
al limit, whi
h is the 
ase if we
ensure that n ≤ Cδ| log ~| for some 0 < Cδ < (2δ)
−1−1. In spite of their singular behaviour,
these 
uto�s 
an be quantized into pseudodi�erential operators Op(χ(n)) des
ribed in [4℄
(the quantization uses a pseudodi�erential 
al
ulus adapted to the energy layer E , drawn
from [29℄). The eigenstate ψ~ is indeed very lo
alized near E , sin
e it satis�es
(3.24) ‖
Op(χ(0))− 1
ψ~‖ = O(~
∞) ‖ψ~‖ .
In the rest of the paper, we also �x a small δ′ > 0, and 
all �Ehrenfest time� the ~-dependent
integer
(3.25) nE(~)
⌊(1− δ′)| log ~|
Noti
e the resemblan
e with the time n1 de�ned in (3.12). The signi�
an
e of this time
s
ale will be dis
ussed in �3.4.5.
The following proposition states that the operators (P ∗
)|α|=nE , almost preserve the en-
ergy lo
alization of ψ~ :
Proposition 3.3. For any L > 0, there exists ~L su
h that, for any ~ ≤ ~L, the Lapla
ian
eigenstate satis�es
(3.26) ∀α, |α| = nE , ‖
Op(χ(nE))− Id
ψ~‖ ≤ ~
L‖ψ~‖ .
We re
ognize here a 
ondition of the form (3.22).
3.4.2. Applying Theorem 3.2: Step 1. We now pre
ise some of the data we will use in the
entropi
 un
ertainty prin
iple, Theorem 3.2. As opposed to the 
hoi
e made in [4℄, we will
use two di�erent partitions π, τ .
• the quantum partitions π and τ are given respe
tively by the families of operators
π = P∨nE = (P ∗
)|α|=nE , τ = T
∨nE = (P
)|α|=nE . Noti
e that these partitions
only di�er by the ordering of the operators Pαi(i) inside the produ
ts. In the
semi
lassi
al limit, these partitions have 
ardinality N = KnE ≍ ~−K0 for some
�xed K0 > 0.
• the isometry will be the propagator at the Ehrenfest time, U = UnE
14 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
• the auxiliarly operator is given as O = Op(χ(nE)), and the error ǫ = ~L, where L
will be 
hosen very large (see �3.4.4).
• the weights v
will be sele
ted in �3.4.4. They will be semi
lassi
ally tempered,
meaning that there exists K1 > 0 su
h that, for ~ small enough, all vα, wα are

ontained in the interval [1, ~−K1].
The entropy and pressures asso
iated with a state ψ ∈ H are given by
hπ(ψ) =
|α|=nE
,(3.27)
pπ,v(ψ) = hπ(ψ)− 2
|α|=nE
ψ‖2 log v
.(3.28)
With respe
t to the se
ond partition, we have
hτ (ψ) =
|α|=nE
,(3.29)
pτ,w(ψ) = hτ (ψ)− 2
|α|=nE
ψ‖2 logw
.(3.30)
We noti
e that the entropy hτ (ψ) exa
tly 
orresponds to the formula (3.16), while hπ(ψ)
is built from the norms
ψ‖2 = ‖Pα0Pα1(1) · · ·Pαn−1(n− 1)ψ‖
If ψ is an eigenfun
tion of U~, the above norm 
an be obtained from (3.13) by ex
hanging
U~ with U
, and repla
ing the sequen
e α = (α0, . . . , αn−1) by ᾱ
= (αn−1, . . . , α0). So the
entropies hπ(ψ) and hτ (ψ) are mapped to one another through the time reversal U~ → U
With these data, we draw from Theorem 3.2 the following
Corollary 3.4. For ~ > 0 small enough 
onsider the data π, τ , U , O as de�ned above.
(3.31) c
O (U)
= max
|α|=|α′|=nE
Op(χ(nE))‖
Then for any normalized state φ satisfying (3.26),
pτ,w(U
φ) + pπ,v(φ) ≥ −2 log
O (U) + h
L−K0−2K1
From (3.26), we see that the above 
orollary applies to the eigenstate ψ~ if ~ is small
enough.
The reason to take the same value nE for the re�ned partitions P
, T ∨nE and the
propagator U
is the following : the produ
ts appearing in c
O (U) 
an be rewritten
(with U ≡ U~):
′ UnE P
= U−nE+1Pα′
U · · ·UPα′0UPαnE−1U · · ·UPα0 = U
ENTROPY OF EIGENFUNCTIONS 15
Thus, the estimate (3.11) with n = 2nE already provides an upper bound for the norms
appearing in (3.31) � the repla
ement of χ by the sharp 
uto� χ(nE) does not harm the
estimate.
To prove Theorem 2.1, we a
tually need to improve the estimate (3.11), as was done in
[4℄, see Theorem 3.5 below. This improvement is done at two levels: we will use the fa
t
that the 
uto�s χ(nE) are sharper than χ, and also the fa
t that the expansion rate of the
geodesi
 �ow (whi
h governs the upper bound in (3.11)) is not uniform, but depends on
the sequen
e α.
Our 
hoi
e for the weights v
will then be guided by the α-dependent upper bounds
given in Theorem 3.5. To state that theorem, we introdu
e some notations.
3.4.3. Coarse-grained unstable Ja
obian. We re
all that, for any energy λ > 0, the geodesi
�ow gt on the energy layer E(λ) = H−1(λ) ⊂ T ∗M is Anosov, so that the tangent spa
e
TρE(λ) at ea
h ρ ∈ T
∗M , H(ρ) > 0 splits into
TρE(λ) = E
u(ρ)⊕Es(ρ)⊕ RXH(ρ)
where Eu (resp. Es) is the unstable (resp. stable) subspa
e. The unstable Ja
obian Ju(ρ)
is de�ned by Ju(ρ) = det
|Eu(g1ρ)
(the unstable spa
es at ρ and g1ρ are equipped with
the indu
ed Riemannian metri
).
This Ja
obian 
an be �dis
retized� as follows in the energy strip Eε ⊃ E . For any pair of
indi
es (α0, α1) ∈ [[1, K]]
, we de�ne
(3.32) Ju1 (α0, α1)
= sup
Ju(ρ) : ρ ∈ T ∗Ωα0 ∩ E
ε, g1ρ ∈ T ∗Ωα1
if the set on the right hand side is not empty, and Ju1 (α0, α1) = e
otherwise, where R > 0
is a �xed large number. For any sequen
e of symbols α of length n, we de�ne
(3.33) Jun(α)
= Ju1 (α0, α1) · · ·J
1 (αn−2, αn−1) .
Although Ju and Ju1 (α0, α1) are not ne
essarily everywhere smaller than unity, there exists
C, λ+, λ− > 0 su
h that, for any n > 0, for any α with |α| = n,
(3.34) C−1 e−n(d−1) λ+ ≤ Jun(α) ≤ C e
−n(d−1) λ− .
One 
an take λ+ = λmax(1+ε), where λmax is the maximal expanding rate in Theorem. 2.1.
We now give our 
entral estimate, easy to draw from [4, Corollary 3.4℄.
Theorem 3.5. Fix small positive 
onstants ε, δ, δ′ and a 
onstant 0 < Cδ < (2δ)
−1 − 1.
Take an open 
over M =
k Ωk of diameter ≤ ε and an asso
iated quantum partition P =
k=1. There exists ~0 su
h that, for any ~ ≤ ~0, for any positive integer n ≤ Cδ| log ~|,
and any pair of sequen
es α, α
of length n,
(3.35) ‖P
′ Op(χ(n))‖ = ‖P
Op(χ(n))‖ ≤ C ~−
−δ enδ
Jun(α) J
The 
onstant C only depends on the Riemannian manifold (M, g). If we take n = nE, this
takes the form
(3.36) ‖P
Op(χ(nE))‖ ≤ C ~−
d−1+cδ
JunE(α) J
(α′) ,
16 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
where c = 2 + 2λ−1max.
The idea of proof in Theorem 3.5 is rather simple, although the te
hni
al implementation
is 
umbersome. We �rst show that for any normalized state ψ, the state Op(χ(n))ψ 
an
be essentially de
omposed into a superposition of ~
−d| suppχ(n)| normalized Lagrangian
states, supported on Lagrangian manifolds transverse to the stable foliation. In fa
t the
Lagrangian states we work with are trun
ated δ�fun
tions, supported on lagrangians of the
form ∪tg
tS∗zM . The a
tion of the operator U
′ = Pα′
U · · ·UPα0 on su
h Lagrangian
states 
an be analyzed through WKB methods, and is simple to understand at the 
lassi
al
level : ea
h appli
ation of the propagator U stret
hes the Lagrangian along the unstable
dire
tion (the rate of stret
hing being des
ribed by the lo
al unstable Ja
obian), whereas
ea
h operator Pk �proje
ts� on a pie
e of Lagrangian of diameter ε. This iteration of
stret
hing and 
utting a

ounts for the exponential de
ay. The αα
-independent fa
tor
on the right of (3.36) results from adding together the 
ontributions of all the initial
Lagrangian states. Noti
e that this prefa
tor is smaller than in Theorem. 3.1 due to the

