Entropy of chaotic eigenstates
1. Overview and Statement of results
These lectures present a recent approach, mainly developed by Nalini Anantharaman and the author, aimed at studying the high-frequency eigenmodes of the Laplace-Beltrami operator on compact riemannian manifolds for which the sectional curvature is everywhere negative. It is well-known that the geodesic flow on such a manifold (which takes place on the unit cotangent bundle ) is strongly chaotic, in the sense that it is uniformly hyperbolic (Anosov). This flow leaves invariant the natural smooth measure on , namely the Liouville measure (which is also the lift of the Lebesgue measure on ). Studying the eigenstates of is thus a part of “quantum chaos”. In these notes we extend the study to more general Schrödinger-like operators in the semiclassical limit, such that the corresponding Hamitonian flow on some compact energy shell has the Anosov property. We also consider the case of quantized Anosov diffeomorphisms on the torus, which are popular toy models in the quantum chaos literature. To set up the problem we first stick to the Laplacian.
For a general riemannian manifold of dimension , there exist no explicit, not even approximate expression for the eigenmodes of the Laplacian. One way to “describe” these modes consists in comparing them (as “quantum” invariants) with “classical” invariants, namely probability measures on , invariant w.r.to the geodesic flow. To this aim, starting from the full sequence of eigenmodes one can construct a family of invariant probability measures on , called semiclassical measures. Each such measure can be associated with a subsequence of eigenmodes which share, in the limit , the same macroscopic localization properties, both on the manifold and in the velocity (or momentum) space: these macroscopic localization properties are “represented” by . One (far-reaching) aim would be a complete classification of the semiclassical measures associated with a given manifold .
This goal is much too ambitious, starting from the fact that the set of invariant measures is itself not always well-understood. We will thus restrict ourselves to the class of manifolds described above, namely manifolds of negative sectional curvature. One advantage is that the classical dynamics is at the same time “irregular” (in the sense of “chaotic”), and “homogeneous”. The geodesic flow on such a manifold is (semi)conjugated with a suspended flow over a simple symbolic dynamics (a subshift of finite type over a finite alphabet), which allows one to explicitly construct many different invariant measures. For instance, such a flow admits infinitely many isolated (unstable) periodic orbits , each of which carries a natural probability invariant measure . The set of periodic orbits is so large that the measures form a dense subset (in the weak-* topology) of the set of invariant probability measures. Hence, it would be interesting to know whether some high-frequency eigenmodes can be asymptotically localized near certain periodic orbits, leading to semiclassical measures of the form
This possibility was named “strong scarring” by Rudnick-Sarnak [RS94], in analogy with a weaker form of “scarring” observed by Heller on some numerically computed eigenmodes [Hel84], namely a “nonrandom enhancement” of the wavefunction in the vicinity of a certain periodic orbit. In the same paper, Rudnick and Sarnak conjectured that such semiclassical measures do not exist for manifolds of negative curvature. More precisely, they formulated the Quantum Unique Ergodicity conjecture
[Quantum Unique Ergodicity] [RS94]
Let be a compact riemannian manifold of negative curvature. Then there exist only one semiclassical measure, namely the Liouville measure .
This conjecture rules out any semiclassical measure of the type 1.1. It also rules out linear combination of the form
The name “quantum unique ergodicity” reminds of a classical notion: a dynamical system (map or flow) is uniquely ergodic if and only if it admits a unique invariant measure. In the present case, the classical system is not uniquely ergodic, but the conjecture is that its quantum analogue conspires to be so.
This conjecture was formulated several years after the proof of a general result describing “almost all” the eigenstates .
Let be a compact riemannian manifold such that the geodesic flow is ergodic w.r.to the Liouville measure . Then, there exists a subsequence of density , such that the subsequence is associated with .111A subsequence is said to be of density iff .
The manifolds encompassed by this theorem include the case of negative curvature, but also more general ones (like manifolds where the curvature is negative outside a flat cylindrical part ). The proof of this theorem is quite “robust”. It has been generalized to many different ergodic systems: Hamiltonian flows ergodic on some compact energy shell [HMR87], broken geodesic flows on some Euclidean domains [GerLei93, ZelZwo96], symplectic diffeomorphism (possibly with discontinuities) on a compact phase space [BDB96]. This result leaves open the possibility of exceptional subsequences (necessarily of density zero) of eigenmodes with different localization properties.