ondition Cδ < (2δ)
−1 − 1.
Remark 3.6. In [4℄ we used the same quantum partition P∨nE for π and τ in Theorem. 3.2.
As a result, we needed to estimate from above the norms ‖P ∗
′ UnE PαOp(χ
(nE))‖ (see [4,
Theorem. 2.6℄). The proof of this estimate was mu
h more involved than the one for
(3.36), sin
e it required to 
ontrol long pie
es of unstable manifolds. By using instead the
two partitions P(n), T (n), we not only prove a more pre
ise lower bound (2.3) on the KS
entropy, but also short-
ir
uit some �ne dynami
al analysis.
3.4.4. Applying Theorem 3.2: Step 2. There remains to 
hoose the weights (v
) to use
in Theorem 3.2. Our 
hoi
e is guided by the following idea: in (3.31), the weights should
balan
e the variations (with respe
t to α,α′) in the norms, su
h as to make all terms in
(3.31) of the same order. Using the upper bounds (3.36), we end up with the following

hoi
e for all α of length nE :
= JunE(α)
−1/2 .
From (3.34), there exists K1 > 0 su
h that, for ~ small enough, all the weights are

ontained in the interval [1, ~−K1], as announ
ed in �3.4.2. Using these weights, the estimate
(3.36) implies the following bound on the 
oe�
ient (3.31):
∀~ < ~0, c
O (U) ≤ C ~
− d−1+cδ
We 
an now apply Corollary 3.4 to the parti
ular 
ase of the eigenstates ψ~. We 
hoose L
su
h that L−K0 − 2K1 > −
d−1+cδ
, so from Corollary 3.4 we draw the following
Proposition 3.7. Let (ψ~)~→0 be our sequen
e of eigenstates (2.2). In the semi
lassi
al
limit, the pressures of ψ~ satisfy
(3.37) pP∨nE ,v(ψ~) + pT ∨nE ,w(ψ~) ≥ −
(d − 1 + cδ)λmax
(1− δ′)
nE +O(1) .
ENTROPY OF EIGENFUNCTIONS 17
If M has 
onstant 
urvature we have log Jn
≤ −n(d − 1)λmax(1 − O(ε)) for all α of
length n, and the above lower bound 
an be written
hP∨nE (ψ~) + hT ∨nE (ψ~) ≥ (d− 1)λmax
1 +O(ε, δ, δ′)
As opposed to (3.21), the above inequality provides a nontrivial lower bound for the quan-
tum entropies at the time nE , whi
h is smaller than the time n1 of (3.12), and will allow
to 
onne
t those entropies to the KS entropy of the semi
lassi
al measure (see below).
3.4.5. Subadditivity until the Ehrenfest time. Even at the relatively small time nE , the

onne
tion between the quantum entropy h(ψ~,P
∨nE) and the 
lassi
al h(µ,P∨nEsm ) is not

ompletely obvious: both are sums of a large number of terms (≍ ~−K0). Before taking
the limit ~ → 0, we will prove that a lower bound of the form (3.37) still holds if we
repla
e nE ≍ | log ~| by some �xed no ∈ N, and P
by the 
orresponding quantum
partition P∨no . The link between quantum pressures at times nE and no is provided by
the following subadditivity property, whi
h is the semi
lassi
al analogue of the 
lassi
al
subadditivity of pressures for invariant measures (see (3.3)).
Proposition 3.8 (Subadditivity). Let δ′ > 0. There is a fun
tion R(no, ~), and a real
number R > 0 independent of δ′, su
h that, for any integer no ≥ 1,
lim sup
|R(no, ~)| ≤ R
and with the following properties. For any small enough ~ > 0, any integers no, n ∈ N with
no + n ≤ nE(~), for any ψ~ normalized eigenstate satisfying (2.2), the following inequality
holds:
pP∨(no+n),v(ψ~) ≤ pP∨no ,v(ψ~) + pP∨n,v(ψ~) +R(no, ~) .
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~).
To prove this proposition, one uses a re�ned version of Egorov's theorem [10℄ to show that
the non�
ommutative dynami
al system formed by (U t
) a
ting (through Heisenberg) on
observables supported near E is (approximately) 
ommutative on time intervals of length
nE(~). Pre
isely, we showed in [4℄ that, provided ε is small enough, for any a, b ∈ C
∀t ∈ [−nE(~), nE(~)], ‖[Op~(a)(t),Op~(b)]‖ = O(~
cδ′), ~ → 0 ,
and the implied 
onstant is uniform with respe
t to t. Within that time interval, the oper-
ators Pαj (j) appearing in the de�nition of the pressures 
ommute up to small semi
lassi
al
errors. This almost 
ommutativity explains why the quantum pressures pP∨n,v(ψ~) satisfy
the same subadditivity property as the 
lassi
al entropy (3.3), for times smaller than nE .
Thanks to this subadditivity, we may �nish the proof of Theorem. 2.1. Fixing no, using
for ea
h ~ the Eu
lidean division nE(~) = q(~)no + r(~) (with r(~) < no), Proposition 3.8
implies that for ~ small enough,
pP∨nE ,v(ψ~)
pP∨no ,v(ψ~)
pP∨r,v(ψ~)
R(no, ~)
18 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). Using (3.37) and the fa
t
that pP∨r,v(ψ~) stays uniformly bounded when ~ → 0, we �nd
(3.38)
pP∨no ,v(ψ~) + pT ∨no ,w(ψ~)
2(d− 1 + cδ)λmax
2(1− δ′)
2R(no, ~)
+Ono(1/nE) .
We are now dealing with quantum partitions P∨no , T ∨no , for n0 ∈ N independent of ~. At
this level the quantum and 
lassi
al entropies are related through the (�nite time) Egorov
theorem, as we had noti
ed in �3.2.2. For any α of length no, the weights ‖Pαψ~‖
both 
onverge to µ
(1lsmMα)
, where we re
all that
1lsmMα = (1l
Mαno−1
◦ gno−1)× . . .× (1lsmMα1
◦ g)× 1lsmMα0
Thus, both entropies hP∨no (ψ~), hT ∨no (ψ~) semi
lassi
ally 
onverge to the 
lassi
al entropy
hno(µ,Psm). As a result, the left hand side of (3.38) 
onverges to
(3.39) 2
hno(µ,Psm)
|α|=no
(1lsmMα )
log Juno(α) .
Sin
e µ is gt-invariant and Juno has the multipli
ative stru
ture (3.33), the se
ond term in
(3.39) 
an be simpli�ed:
|α|=no
(1lsmMα )
log Juno(α) = (no − 1)
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1) .
We have thus obtained the lower bound
(3.40)
hno(µ,Psm)
no − 1
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1)−
(d− 1 + cδ)λmax
2(1− δ′)
At this stage we may forget about δ and δ′. The above lower bound does not depend on
the derivatives of the fun
tions 1lsmMα , so the same bound 
arries over if we repla
e 1l
the 
hara
teristi
 fun
tions 1lMα . We 
an �nally let no tend to +∞, then let the diameter
ε tend to 0. The left hand side 
onverges to hKS(µ) while, from the de�nition (3.32), the
sum in the right hand side of (3.40) 
onverges to the integral
log Ju(ρ)dµ(ρ) as ε → 0,
whi
h proves (2.3).
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 hyperboli
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 properties of eigenfun
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20 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
[31℄ S.A. Wolpert, The modulus of 
ontinuity for Γ0(m)/H semi-
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al limits, Commun. Math.
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CMLS, É
ole Polyte
hnique, 91128 Palaiseau, Fran
e
E-mail address : nalini�math.polyte
hnique.fr
Mathemati
al Institute, University of Bonn, Beringstraÿe 1,D-53115 Bonn, Germany
E-mail address : ko
h�math.uni-bonn.de
Servi
e de Physique Théorique, CEA/DSM/PhT, Unité de re
her
he asso
iée au CNRS,
CEA/Sa
lay, 91191 Gif-sur-Yvette, Fran
e
E-mail address : snonnenma
her�
ea.fr
	1. Motivations
	2. Main result.
	 Acknowledgements
	3. Outline of the proof
	3.1. Definition of the metric entropy
	3.2. From classical to quantum dynamical entropy
	3.3. Entropic uncertainty principle
	3.4. Applying the entropic uncertainty principle to the Laplacian eigenstates
	References

  We study the high--energy limit for eigenfunctions of the laplacian, on a
compact negatively curved manifold. We review the recent result of
Anantharaman-Nonnenmacher giving a lower bound on the Kolmogorov-Sinai entropy
of semiclassical measures, and improve this lower bound in the case of variable
negative curvature.

ENTROPY OF EIGENFUNCTIONS
NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER
Abstra
t. We study the high�energy limit for eigenfun
tions of the lapla
ian, on a

ompa
t negatively 
urved manifold. We review the re
ent result of Anantharaman�
Nonnenma
her [4℄ giving a lower bound on the Kolmogorov�Sinai entropy of semi
lassi
al
measures. The bound proved here improves the result of [4℄ in the 
ase of variable negative

urvature.
1. Motivations
The theory of quantum 
haos tries to understand how the 
haoti
 behaviour of a 
lassi-

al Hamiltonian system is re�e
ted in its quantum 
ounterpart. For instan
e, let M be a

ompa
t Riemannian C∞ manifold, with negative se
tional 
urvatures. The geodesi
 �ow
has the Anosov property, whi
h is 
onsidered as the ideal 
haoti
 behaviour in the theory
of dynami
al systems. The 
orresponding quantum dynami
s is the unitary �ow gener-
ated by the Lapla
e-Beltrami operator on L2(M). One expe
ts that the 
haoti
 properties
of the geodesi
 �ow in�uen
e the spe
tral theory of the Lapla
ian. The Random Matrix

onje
ture [7℄ asserts that the large eigenvalues should, after proper unfolding, statisti-

ally resemble those of a large random matrix, at least for a generi
 Anosov metri
. The
Quantum Unique Ergodi
ity 
onje
ture [26℄ (see also [6, 30℄) des
ribes the 
orresponding
eigenfun
tions ψk: it 
laims that the probability measure |ψk(x)|
2dx should approa
h (in
the weak topology) the Riemannian volume, when the eigenvalue tends to in�nity. In fa
t
a stronger property should hold for the Wigner transform Wψ, a fun
tion on the 
otangent
bundle T ∗M , (the 
lassi
al phase spa
e) whi
h simultaneously des
ribes the lo
alization of
the wave fun
tion ψ in position and momentum.
We will adopt a semi
lassi
al point of view, that is 
onsider the eigenstates of eigenvalue
unity of the semi
lassi
al Lapla
ian −~2△, thereby repla
ing the high-energy limit by the
semi
lassi
al limit ~ → 0. We denote by (ψk)k∈N an orthonormal basis of L
2(M) made of
eigenfun
tions of the Lapla
ian, and by (− 1
)k∈N the 
orresponding eigenvalues:
(1.1) − ~2k△ψk = ψk, with ~k+1 ≤ ~k .
We are interested in the high-energy eigenfun
tions of −△, in other words the semi
lassi
al
limit ~k → 0.
The Wigner distribution asso
iated to an eigenfun
tion ψk is de�ned by
Wk(a) = 〈Op~k(a)ψk, ψk〉L2(M), a ∈ C
∗M) .
Here Op
is a quantization pro
edure, set at the s
ale (wavelength) ~k, whi
h asso
iates to
any smooth phase spa
e fun
tion a (with ni
e behaviour at in�nity) a bounded operator on
http://arxiv.org/abs/0704.1564v1
2 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
L2(M). See for instan
e [13℄ or [14℄ for various quantizations Op
. On a manifold,
one 
an use lo
al 
oordinates to de�ne Op in a �nite system of 
harts, then glue the obje
ts
de�ned lo
ally thanks to a smooth partition of unity [11℄. For standard quantizations Op
the Wigner distribution is of the form Wk(x, ξ) dx dξ, where Wk(x, ξ) is a smooth fun
tion
on T ∗M , 
alled the Wigner transform of ψ. If a is a fun
tion on the manifold M , Op

an be taken as the multipli
ation by a, and thus we have Wk(a) =
a(x)|ψk(x)|
2dx: the
Wigner transform is thus a mi
rolo
al lift of the density |ψk(x)|
. Although the de�nition
ofWk depends on a 
ertain number of 
hoi
es, like the 
hoi
e of lo
al 
oordinates, or of the
quantization pro
edure (Weyl, anti-Wi
k, �right� or �left� quantization...), its asymptoti
behaviour when ~k −→ 0 does not. A

ordingly, we 
all semi
lassi
al measures the limit
points of the sequen
e (Wk)k∈N, in the distribution topology.
In the semi
lassi
al limit, �quantum me
hani
s 
onverges to 
lassi
al me
hani
s�. We
will denote |·|x the norm on T
xM given by the metri
. The geodesi
 �ow (g
t)t∈R is the
Hamiltonian �ow on T ∗M generated by the Hamiltonian H(x, ξ) =
|ξ|2x
. A quantization
of this Hamiltonian is given by the res
aled Lapla
ian −~
, whi
h generates the unitary
�ow (U t
) = (exp(it~△
)) a
ting on L2(M). The semi
lassi
al 
orresponden
e of the �ows
) and (gt) is expressed through the Egorov Theorem :
Theorem 1.1. Let a ∈ C∞c (T
∗M). Then, for any given t in R,
(1.2) ‖U−t
(a)U t
(a ◦ gt)‖L2(M) = O(~) , ~ → 0 .
The 
onstant implied in the remainder grows (often exponentially) with t, whi
h rep-
resents a notorious problem when one wants to study the large time behaviour of (U t
Typi
ally, the quantum-
lassi
al 
orresponden
e will break down for times t of the order
of the Ehrenfest time (3.25).
Using (1.2) and other standard semi
lassi
al arguments, one shows the following :
Proposition 1.2. Any semi
lassi
al measure is a probability measure 
arried on the energy
layer E = H−1(1
) (whi
h 
oin
ides with the unit 
otangent bundle S∗M). This measure is
invariant under the geodesi
 �ow.
Let us 
all M the set of gt-invariant probability measures on E . This set is 
onvex
and 
ompa
t for the weak topology. If the geodesi
 �ow has the Anosov property � for
instan
e if M has negative se
tional 
urvature � that set is very large. The geodesi
�ow has 
ountably many periodi
 orbits, ea
h of them 
arrying an invariant probability
measure. There are many other invariant measures, like the equilibrium states obtained
by variational prin
iples [19℄, among them the Liouville measure µLiouv, and the measure
of maximal entropy. Note that, for all these examples of measures, the geodesi
 �ow
a
ts ergodi
ally, meaning that these examples are extremal points in M. Our aim is to
determine, at least partially, the set Msc formed by all possible semi
lassi
al measures. By
its de�nition, Msc is a 
losed subset of M, in the weak topology.
For manifolds su
h that the geodesi
 �ow is ergodi
 with respe
t to the Liouville measure,
it has been known for some time that almost all eigenfun
tions be
ome equidistributed over
E , in the semi
lassi
al limit. This property is dubbed as Quantum Ergodi
ity :
ENTROPY OF EIGENFUNCTIONS 3
Theorem 1.3. [27, 32, 11℄ Let M be a 
ompa
t Riemannian manifold, assume that the
a
tion of the geodesi
 �ow on E = S∗M is ergodi
 with respe
t to the Liouville measure.
Let (ψk)k∈N be an orthonormal basis of L
2(M) 
onsisting of eigenfun
tions of the Lapla
ian
(1.1), and let (Wk) be the asso
iated Wigner distributions on T
Then, there exists a subset S ⊂ N of density 1, su
h that
(1.3) Wk −→µLiouv, k → ∞, k ∈ S.
The question of existen
e of �ex
eptional� subsequen
es of eigenstates with a di�erent
behaviour is still open. On a negatively 
urved manifold, the geodesi
 �ow satis�es the
ergodi
ity assumption, and in fa
t mu
h stronger properties : mixing, K�property, et
.
For su
h manifolds, it has been postulated in the Quantum Unique Ergodi
ity 
onje
ture
[26℄ that the full sequen
e of eigenstates be
omes semi
lassi
ally equidistributed over E :
one 
an take S = N in the limit (1.3). In other words, this 
onje
ture states that there
exists a unique semi
lassi
al measure, and Msc = {µLiouv}.
So far the most pre
ise results on this question were obtained for manifolds M with

onstant negative 
urvature and arithmeti
 properties: see Rudni
k�Sarnak [26℄, Wolpert
[31℄. In that very parti
ular situation, there exists a 
ountable 
ommutative family of
self�adjoint operators 
ommuting with the Lapla
ian : the He
ke operators. One may
thus de
ide to restri
t the attention to 
ommon bases of eigenfun
tions, often 
alled �arith-
meti
� eigenstates, or He
ke eigenstates. A few years ago, Lindenstrauss [24℄ proved that
any sequen
e of arithmeti
 eigenstates be
ome asymptoti
ally equidistributed. If there is
some degenera
y in the spe
trum of the Lapla
ian, note that it 
ould be possible that the
Quantum Unique Ergodi
ity 
onje
tured by Rudni
k and Sarnak holds for one orthonormal
basis but not for another. On su
h arithmeti
 manifolds, it is believed that the spe
trum
of the Lapla
ian has bounded multipli
ity: if this is really the 
ase, then the semi
lassi
al
equidistribution easily extends to any sequen
e of eigenstates.
Nevertheless, one may be less optimisti
 when extending the Quantum Unique Ergod-
i
ity 
onje
ture to more general systems. One of the simplest example of a symple
ti
Anosov dynami
al system is given by linear hyperboli
 automorphisms of the 2-torus, e.g.
Arnold's �
at map�
. This system 
an be quantized into a sequen
e ofN×N unitary
matri
es � the propagators, where N ∼ ~−1 [18℄. The eigenstates of these matri
es satisfy
a Quantum Ergodi
ity theorem similar with Theorem 1.3, meaning that almost all eigen-
states be
ome equidistributed on the torus in the semi
lassi
al limit [9℄. Besides, one 
an

hoose orthonormal eigenbases of the propagators, su
h that the whole sequen
e of eigen-
states is semi
lassi
ally equidistributed [22℄. Still, be
ause the spe
tra of the propagators
are highly degenerate, one 
an also 
onstru
t sequen
es of eigenstates with a di�erent limit
measure [15℄, for instan
e, a semi
lassi
al measure 
onsisting in two ergodi
 
omponents:
half of it is the Liouville measure, while the other half is a Dira
 peak on a single (unsta-
ble) periodi
 orbit. It was also shown that this half-lo
alization is maximal for this model
[16℄ : a semi
lassi
al measure 
annot have more than half its mass 
arried by a 
ountable
union of periodi
 orbits. The same type of half-lo
alized eigenstates were 
onstru
ted by
two of the authors for another solvable model, namely the �Walsh quantization� of the
4 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
baker's map on the torus [3℄; for that model, there exist ergodi
 semi
lassi
al measures of
purely fra
tal type (that is, without any Liouville 
omponent). Another type of semi
las-
si
al measure was re
ently obtained by Kelmer for quantized hyperboli
 automorphisms
on higher-dimensional tori [20℄: it 
onsists in the Lebesgue measure on some invariant