The QUE conjecture was motivated by partial results concerning a much more restricted class of manifolds, namely compact quotients of the hyperbolic disk for which is an arithmetic subgroup of 222More precisely, is derived from an Eichler order in a quaternion algebra.. Such manifolds admit a commutative algebra of selfadjoint Hecke operators which all commute with the Laplacian. It thus makes sense to preferably consider an eigenbasis made of joint eigenmodes of the Laplacian and the Hecke operators (called Hecke eigenmodes), and the associated semiclassical measures (called Hecke semiclassical measures)333It is widely believed that the spectrum of on such a manifold is simple, in which case the restriction to Hecke eigenmodes is not necessary.. Rudnick-Sarnak proved that the only Hecke semiclassical measure of the form 1.2 is the Liouville measure (). Finally, Lindenstrauss [Lin06] showed that for such manifolds, the only Hecke semiclassical measure is the Liouville measure, thus proving an arithmetic form of QUE. He used as an intermediate step a lower bound for the Kolmogorov-Sinai (KS) entropy of Hecke semiclassical measures, which he proved in a joint work with Bourgain.
[BourLin03] Let be an arithmetic quotient . Consider a Hecke semiclassical measure . Then for any any small , the measure of the tube of diameter around the stretch of trajectory is bounded by
As a consequence, for almost any ergodic component of , one has
As we will see below, the KS entropy is an affine quantity, therefore also satisfies the same lower bound.
1.2. Entropy as a measure of localization
In the previous section we already noticed a relationship between phase space localization and entropy: a uniform lower bound on the measure of thin tubes implies a positive lower bound on the entropy of the measure. For this reason, it is meaningful to consider the KS entropy of a given invariant measure as a “quantitative indicator of localization” of that measure. In section 3.1 we will give a precise definition of the entropy. For now, let us only provide a few properties valid in the case of Anosov flows [KatHas95, Chap. 4 ].
is a real function defined on the set of invariant probability measures. It takes values in a finite interval and is upper semicontinuous. In information-theoretic language, it measures the average complexity of the flow w.r.to that measure. The maximum entropy is also the topological entropy of the flow on , which is a standard measure of the complexity of the flow.
a measure supported on a single periodic orbit has zero entropy.
Since the flow is Anosov, at each point the tangent space splits into , respectively the unstable, stable subspaces and the flow direction. Each of these subspaces is flow-invariant. Let us call the unstable Jacobian at the point . Then, any invariant measure satisfies
The equality is reached iff . In constant curvature, one has .
the entropy is affine: .
Apart from the result of Bourgain-Lindenstrauss (relative to arithmetic surfaces), the first result on the entropy of semiclassical measures was obtained by Anantharaman:
[Ana08] Let be a manifold of negative sectional curvature. Then, there exists such that any semiclassical measure satisfies
Furthermore, the flow restricted to the support of has a nontrivial complexity: its topological entropy satisfies
where is the minimal volume expanding rate of the unstable manifold.
The lower bound is not very explicit and is rather “small”. This is to be opposed to the lower bound controlling the complexity of the flow on , given in terms of the hyperbolicity of the flow. The lower bound on the KS entropy was improved by Anantharaman, Koch and the author:
where is the maximal expansion rate of the flow.
In the particular case where has constant curvature , this bound reads
In the constant curvature case, the above bound roughly means that high-frequency eigenmodes of the Laplacian are at least half-delocalized. Still, the bound (1.5) is not very satisfactory when the curvature varies much across ; since may be as small as , the right hand side in (1.5) can become negative (therefore trivial) in case . The following lower bound seems more natural:
Let be a manifold of negative sectional curvature. Then, any semiclassical measure satisfies
This bound is identical with (1.6) in the case of curvature . Using a nontrivial extension of the methods developed in [AnaKoNo06], it has been recently proved by G.Rivière in the case of surfaces () of nonpositive curvature [Riv08, Riv09], and also by B.Gutkin for a certain class of quantized interval maps [Gut08].
1.3. Generalization to Anosov Hamiltonian flows and symplectic maps
The conjecture 6 is weaker than the QUE conjecture 1. We expect the bound (1.7) to apply as well to more general classes of quantized chaotic dynamical systems, like Anosov Hamiltonian flows or symplectic diffeomorphisms on a compact phase space. In these notes we will extend the bound (1.5) to these more general Anosov systems (the first instance of this entropic bound actually appeared when studying the Walsh-quantized baker’s map [AnaNo07-1]). The central result of these notes is the following theorem.
i) Let be a Hamilton function on some phase space with principal symbol , such that the energy shell is compact, and the Hamiltonian flow on is Anosov (see 2.2). Let be the -quantization of . Then, any semiclassical measure associated with a sequence of null eigenmodes of satisfies the following entropic bound:
ii) Let be the -dimensional torus, equipped with its standard symplectic structure. Let be an Anosov diffeomorphism, which can be quantized into a family of unitary propagators defined on (finite dimensional) quantum Hilbert spaces . Then, any semiclassical measure associated with a sequence of eigenmodes of satisfies the entropic bound
In §5 we will state more precisely what we meant by a “quantized torus diffeomorphism”. Let us mention that the same proof could apply as well to Anosov symplectic maps on more general symplectic manifolds admitting some form of quantization. We restricted the statement to the -torus because simple Anosov diffeomorphisms on can be constructed, and their quantization is by now rather standard. As we explain below, their study has revealed interesting features regarding the QUE conjecture.
1.4. Counterexamples to QUE for Anosov diffeomorphisms
For the simplest Anosov diffeos on , namely the hyperbolic symplectomorphisms of the torus (colloquially known as “quantum Arnold’s cat map”), the QUE conjecture is known to fail. Indeed, in [FNDB03] counterexamples to QUE for “cat maps” on the -dimensional torus were exhibited, in the form of explicit semiclassical sequences of eigenstates of the quantized map, associated with semiclassical measures of type (1.2) with . In [FNDB04] we also showed that, for this particular map, semiclassical measures of the form (1.2) necessarily satisfy .
In the case of toral symplectomorphisms on higher dimensional tori , Kelmer [Kelm07, Kelm10] has exhibited semiclassical measures in the form of the Lebesgue measure on certain co-isotropic affine subspaces of the torus invariant through the map:
For another example of a chaotic map (the baker’s map quantized à la Walsh), we were able to construct semiclassical measures of purely fractal (self-similar) nature.
It is worth mentioning an interesting result obtained by S.Brooks [Broo08] in the case of the “quantum Arnold’s cat map”. Brooks takes into accout the possibility to split any invariant measure into ergodic components:
where the probability measure , defined for -almost every point , is ergodic. The affineness of the KS entropy ensures that
so to get a lower bound on it is sufficient to show that “high-entropy” components have a positive weight in . Brooks’s result reads as follows:
[Broo08]Let be a linear hyperbolic symplectomorphism, with positive Lyapunov exponent ( is also equal to the topological entropy of on ). Fix any , and consider any associated semiclassical mesure . Then the following inequality holds:
This result directly implies (through (1.10)) the bound , but it also implies (by sending ) the above-mentioned fact that the weight of atomic components of is smaller or equal to the weight of its Lebesgue component.
1.5. Plan of the paper
These lectures reproduce most of the proofs of [AnaNo07-2, AnaKoNo06] dealing with eigenstates of the Laplacian on manifolds of negative curvature. Yet, we extend the proofs in order to deal with more general Hamiltonian flows of Anosov type (for instance, adding some small potential to the free motion on ). This can be done at the price of using more general, “microlocal” partitions of unity, as opposed to the “local” partition of unity used in [AnaNo07-2, AnaKoNo06] (which was given in terms of functions only depending on the position variable). This microlocal setting is somehow more natural, since it does not depend on the way unstable manifolds project down to the manifold . It is also more natural in the case of Anosov maps.
In §2 we recall the semiclassical tools we will need, starting with the -pseudodifferential calculus on a compact manifold, and including some exotic classes of symbols. We also define the main object of study, namely the semiclassical measures associated with sequences of null eigenstates of a family of Hamiltonians . In the central section §3 we provide the proof of Thm. 7, i), that is in the case of an Anosov Hamiltonian flow on a compact manifold . We first recall the definition of entropies and pressures associated with invariant measures. We then introduce microlocal quantum partitions in §3.2, and their refinements used to define quantum entropies and pressures associated with the eigenstates . We try to provide some geometric intuition on the operators defining these partitions. We then state the central hyperbolic dispersive estimate on the norms of these operators, deferring the proof to §4. We then introduce several versions of “entropic uncertainty principles”, from the simplest to the most complex, microlocal form (Prop. 29) . We then apply this microlocal EUP in order to bound from below quantum and classical pressures associated with our eigenmodes. §4 is devoted to the proof of the hyperbolic dispersive estimate. Here we adapt the proof of [NoZw09], which is valid in more general situations than the case of geodesic flows. Finally, in §5 we briefly recall the framework of quantized maps on the torus, and provide the details necessary to obtain Thm.7, ii).
I am grateful to D.Jakobson and I.Polterovich who invited me to give this minicourse in Montréal and to write these notes. Most of the material of these notes were obtained through from collaborations with N.Anantharaman, H.Koch and M.Zworski. I have also been partially supported by the project ANR-05-JCJC-0107091 of the Agence Nationale de la Recherche.