o-isotropi
 subspa
e of the torus.
For these Anosov models on tori, the 
onstru
tion of ex
eptional eigenstates strongly
uses nongeneri
 algebrai
 properties of the 
lassi
al and quantized systems, and 
annot be
generalized to nonlinear systems.
2. Main result.
In order to understand the set Msc, we will attempt to 
ompute the Kolmogorov�Sinai
entropies of semi
lassi
al measures. We work on a 
ompa
t Riemannian manifold M of
arbitrary dimension, and assume that the geodesi
 �ow has the Anosov property. A
tually,
our method 
an without doubt be adapted to more general Anosov Hamiltonian systems.
The Kolmogorov�Sinai entropy, also 
alled metri
 entropy, of a (gt)-invariant probability
measure µ is a nonnegative number hKS(µ) that des
ribes, in some sense, the 
omplexity of
a µ-typi
al orbit of the �ow. The pre
ise de�nition will be given later, but for the moment
let us just give a few fa
ts. A measure 
arried on a 
losed geodesi
 has vanishing entropy.
In 
onstant 
urvature, the entropy is maximal for the Liouville measure. More generally,
for any Anosov �ow, the energy layer E is foliated into unstable manifolds of the �ow. An
upper bound on the entropy of an invariant probability measure is then provided by the
Ruelle inequality:
(2.1) hKS(µ) ≤
log Ju(ρ)dµ(ρ)
In this inequality, Ju(ρ) is the unstable Ja
obian of the �ow at the point ρ ∈ E , de�ned
as the Ja
obian of the map g−1 restri
ted to the unstable manifold at the point g1ρ (note
that the average of log Ju over any invariant measure is negative). The equality holds in
(2.1) if and only if µ is the Liouville measure on E [23℄. If M has dimension d and has

onstant se
tional 
urvature −1, the above inequality just reads hKS(µ) ≤ d− 1.
Finally, an important property of the metri
 entropy is that it is an a�ne fun
tional on
M. A

ording to the Birkho� ergodi
 theorem, for any µ ∈ M and for µ�almost every
ρ ∈ E , the weak limit
µρ = lim
|t|−→∞
δgsρds
exists, and is an ergodi
 probability measure. We 
an then write
µρdµ(ρ),
whi
h realizes the ergodi
 de
omposition of µ. The a�neness of the KS entropy means
hKS(µ) =
hKS(µ
ρ)dµ(ρ).
ENTROPY OF EIGENFUNCTIONS 5
An obvious 
onsequen
e is the fa
t that the range of hKS on M is an interval [0, hmax].
In the whole arti
le, we 
onsider a 
ertain subsequen
e of eigenstates (ψkj )j∈N of the
Lapla
ian, su
h that the 
orresponding sequen
e of Wigner distributions (Wkj) 
onverges
to a semi
lassi
al measure µ. In the following, the subsequen
e (ψkj )j∈N will simply be
denoted by (ψ~)~→0, using the slightly abusive notation ψ~ = ψ~kj for the eigenstate ψkj .
Ea
h eigenstate ψ~ thus satis�es
(2.2) (−~2 △−1)ψ~ = 0 .
In [2℄ the �rst author proved that the entropy of any µ ∈ Msc is stri
tly positive. In [4℄,
more expli
it lower bounds were obtained. The aim of this paper is to improve the lower
bounds of [4℄ into the following
Theorem 2.1. Let µ be a semi
lassi
al measure asso
iated to the eigenfun
tions of the
Lapla
ian on M . Then its metri
 entropy satis�es
(2.3) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
(d− 1)
λmax ,
where d = dimM and λmax = limt→±∞
log supρ∈E |dg
ρ| is the maximal expansion rate of
the geodesi
 �ow on E .
In parti
ular, if M has 
onstant se
tional 
urvature −1, we have
(2.4) hKS(µ) ≥
In dimension d, we always have
log Ju(ρ)dµ(ρ)
≤ (d− 1)λmax ,
so the above bound is an improvement over the one obtained in [4℄,
(2.5) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
− (d− 1)λmax .
In the 
ase of 
onstant or little-varying 
urvature, the bound (2.4) is mu
h sharper than
the one proved in [2℄. On the other hand, if the 
urvature varies a lot (still being negative
everywhere), the right hand side of (2.3) may a
tually be negative, in whi
h 
ase the
bound is trivial. We believe this �problem� to be a te
hni
al short
oming of our method,
and a
tually 
onje
ture the following bound:
(2.6) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
Extended to the 
ase of the quantized torus automorphisms or the Walsh-quantized baker's
map, this bound is saturated for the half-lo
alized semi
lassi
al measures 
onstru
ted in
[15℄, as well as those obtained in [20, 3℄. This bound allows 
ertain ergodi
 
omponents
to be 
arried by 
losed geodesi
s, as long as other 
omponents have positive entropy. This
may be 
ompared with the following result obtained by Bourgain and Lindenstrauss in the

ase of arithmeti
 surfa
es :
6 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
Theorem 2.2. [8℄ Let M be a 
ongruen
e arithmeti
 surfa
e, and (ψj) an orthonormal
basis of eigenfun
tions for the Lapla
ian and the He
ke operators.
Let µ be a 
orresponding semi
lassi
al measure, with ergodi
 de
omposition µ =
µρdµ(ρ).
Then, for µ-almost all ergodi
 
omponents we have hKS(µ
ρ) ≥ 1
As dis
ussed above, the Liouville measure is the only one satisfying hKS(µ) =
log Ju(ρ) dµ(ρ)
[23℄, so the Quantum Unique Ergodi
ity would be proven in one 
ould repla
e 1/2 by 1 on
the right hand side of (2.6). However, we believe that (2.6) is the optimal result that 
an
be obtained without using mu
h more pre
ise information, like for instan
e a sharp 
ontrol
on the spe
tral degenera
ies, or �ne information on the lengths of 
losed geodesi
s.
Indeed, in the above mentioned examples of Anosov systems where the Quantum Unique
Ergodi
ity 
onje
ture is wrong and the bound (2.6) sharp, the quantum spe
trum has very
high degenera
ies, whi
h 
ould be responsible for the possibility to 
onstru
t ex
eptional
eigenstates. Su
h high degenera
ies are not expe
ted in the 
ase of the Lapla
ian on a neg-
atively 
urved manifold. For the moment, however, there is no 
lear understanding of the
pre
ise relation between spe
tral degenera
ies and failure of Quantum Unique Ergodi
ity.
A
knowledgements. N.A and S.N. were partially supported by the Agen
e Nationale
de la Re
her
he, under the grant ANR-05-JCJC-0107-01. They bene�ted from numerous
dis
ussions with Y. Colin de Verdière and M. Zworski. S.N. is grateful to the Mathemati
al
Department in Bonn for its hospitality in De
ember 2006.
3. Outline of the proof
We start by re
alling the de�nition and some properties of the metri
 entropy asso
iated
with a probability measure on T ∗M , invariant through the geodesi
 �ow. In �3.2 we extend
the notion of entropy to the quantum framework. Our approa
h is semi
lassi
al, so we want
the 
lassi
al and quantum entropies to be 
onne
ted in some way when ~ → 0. The weights
appearing in our quantum entropy are estimated in Thm. 3.1, whi
h was proven and used
in [2℄. In �3.2.1 we also 
ompare our quantum entropy with several �quantum dynami
al
entropies� previously de�ned in the literature. The proof of Thm. 2.1 a
tually starts in
�3.3, where we present the algebrai
 tool allowing us to take advantage of our estimates
(3.9) (or their optimized version given in Thm. 3.5), namely an �entropi
 un
ertainty
prin
iple� spe
i�
 of the quantum framework. From �3.4 on, we apply this �prin
iple� to
the quantum entropies appearing in our problem, and pro
eed to prove Thm. 2.1. Although
the method is basi
ally the same as in [4℄, several small modi�
ations allow to �nally obtain
the improved lower bound (2.3), and also simplify some intermediate proofs, as explained
in Remark 3.6.
3.1. De�nition of the metri
 entropy. In this paper we will meet several types of
entropies, all of whi
h are de�ned using the fun
tion η(s) = −s log s, for s ∈ [0, 1]. We
start with the Kolmogorov-Sinai entropy of the geodesi
 �ow with respe
t to an invariant
probability measure.
Let µ be a probability measure on the 
otangent bundle T ∗M . Let P = (E1, . . . , EK) be
a �nite measurable partition of T ∗M : T ∗M =
i=1Ei. We will denote the set of indi
es
ENTROPY OF EIGENFUNCTIONS 7
{1, . . . , K} = [[1, K]]. The Shannon entropy of µ with respe
t to the partition P is de�ned
hP(µ) = −
µ(Ek) logµ(Ek) =
µ(Ek)
For any integer n ≥ 1, we denote by P∨n the partition formed by the sets
(3.1) E
= Eα0 ∩ g
−1Eα1 . . . ∩ g
−n+1Eαn−1 ,
where α = (α0, . . . , αn−1) 
an be any sequen
e in [[1, K]]
(su
h a sequen
e is said to be
of length |α| = n). The partition P∨n is 
alled the n-th re�nement of the initial partition
P = P∨1. The entropy of µ with respe
t to P∨n is denoted by
(3.2) hn(µ,P) = hP∨n(µ) =
α∈[[1,K]]n
If µ is (gt)�invariant, it follows from the 
onvexity of the logarithm that
(3.3) ∀n,m ≥ 1, hn+m(µ,P) ≤ hn(µ,P) + hm(µ,P),
in other words the sequen
e (hn(µ,P))n∈N is subadditive. The entropy of µ with respe
t
to the a
tion of the geodesi
 �ow and to the partition P is de�ned by
(3.4) hKS(µ,P) = lim
hn(µ,P)
= inf
hn(µ,P)
Ea
h weight µ(E
) measures the µ�probability to visit su