2. Preliminaries and problematics
2.1. Semiclassical calculus on
The original application of the methods presented below concern the Laplace-Beltrami operator on a smooth compact manifold of negative sectional curvature. To deal with this problem, one needs to define a certain number of auxiliary operators on , which are -pseudodifferential operators on (DOs), or -Fourier integral operators on . We will only recall the definition and construction of the former class.
The Hamiltonians mentioned in Theorem 7 also belong to some class of -pseudodifferential operators, but the manifold on which they are defined is not necessarily compact any more. In this setting, the smooth manifold can be taken as the Euclidean space , or be Euclidean near infinity, that is , where is the ball of radius in , and is a compact manifold, the boundary of which is smoothly glued to .
2.1.1. Symbol classes on and -pseudodifferential calculus
Let us construct an -quantization procedure on a Riemannian manifold . To a certain class of well-behaved functions on (the physical observables, referred to as symbols in mathematics) one can associate, through a well-defined quantization procedure , a corresponding set of operators acting on . By “well-behaved” one generally refers to certain conditions on the regularity and growth of the function. There are many different types of classes of “well-behaved symbols”; we will be using the class
Here we use the “japanese brackets” notation . The estimates are supposed to hold uniformly for and . The seminorms can be defined locally on coordinate charts of ; due to the factor , this class is invariant w.r.to changes of coordinate charts on , and thus makes sense intrinsically on the manifold .
Some (important) symbols in this class are of the form
In that case, is called the principal symbol of .
For , a symbol can be quantized using the Weyl quantization: it acts on through as the integral operator
If is a real function, this operator is essentially selfadjoint on .
If is a more complicated manifold, one can quantize by first splitting it into pieces localized on various coordinate charts , through a finite partition of unity , :
Each component can be considered as a function on , and be quantized through (2.1), producing an operator ) acting on . A wavefunction will be cut into pieces , where the cutoffs satisfy 444Throughout the text we will ofen encouter such “embedded cutoffs”. The property will be denoted by .. Our final quantization is then defined as:
The image of the class through quantization is an operator algebra acting on , denoted by . This algebra has “nice” properties in the semiclassical limit . The product of two such operators behaves as a “decoration” of the usual multiplication:
where admits an asymptotic expansion of the form
Here the first component , while each is a linear combination of derivatives , with . In the case and is the Weyl quantization (2.1), is called the Moyal product. In the case , can be extended into a continuous operator on , and the sharp Gårding inequality ensures that
The quantization procedure is obviously not unique: it depends on the choice of coordinates on each chart, on the choice of quantization , on the choice of cutoffs . Fortunately, this non-uniqueness becomes irrelevant in the semiclassical limit.
In the semiclassical limit , two -quantizations differ at most at subprincipal order:
2.2. From the Laplacian to more general quantum Hamiltonians
2.2.1. Rescaling the Laplacian
One of our objectives is to study an eigenbasis of the Laplace-Beltrami operator on some compact Riemannian manifold . To deal with the high-frequency limit , it turns out convenient to use a “quantum mechanics” point of view, namely rewrite the eigenmode equation
in the form
This way, appears as an effective Planck’s constant (which is of the order of the wavelength of the state ). The rescaled Laplacian operator
is the -quantization of a certain classical Hamiltonian
The principal symbol generates (through Hamilton’s equations) the motion of a free particle on . In particular, the Hamilton flow restricted to the energy shell is the geodesic flow (in the following, we will often denote by this energy shell).
The Laplacian eigenmodes will often be denoted by instead of , with the convention that the state satisfies the eigenvalue equation
We will call a countable set of scales , with only accumulation point at the origin. A sequence of states indexed by will be denoted by , or sometimes, omitting the reference to a specific , by .
2.2.2. Anosov Hamiltonian flows
In Theorem 7 we deal with more general Hamiltonians on , where is compact or could also be the Euclidean space . The (real) symbol is assumed to belong to a class , and admit the expansion
that is is the principal symbol of . We could as well consider more general symbol classes, see for instance [EvZw09, Sec. 4.3] in the case . We assume that
the energy shell is compact, so that is compact as well for small enough.
the Hamiltonian flow restricted to the energy shell does not admit fixed points, and is of Anosov type.
The Hamiltonian is quantized into an operator . The first assumption above implies that, for small enough, the spectrum of near zero is purely discrete. We will focus on sequences of normalized null eigenstates :
If is a “quasi-null” eigenstate of , that is if with , then it is a null eigenstate of , which admits the same principal symbol as . As a result, Thm. 7 is also valid for such sequences of states.