essively Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 through the geodesi
 �ow. Roughly speaking, the entropy measures
the exponential de
ay of these probabilities when n gets large. It is easy to see that
hKS(µ,P) ≥ β if there exists C su
h that µ(Eα) ≤ C e
, for all n and all α ∈ [[1, K]]
Finally, the Kolmogorov-Sinai entropy of µ with respe
t to the a
tion of the geodesi
�ow is de�ned as
(3.5) hKS(µ) = sup
hKS(µ,P),
the supremum running over all �nite measurable partitions P. The 
hoi
e to 
onsider the
time 1 of the geodesi
 �ow in the de�nition (3.1) may seem arbitrary, but the entropy has
a natural s
aling property : the entropy of µ with respe
t to the �ow (gat) is |a|�times its
entropy with respe
t to (gt).
Assume µ is 
arried on the energy layer E . Due to the Anosov property of the geodesi
�ow on E , it is known that the supremum (3.5) is rea
hed as soon as the diameter of the
partition P ∩ E (that is, the maximum diameter of its elements Ek ∩ E) is small enough.
Furthermore, let us assume (without loss of generality) that the inje
tivity radius of M is
larger than 1. Then, we may restri
t our attention to partitions P obtained by lifting on
E a partition of the manifoldM , that is take M =
k=1Mk and then Ek = T
∗Mk. In fa
t,
if the diameter of Mk in M is of order ε, then the diameter of the partition P
∨2 ∩E in E is
also of order ε. This spe
ial 
hoi
e of our partition is not 
ru
ial, but it simpli�es 
ertain
aspe
ts of the analysis.
8 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The existen
e of the limit in (3.4), and the fa
t that it 
oin
ides with the in�mum, follow
from a standard subadditivity argument. It has a 
ru
ial 
onsequen
e : if (µi) is a sequen
e
of (gt)�invariant probability measures on T ∗M , weakly 
onverging to a probability µ, and
if µ does not 
harge the boundary of the partition P, we have
hKS(µ,P) ≥ lim sup
hKS(µi,P) .
In parti
ular, assume that for i large enough, the following estimates hold :
(3.6) ∀n ≥ 1, ∀α ∈ [[1, K]]
, µi(Eα) ≤ Ci e
−βn ,
with β independent of i. This implies for i large enough hKS(µi,P) ≥ β, and this estimate
goes to the limit to yield hKS(µ) ≥ β.
3.2. From 
lassi
al to quantum dynami
al entropy. Sin
e our semi
lassi
al measure
µ is de�ned as a limit of Wigner distributions W~, a naive idea would be to estimate
from below the KS entropy of W~ and then take the limit ~ → 0. This idea 
annot work
dire
tly, be
ause the Wigner transformsW~ are neither positive, nor are they (g
t)�invariant.
Therefore, one 
annot dire
tly use the (formal) integrals W~(Eα) =
W~(x, ξ) dx dξ to

ompute the entropy of the semi
lassi
al measure.
Instead, the method initiated by the �rst author in [2℄ is based on the following remarks.
Ea
h integral W~(Eα) 
an also be written as W~(1lEα) =
W~ 1lEα , where 1lEα is the

hara
teristi
 fun
tion on the set E
, that is
(3.7) 1lEα = (1lEαn−1 ◦ g
n−1)× . . .× (1lEα1 ◦ g)× 1lEα0 .
Remember we took Ek = T
∗Mk, where the Mk form a partition of M .
From the de�nition of the Wigner distribution, this integral 
orresponds formally to the
overlap 〈ψ~,Op~(1lEα )ψ~〉. Yet, the 
hara
teristi
 fun
tions 1lEα have sharp dis
ontinuities,
so their quantizations 
annot be in
orporated in a ni
e pseudodi�erential 
al
ulus. Besides,
the set E
is not 
ompa
tly supported, and shrinks in the unstable dire
tion when n =
|α| −→ +∞, so that the operator Op
(1lEα ) is very problemati
.
We also note that an overlap of the form 〈ψ~,Op~(1lEα)ψ~〉 is a hybrid expression: this is
a quantum matrix element of an operator de�ned in terms of the 
lassi
al evolution (3.7).
From the point of view of quantum me
hani
s, it is more natural to 
onsider, instead, the
operator obtained as the produ
t of Heisenberg-evolved quantized fun
tions, namely
(3.8) (U−n+1
Pαn−1U
) (U−n+2
Pαn−2U
) · · · (U−1
Pα1U~)Pα0 .
Here we used the shorthand notation Pk = 1lMk , k ∈ [[1, K]] (multipli
ation operators). To
remedy the fa
t that the fun
tions 1lMk are not smooth, whi
h would prevent us from using
a semi
lassi
al 
al
ulus, we apply a 
onvolution kernel to smooth them, obtain fun
tions
1lsmMk ∈ C
∞(M), and 
onsider Pk
= 1lsmMk (we 
an do this keeping the property
k=1 1l
In the following, we will use the notation A(t)
= U−t
for the Heisenberg evolution
of the operator A though the S
hrödinger �ow U t
= exp(−it~△
). The norm ‖•‖ will denote
ENTROPY OF EIGENFUNCTIONS 9
either the Hilbert norm on L2(M), or the 
orresponding operator norm. The subsequent
�purely quantum� norms were estimated in [2, Thm. 1.3.3℄:
Theorem 3.1. (The main estimate [2℄) Set as above Pk
= 1lsmMk . For every K > 0,
there exists ~K > 0 su
h that, uniformly for all ~ < ~K, for all n ≤ K| log ~|, for all
(α0, . . . , αn−1) ∈ [[1, K]]
(3.9) ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 ψ~‖ ≤ 2(2π~)
−d/2 e−
n(1 +O(ε))n.
The exponent Λ is given by the �smallest expansion rate�:
Λ = − sup
log Ju(ρ)dν(ρ) = inf
λ+i (γ).
The in�mum on the right hand side runs over the set of 
losed orbits on E , and the λ+i denote
the positive Lyapunov exponents along the orbit, that is the logarithms of the expanding
eigenvalues of the Poin
aré map, divided by the period of the orbit. The parameter ε > 0
is an upper bound on the diameters of the supports of the fun
tions 1lsmMk in M .
From now on we will 
all the produ
t operator
(3.10) P
= Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 , α ∈ [[1, K]]
To prove the above estimate, one a
tually 
ontrols the operator norm
(3.11) ‖P
(χ)‖ ≤ 2(2π~)−d/2 e−
n(1 +O(ε))n ,
where χ ∈ C∞c (E
ε) is an energy 
uto� su
h that χ = 1 near E , supported inside a neigh-
bourhood Eε = H−1([1
− ε, 1
+ ε]) of E .
In quantum me
hani
s, the matrix element 〈ψ~, Pαψ~〉 looks like the �probability�, for a
parti
le in the state ψ~, to visit su

essively the phase spa
e regions Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 of the S
hrödinger �ow. Theorem 3.1 implies that this �probability�
de
ays exponentially fast with n, with rate Λ
, but this de
ay only starts around the time
(3.12) n1
d| log ~|
whi
h is a kind of �Ehrenfest time� (see (3.25) for another de�nition of Ehrenfest time).
Yet, be
ause the matrix elements 〈ψ~, Pαψ~〉 are not real in general, they 
an hardly be
used to de�ne a �quantum measure�. Another possibility to de�ne the probability for the
parti
le to visit the sets Eαk at times k, is to take the squares of the norms appearing in
(3.9):
(3.13) ‖P
2 = ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0ψ~‖
Now we require the smoothed 
hara
teristi
 fun
tions 1lsmMi to satisfy the identity
(3.14)
1lsmMk(x)
= 1 for any point x ∈M .
10 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
We denote by Psm the smooth partition of M made by the fun
tions
(1lsmMk)
. The

orresponding set of multipli
ation operators (Pk)
= Pq forms a �quantum partition of
unity� :
(3.15)
P 2k = IdL2 .
For any n ≥ 1, we re�ne the quantum partition Pq into (Pα)|α|, as in (3.10). The weights
(3.13) exa
tly add up to unity, so it makes sense to 
onsider the entropy
(3.16) hn(ψ~,Pq)
α∈[[1,K]]n
3.2.1. Conne
tion with other quantum entropies. This entropy appears to be a parti
ular

ase of the �general quantum entropies� des
ribed by Sªom
zy«ski and �y
zkowski [28℄,
who already had in mind appli
ations to quantum 
haos. In their terminology, a family of
bounded operators π = (πk)
k=1 on a Hilbert spa
e H satisfying
(3.17)
π∗k πk = IdH
provides an �instrument� whi
h, to ea
h index k ∈ [[1,N ]], asso
iates the following map on
density matri
es:
ρ 7→ I(k)ρ = πk ρ π
k , a nonnegative operator with tr(I(k)ρ) ≤ 1 .
From a unitary propagator U and its adjoint a
tion Uρ = UρU−1, they propose to 
onstru
t
the re�ned instrument
I(α)ρ
= I(αn−1) ◦ · · · U ◦ I(α1) ◦ U ◦ I(α0)ρ = U
−n+1 π
Un−1 , α ∈ [[1,N ]]
where we used (3.10) to re�ne the operators πk into πα. We obtain the probability weights
(3.18) tr(I(α)ρ) = tr(π
) , α ∈ [[1,N ]]
For any U-invariant density ρ, these weights provide an entropy
(3.19) hn(ρ, I) =
α∈[[1,N ]]n
tr(I(α)ρ)
One easily 
he
ks that our quantum partition Pq = (Pk)
k=1 satis�es (3.17), and that if
one takes ρ = |ψ~〉〈ψ~| the weights tr(I(α)ρ) exa
tly 
orrespond to our weights ‖Pαψ‖
Hen
e, the entropy (3.19) 
oin
ides with (3.16).
Around the same time, Ali
ki and Fannes [1℄ used the same quantum partition (3.17)
(whi
h they 
alled ��nite operational partitions of unity�) to de�ne a di�erent type of
entropy, now 
alled the �Ali
ki-Fannes entropy� (the de�nition extends to general C∗-
dynami
al systems). For ea
h n ≥ 1 they extend the weights (3.18) to �o�-diagonal entries�
to form a N n ×N n density matrix ρn:
(3.20) [ρn]α′,α = tr(πα′ ρ π
), α,α′ ∈ [[1,N ]]
ENTROPY OF EIGENFUNCTIONS 11
The AF entropy of the system (U , ρ) is then de�ned as follows: take the Von Neumann
entropy of these density matri
es, hAFn (ρ, π) = tr η(ρn), then take lim supn→∞
hAFn (ρ, π)
and �nally take the supremum over all possible �nite operational partitions of unity π.
We mention that tra
es of the form (3.20) also appear in the �quantum histories� ap-
proa
h to quantum me
hani
s (see e.g. [17℄, and [28, Appendix D℄ for referen
es).
3.2.2. Naive treatment of the entropy hn(ψ~,Pq). For �xed |α| > 0, the Egorov theorem
shows that ‖P

onverges to the 
lassi
al weight µ
(1lsmMα )
when ~ → 0, so for �xed
n > 0 the entropy hn(ψ~,Pq) 
onverges to hn(µ,Psm), de�ned as in (3.2), the 
hara
teristi
fun
tions 1lMk being repla
ed by their smoothed versions (1l
)2. On the other hand, from
the estimate (3.11) the entropies hn(ψ~,Pq) satisfy, for ~ small enough,
(3.21) hn(ψ~,Pq) ≥ n
Λ+O(ε)
− d| log ~|+O(1) ,
for any time n ≤ K| log ~|. For large times n ≈ K| log ~|, this provides a lower bound
hn(ψ~,Pq) ≥
Λ +O(ε)
+O(1/| log ~|) ,
whi
h looks very promising sin
e K 
an be taken arbitrary large: we 
ould be tempted to
take the semi
lassi
al limit, and dedu
e a lower bound hKS(µ) ≥ Λ.
Unfortunately, this does not work, be
ause in the range {n > n1} where the estimate
(3.21) is useful, the Egorov theorem breaks down, the weights (3.13) do not approximate
the 
lassi
al weights µ
(1lsmMα )
, and there is no relationship between hn(ψ,Pq) and the

lassi
al entropies hn(µ,Psm).
This breakdown of the quantum-
lassi
al 
orresponden
e around the Ehrenfest time is
ubiquitous for 
haoti
 dynami
s. It has been observed before when studying the 
onne
tion
between the Ali
ki-Fannes entropy for the quantized torus automorphisms and the KS
entropy of the 
lassi
al dynami
s [5℄: the quantum entropies hAFn (ψ~,Pq) follow the 
lassi
al
hn(µ,Psm) until the Ehrenfest time (and therefore grow linearly with n), after whi
h they
�saturate�, to produ
e a vanishing entropy lim supn→∞
hAFn (ψ~,Pq).
To prove the Theorem 2.1, we will still use the estimates (3.11), but in a more subtle
way, namely by referring to an entropi
 un
ertainty prin
iple.
3.3. Entropi
 un
ertainty prin
iple. The theorem below is an adaptation of the en-
tropi
 un
ertainty prin
iple 
onje
tured by Deuts
h and Kraus [12, 21℄ and proved by
Massen and U�nk [25℄. These authors were investigating the theory of measurement in
quantum me
hani
s. Roughly speaking, this result states that if a unitary matrix has
�small� entries, then any of its eigenve
tors must have a �large� Shannon entropy.
Let (H, 〈., .〉) be a 
omplex Hilbert spa
e, and denote ‖ψ‖ =
〈ψ, ψ〉 the asso
iated
norm. Consider a quantum partition of unity (πk)
k=1 on H as in (3.17). If ‖ψ‖ = 1,
we de�ne the entropy of ψ with respe
t to the partition π as in (3.16), namely hπ(ψ) =
12 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
‖πk ψ‖
. We extend this de�nition by introdu
ing the notion of pressure, asso
i-
ated to a family v = (vk)k=1,...,N of positive real numbers: the pressure is de�ned by
pπ,v(ψ)
‖πk ψ‖
‖πk ψ‖
2 log v2k.
In Theorem 3.2, we a
tually need two partitions of unity (πk)
k=1 and (τj)
j=1, and two
families of weights v = (vk)
k=1, w = (wj)
j=1, and 
onsider the 
orresponding pressures
pπ,v(ψ), pτ,w(ψ). Besides the appearan
e of the weights v, w, we bring another modi�
ation
to the statement in [25℄ by introdu
ing an auxiliary operator O.
Theorem 3.2. [4, Thm. 6.5℄ Let O be a bounded operator and U be an isometry on H.
De�ne c
(v,w)
O (U)
= supj,k wj vk ‖τj U π
kO‖, and V = maxk vk, W = maxj wj.
Then, for any ǫ ≥ 0, for any normalized ψ ∈ H satisfying
(3.22) ∀k = 1, . . . ,N , ‖(Id−O) πk ψ‖ ≤ ǫ ,
the pressures pτ,w
, pπ,v
satisfy
+ pπ,v
≥ −2 log
(v,w)
O (U) +N V W ǫ
Example 1. The original result of [25℄ 
orresponds to the 
ase where H = CN , O = Id,
ǫ = 0, N = M, vk = wj = 1, and the operators πk = τk are the orthogonal proje
tors on
some orthonormal basis (ek)
k=1 of H. In this 
ase, the theorem asserts that
hπ(U ψ) + hπ(ψ) ≥ −2 log c(U)
where c(U) = supj,k |〈ek,Uej〉| is the supremum of all matrix elements of U in the orthonor-
mal basis (ek). As a spe
ial 
ase, one gets hπ(ψ) ≥ − log c(U) if ψ is an eigenfun
tion of
3.4. Applying the entropi
 un
ertainty prin
iple to the Lapla
ian eigenstates.
In this se
tion we explain how to use Theorem 3.2 in order to obtain nontrivial information
on the quantum entropies (3.16) and then hKS(µ). For this we need to de�ne the data
to input in the theorem. Ex
ept the Hilbert spa
e H = L2(M), all other data depend on
the semi
lassi
al parameter ~: the quantum partition π, the operator O, the positive real
number ǫ, the weights (vj), (wk) and the unitary operator U .
As explained in se
tion 3.2, we partition M into M = ⊔Kk=1Mk, 
onsider open sets
Ωk ⊃⊃Mk (whi
h we assume to have diameters ≤ ε), and 
onsider smoothed 
hara
teristi
fun
tions 1lsmMk supported respe
tively inside Ωk, and satisfying the identity (3.14). The
asso
iated multipli
ation operators on H are form a quantum partition (Pk)
k=1, whi
h we
had 
alled Pq. To alleviate notations, we will drop the subs
ript q.
From (3.15), and using the unitarity of U~, one realizes that for any n ≥ 1, the families
of operators P∨n = (P ∗
)|α|=n and T
∨n = (P
)|α|=n (see (3.10)) make up two quantum
partitions of unity as in (3.17), of 
ardinal Kn.
ENTROPY OF EIGENFUNCTIONS 13
3.4.1. Sharp energy lo
alization. In the estimate (3.11), we introdu
ed an energy 
uto� χ
on a �nite energy strip Eε, with χ ≡ 1 near E . This 
uto� does not appear in the estimate
(3.9), be
ause, when applied to the eigenstate ψ~, the operator Op~(χ) essentially a
ts like
the identity.
The estimate (3.11) will a
tually not su�
e to prove Theorem 2.1. We will need to
optimize it by repla
ing χ in (3.11) with a �sharp� energy 
uto�. For some �xed (small)
δ ∈ (0, 1), we 
onsider a smooth fun
tion χδ ∈ C
∞(R; [0, 1]), with χδ(t) = 1 for |t| ≤ e
and χδ(t) = 0 for |t| ≥ 1. Then, we res
ale that fun
tion to obtain the following family of
~-dependent 
uto�s near E :
(3.23) ∀~ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗M, χ(n)(ρ; ~)
e−nδ ~−1+δ(H(ρ)− 1/2)
The 
uto� χ(n) is supported in a tubular neighbourhood of E of width 2~1−δ enδ. We will
always assume that this width is << ~1/2 in the semi
lassi
al limit, whi
h is the 
ase if we
ensure that n ≤ Cδ| log ~| for some 0 < Cδ < (2δ)
−1−1. In spite of their singular behaviour,
these 
uto�s 
an be quantized into pseudodi�erential operators Op(χ(n)) des
ribed in [4℄
(the quantization uses a pseudodi�erential 
al
ulus adapted to the energy layer E , drawn
from [29℄). The eigenstate ψ~ is indeed very lo
alized near E , sin
e it satis�es
(3.24) ‖
Op(χ(0))− 1
ψ~‖ = O(~
∞) ‖ψ~‖ .
In the rest of the paper, we also �x a small δ′ > 0, and 
all �Ehrenfest time� the ~-dependent
integer
(3.25) nE(~)
⌊(1− δ′)| log ~|
Noti
e the resemblan
e with the time n1 de�ned in (3.12). The signi�
an
e of this time
s
ale will be dis
ussed in �3.4.5.
The following proposition states that the operators (P ∗
)|α|=nE , almost preserve the en-
ergy lo
alization of ψ~ :
Proposition 3.3. For any L > 0, there exists ~L su
h that, for any ~ ≤ ~L, the Lapla
ian
eigenstate satis�es
(3.26) ∀α, |α| = nE , ‖
Op(χ(nE))− Id
ψ~‖ ≤ ~
L‖ψ~‖ .
We re
ognize here a 
ondition of the form (3.22).
3.4.2. Applying Theorem 3.2: Step 1. We now pre
ise some of the data we will use in the
entropi
 un
ertainty prin
iple, Theorem 3.2. As opposed to the 
hoi
e made in [4℄, we will
use two di�erent partitions π, τ .
• the quantum partitions π and τ are given respe
tively by the families of operators
π = P∨nE = (P ∗
)|α|=nE , τ = T
∨nE = (P
)|α|=nE . Noti
e that these partitions
only di�er by the ordering of the operators Pαi(i) inside the produ
ts. In the
semi
lassi
al limit, these partitions have 
ardinality N = KnE ≍ ~−K0 for some
�xed K0 > 0.
• the isometry will be the propagator at the Ehrenfest time, U = UnE
14 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
• the auxiliarly operator is given as O = Op(χ(nE)), and the error ǫ = ~L, where L
will be 
hosen very large (see �3.4.4).
• the weights v
will be sele
ted in �3.4.4. They will be semi
lassi
ally tempered,
meaning that there exists K1 > 0 su
h that, for ~ small enough, all vα, wα are

ontained in the interval [1, ~−K1].
The entropy and pressures asso
iated with a state ψ ∈ H are given by
hπ(ψ) =
|α|=nE
,(3.27)
pπ,v(ψ) = hπ(ψ)− 2
|α|=nE
ψ‖2 log v
.(3.28)
With respe
t to the se
ond partition, we have
hτ (ψ) =
|α|=nE
,(3.29)
pτ,w(ψ) = hτ (ψ)− 2
|α|=nE
ψ‖2 logw
.(3.30)
We noti
e that the entropy hτ (ψ) exa
tly 
orresponds to the formula (3.16), while hπ(ψ)
is built from the norms
ψ‖2 = ‖Pα0Pα1(1) · · ·Pαn−1(n− 1)ψ‖
If ψ is an eigenfun
tion of U~, the above norm 
an be obtained from (3.13) by ex
hanging
U~ with U
, and repla
ing the sequen
e α = (α0, . . . , αn−1) by ᾱ
= (αn−1, . . . , α0). So the
entropies hπ(ψ) and hτ (ψ) are mapped to one another through the time reversal U~ → U
With these data, we draw from Theorem 3.2 the following
Corollary 3.4. For ~ > 0 small enough 
onsider the data π, τ , U , O as de�ned above.
(3.31) c
O (U)
= max
|α|=|α′|=nE
Op(χ(nE))‖
Then for any normalized state φ satisfying (3.26),
pτ,w(U
φ) + pπ,v(φ) ≥ −2 log
O (U) + h
L−K0−2K1
From (3.26), we see that the above 
orollary applies to the eigenstate ψ~ if ~ is small
enough.
The reason to take the same value nE for the re�ned partitions P
, T ∨nE and the
propagator U
is the following : the produ
ts appearing in c
O (U) 
an be rewritten
(with U ≡ U~):
′ UnE P
= U−nE+1Pα′
U · · ·UPα′0UPαnE−1U · · ·UPα0 = U
ENTROPY OF EIGENFUNCTIONS 15
Thus, the estimate (3.11) with n = 2nE already provides an upper bound for the norms
appearing in (3.31) � the repla
ement of χ by the sharp 
uto� χ(nE) does not harm the
estimate.
To prove Theorem 2.1, we a
tually need to improve the estimate (3.11), as was done in
[4℄, see Theorem 3.5 below. This improvement is done at two levels: we will use the fa
t
that the 
uto�s χ(nE) are sharper than χ, and also the fa
t that the expansion rate of the
geodesi
 �ow (whi
h governs the upper bound in (3.11)) is not uniform, but depends on
the sequen
e α.
Our 
hoi
e for the weights v
will then be guided by the α-dependent upper bounds
given in Theorem 3.5. To state that theorem, we introdu
e some notations.
3.4.3. Coarse-grained unstable Ja
obian. We re
all that, for any energy λ > 0, the geodesi
�ow gt on the energy layer E(λ) = H−1(λ) ⊂ T ∗M is Anosov, so that the tangent spa
e
TρE(λ) at ea
h ρ ∈ T
∗M , H(ρ) > 0 splits into
TρE(λ) = E
u(ρ)⊕Es(ρ)⊕ RXH(ρ)
where Eu (resp. Es) is the unstable (resp. stable) subspa
e. The unstable Ja
obian Ju(ρ)
is de�ned by Ju(ρ) = det
|Eu(g1ρ)
(the unstable spa
es at ρ and g1ρ are equipped with
the indu
ed Riemannian metri
).
This Ja
obian 
an be �dis
retized� as follows in the energy strip Eε ⊃ E . For any pair of
indi
es (α0, α1) ∈ [[1, K]]
, we de�ne
(3.32) Ju1 (α0, α1)
= sup
Ju(ρ) : ρ ∈ T ∗Ωα0 ∩ E
ε, g1ρ ∈ T ∗Ωα1
if the set on the right hand side is not empty, and Ju1 (α0, α1) = e
otherwise, where R > 0
is a �xed large number. For any sequen
e of symbols α of length n, we de�ne
(3.33) Jun(α)
= Ju1 (α0, α1) · · ·J
1 (αn−2, αn−1) .
Although Ju and Ju1 (α0, α1) are not ne
essarily everywhere smaller than unity, there exists
C, λ+, λ− > 0 su
h that, for any n > 0, for any α with |α| = n,
(3.34) C−1 e−n(d−1) λ+ ≤ Jun(α) ≤ C e
−n(d−1) λ− .
One 
an take λ+ = λmax(1+ε), where λmax is the maximal expanding rate in Theorem. 2.1.
We now give our 
entral estimate, easy to draw from [4, Corollary 3.4℄.
Theorem 3.5. Fix small positive 
onstants ε, δ, δ′ and a 
onstant 0 < Cδ < (2δ)
−1 − 1.
Take an open 
over M =
k Ωk of diameter ≤ ε and an asso
iated quantum partition P =
k=1. There exists ~0 su
h that, for any ~ ≤ ~0, for any positive integer n ≤ Cδ| log ~|,
and any pair of sequen
es α, α
of length n,
(3.35) ‖P
′ Op(χ(n))‖ = ‖P
Op(χ(n))‖ ≤ C ~−
−δ enδ
Jun(α) J
The 
onstant C only depends on the Riemannian manifold (M, g). If we take n = nE, this
takes the form
(3.36) ‖P
Op(χ(nE))‖ ≤ C ~−
d−1+cδ
JunE(α) J
(α′) ,
16 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
where c = 2 + 2λ−1max.
The idea of proof in Theorem 3.5 is rather simple, although the te
hni
al implementation
is 
umbersome. We �rst show that for any normalized state ψ, the state Op(χ(n))ψ 
an
be essentially de
omposed into a superposition of ~
−d| suppχ(n)| normalized Lagrangian
states, supported on Lagrangian manifolds transverse to the stable foliation. In fa
t the
Lagrangian states we work with are trun
ated δ�fun
tions, supported on lagrangians of the
form ∪tg
tS∗zM . The a
tion of the operator U
′ = Pα′
U · · ·UPα0 on su
h Lagrangian
states 
an be analyzed through WKB methods, and is simple to understand at the 
lassi
al
level : ea
h appli
ation of the propagator U stret
hes the Lagrangian along the unstable
dire
tion (the rate of stret
hing being des
ribed by the lo
al unstable Ja
obian), whereas
ea
h operator Pk �proje
ts� on a pie
e of Lagrangian of diameter ε. This iteration of
stret
hing and 
utting a

ounts for the exponential de
ay. The αα
-independent fa
tor
on the right of (3.36) results from adding together the 
ontributions of all the initial
Lagrangian states. Noti
e that this prefa
tor is smaller than in Theorem. 3.1 due to the

ondition Cδ < (2δ)
−1 − 1.
Remark 3.6. In [4℄ we used the same quantum partition P∨nE for π and τ in Theorem. 3.2.
As a result, we needed to estimate from above the norms ‖P ∗
′ UnE PαOp(χ
(nE))‖ (see [4,
Theorem. 2.6℄). The proof of this estimate was mu
h more involved than the one for
(3.36), sin
e it required to 
ontrol long pie
es of unstable manifolds. By using instead the
two partitions P(n), T (n), we not only prove a more pre
ise lower bound (2.3) on the KS
entropy, but also short-
ir
uit some �ne dynami
al analysis.
3.4.4. Applying Theorem 3.2: Step 2. There remains to 
hoose the weights (v
) to use
in Theorem 3.2. Our 
hoi
e is guided by the following idea: in (3.31), the weights should
balan
e the variations (with respe
t to α,α′) in the norms, su
h as to make all terms in
(3.31) of the same order. Using the upper bounds (3.36), we end up with the following

hoi
e for all α of length nE :
= JunE(α)
−1/2 .
From (3.34), there exists K1 > 0 su
h that, for ~ small enough, all the weights are

ontained in the interval [1, ~−K1], as announ
ed in �3.4.2. Using these weights, the estimate
(3.36) implies the following bound on the 
oe�
ient (3.31):
∀~ < ~0, c
O (U) ≤ C ~
− d−1+cδ
We 
an now apply Corollary 3.4 to the parti
ular 
ase of the eigenstates ψ~. We 
hoose L
su
h that L−K0 − 2K1 > −
d−1+cδ
, so from Corollary 3.4 we draw the following
Proposition 3.7. Let (ψ~)~→0 be our sequen
e of eigenstates (2.2). In the semi
lassi
al
limit, the pressures of ψ~ satisfy
(3.37) pP∨nE ,v(ψ~) + pT ∨nE ,w(ψ~) ≥ −
(d − 1 + cδ)λmax
(1− δ′)
nE +O(1) .
ENTROPY OF EIGENFUNCTIONS 17
If M has 
onstant 
urvature we have log Jn
≤ −n(d − 1)λmax(1 − O(ε)) for all α of
length n, and the above lower bound 
an be written
hP∨nE (ψ~) + hT ∨nE (ψ~) ≥ (d− 1)λmax
1 +O(ε, δ, δ′)
As opposed to (3.21), the above inequality provides a nontrivial lower bound for the quan-
tum entropies at the time nE , whi
h is smaller than the time n1 of (3.12), and will allow
to 
onne
t those entropies to the KS entropy of the semi
lassi
al measure (see below).
3.4.5. Subadditivity until the Ehrenfest time. Even at the relatively small time nE , the

onne
tion between the quantum entropy h(ψ~,P
∨nE) and the 
lassi
al h(µ,P∨nEsm ) is not

ompletely obvious: both are sums of a large number of terms (≍ ~−K0). Before taking
the limit ~ → 0, we will prove that a lower bound of the form (3.37) still holds if we
repla
e nE ≍ | log ~| by some �xed no ∈ N, and P
by the 
orresponding quantum
partition P∨no . The link between quantum pressures at times nE and no is provided by
the following subadditivity property, whi
h is the semi
lassi
al analogue of the 
lassi
al
subadditivity of pressures for invariant measures (see (3.3)).
Proposition 3.8 (Subadditivity). Let δ′ > 0. There is a fun
tion R(no, ~), and a real
number R > 0 independent of δ′, su
h that, for any integer no ≥ 1,
lim sup
|R(no, ~)| ≤ R
and with the following properties. For any small enough ~ > 0, any integers no, n ∈ N with
no + n ≤ nE(~), for any ψ~ normalized eigenstate satisfying (2.2), the following inequality
holds:
pP∨(no+n),v(ψ~) ≤ pP∨no ,v(ψ~) + pP∨n,v(ψ~) +R(no, ~) .
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~).
To prove this proposition, one uses a re�ned version of Egorov's theorem [10℄ to show that
the non�
ommutative dynami
al system formed by (U t
) a
ting (through Heisenberg) on
observables supported near E is (approximately) 
ommutative on time intervals of length
nE(~). Pre
isely, we showed in [4℄ that, provided ε is small enough, for any a, b ∈ C
∀t ∈ [−nE(~), nE(~)], ‖[Op~(a)(t),Op~(b)]‖ = O(~
cδ′), ~ → 0 ,
and the implied 
onstant is uniform with respe
t to t. Within that time interval, the oper-
ators Pαj (j) appearing in the de�nition of the pressures 
ommute up to small semi
lassi
al
errors. This almost 
ommutativity explains why the quantum pressures pP∨n,v(ψ~) satisfy
the same subadditivity property as the 
lassi
al entropy (3.3), for times smaller than nE .
Thanks to this subadditivity, we may �nish the proof of Theorem. 2.1. Fixing no, using
for ea
h ~ the Eu
lidean division nE(~) = q(~)no + r(~) (with r(~) < no), Proposition 3.8
implies that for ~ small enough,
pP∨nE ,v(ψ~)
pP∨no ,v(ψ~)
pP∨r,v(ψ~)
R(no, ~)
18 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). Using (3.37) and the fa
t
that pP∨r,v(ψ~) stays uniformly bounded when ~ → 0, we �nd
(3.38)
pP∨no ,v(ψ~) + pT ∨no ,w(ψ~)
2(d− 1 + cδ)λmax
2(1− δ′)
2R(no, ~)
+Ono(1/nE) .
We are now dealing with quantum partitions P∨no , T ∨no , for n0 ∈ N independent of ~. At
this level the quantum and 
lassi
al entropies are related through the (�nite time) Egorov
theorem, as we had noti
ed in �3.2.2. For any α of length no, the weights ‖Pαψ~‖
both 
onverge to µ
(1lsmMα)
, where we re
all that
1lsmMα = (1l
Mαno−1
◦ gno−1)× . . .× (1lsmMα1
◦ g)× 1lsmMα0
Thus, both entropies hP∨no (ψ~), hT ∨no (ψ~) semi
lassi
ally 
onverge to the 
lassi
al entropy
hno(µ,Psm). As a result, the left hand side of (3.38) 
onverges to
(3.39) 2
hno(µ,Psm)
|α|=no
(1lsmMα )
log Juno(α) .
Sin
e µ is gt-invariant and Juno has the multipli
ative stru
ture (3.33), the se
ond term in
(3.39) 
an be simpli�ed:
|α|=no
(1lsmMα )
log Juno(α) = (no − 1)
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1) .
We have thus obtained the lower bound
(3.40)
hno(µ,Psm)
no − 1
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1)−
(d− 1 + cδ)λmax
2(1− δ′)
At this stage we may forget about δ and δ′. The above lower bound does not depend on
the derivatives of the fun
tions 1lsmMα , so the same bound 
arries over if we repla
e 1l
the 
hara
teristi
 fun
tions 1lMα . We 
an �nally let no tend to +∞, then let the diameter
ε tend to 0. The left hand side 
onverges to hKS(µ) while, from the de�nition (3.32), the
sum in the right hand side of (3.40) 
onverges to the integral
log Ju(ρ)dµ(ρ) as ε → 0,
whi
h proves (2.3).
Referen
es
[1℄ R. Ali
ki and M. Fannes, De�ning quantum dynami
al entropy, Lett. Math. Phys. 32 75�82
(1994)
[2℄ N. Anantharaman, Entropy and the lo
alization of eigenfun
tions, to appear in Ann. Math.
[3℄ N. Anantharaman and S. Nonnenma
her, Entropy of semi
lassi
al measures of the Walsh-
quantized baker's map, Ann. H. Poin
aré 8, 37�74 (2007)
[4℄ N. Anantharaman, S. Nonnenma
her, Half�delo
alization of eigenfun
tions of the lapla
ian on
an Anosov manifold, http://hal.ar
hives-ouvertes.fr/hal-00104963
[5℄ F. Benatti, V. Cappellini, M. De Co
k, M. Fannes and D. Van Peteghem, Classi
al Limit of
Quantum Dynami
al Entropies, Rev. Math. Phys. 15, 1�29 (2003)
[6℄ M.V. Berry, Regular and irregular semi
lassi
al wave fun
tions, J.Phys. A 10, 2083�2091 (1977)
http://hal.archives-ouvertes.fr/hal-00104963
ENTROPY OF EIGENFUNCTIONS 19
[7℄ O. Bohigas, Random matrix theory and 
haoti
 dynami
s, in M.J. Giannoni, A. Voros and J. Zinn-
Justin eds., Chaos et physique quantique, (É
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CMLS, É
ole Polyte
hnique, 91128 Palaiseau, Fran
e
E-mail address : nalini�math.polyte
hnique.fr
Mathemati
al Institute, University of Bonn, Beringstraÿe 1,D-53115 Bonn, Germany
E-mail address : ko
h�math.uni-bonn.de
Servi
e de Physique Théorique, CEA/DSM/PhT, Unité de re
her
he asso
iée au CNRS,
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lay, 91191 Gif-sur-Yvette, Fran
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E-mail address : snonnenma
her�
ea.fr
	1. Motivations
	2. Main result.
	 Acknowledgements
	3. Outline of the proof
	3.1. Definition of the metric entropy
	3.2. From classical to quantum dynamical entropy
	3.3. Entropic uncertainty principle
	3.4. Applying the entropic uncertainty principle to the Laplacian eigenstates
	References