2.3. -dependent singular’ observables
In the following we will have to use some classes of “singular” -dependent symbols.
2.3.1. “Isotropically singular” observables
For , we will consider the class
Such functions can strongly oscillate on scales . The corresponding operators belong to an algebra which can still be analyzed using an -expansion of the type (2.4). The main difference is that the higher-order terms . Similarly, the Garding inequality reads, for :
where the implicit constant depends on a certain seminorm of .
2.3.2. “Anisotropically singular” observables
We will also need to quantize observables which are “very singular” along certain directions, away from some specific submanifold (see for instance [SjoZwo99] for a presentation). Consider a compact co-isotropic manifold of dimension (with ). Near each point , there exist local canonical coordinates such that . For some index , we define as follows a class of smooth symbols :
for any family of smooth vector fields tangent to and of smooth vector fields , we have in any neighbourhood of :
away from we require .
Such a symbol can be split into components localized in neighbourhoods , plus an “external” piece vanishing near . Each piece is Weyl-quantized in local adapted canonical coordinates on (as in (2.1)), and then brought back to the original coordinates using Fourier integral operators. On the other hand, is quantized as in (2.2). Finally, is obtained by summing the various contributions. The resulting class of operators is denoted by .
2.3.3. Sharp energy cutoffs
We will mostly use this quantization relative to the energy layer , in order to define a family of sharp energy cutoffs. Namely, for some small we will start from a cutoff such that for , for . From there, we define, for each and each , the rescaled function by
The functions are “sharp” energy cutoffs, they belong to the class . We will always consider , where the constant , such that is microscopic.
These cutoffs can be quantized in two ways:
we may directly quantize the function , into .
or we can consider, using functional calculus, the operators . These operators (which generally differ from the previous ones) also belongs to .
The sequence is an increasing sequence of embedded cutoffs: for each , we have (equivalently, ). More precisely, we have here
This distance between the supports implies the following
For any symbol and any , one has
The same property holds if we replace by .
Using the calculus of the class , one can use the ellipticity of away from to show that, if is a sequence of null eigenstates of , then
That is, in the semiclassical limit the eigenstate is microlocalized inside the energy layer of width around .
2.4. Semiclassical measures
The -semiclassical calculus allows us to define what we mean by “phase space distribution of the eigenstate ”, through the notion of semiclassical measure. A Borel measure on the phase space can be fully characterized by the set of its values
over smooth test functions . For each semiclassical scale , one can quantize a test function into a test operator (which is, as mentioned above, a continuous operator on ). To any normalized state we can then associate the linear functional
is a distribution on , which encodes the localization properties of the state in the phase space, at the scale . Let us give an example. Using some local coordinate chart near and a function , we can define a Gaussian wavepacket by
Here is a smooth cutoff equal to unity near , which vanishes outside the coordinate chart, is a normalization factor. When , the distribution associated with this wavefunction gets very peaked around the point . If we had used the quantization at the scale , the measure would have been peaked around instead.
Since the distribution is defined by duality w.r.to the quantization , it depends on the precise quantization scheme . In the case and is the Weyl quantization, the distribution is called the Wigner distribution associated with the state and the scale . Fortunately, as shown by proposition 10, this scheme-dependence is irrelevant in the semiclassical limit.
For any , consider the distributions , defined by duality with two -quantizations , . Then, the following estimate holds in the semiclassical limit, uniformly w.r.to :
Let be a set of scales. For a given family of -normalized states , we consider the sequence of distributions on . It is always possible to extract a subset of scales , such that
with a certain distribution on . One can show that is a Radon measure on [EvZw09, Thm 5.2]. From the above remarks, does not depend on the precise scheme of quantization.
The measure is called the semiclassical measure associated with the subsequence . It is also a semiclassical measure associated with the sequence .
From now on, we will assume that is a null eigenstate of the quantum Hamiltonian in §2.2: we will then call a semiclassical measure of the Hamiltonian .
Any semiclassical measure associated with a sequence of eigenstates of the Hamiltonian is a probability measure supported on the energy layer , which is invariant w.r.to the geodesic flow on .
Possibly after extracting a subsequence, we assume that is the semiclassical measure associated with a sequence of eigenstates . The support property of comes from the fact that the operator is elliptic outside . As a result, for any vanishing near , one can construct a symbol such that
Applying this equality to the eigenstates , we get , proving the support property of .
To prove the flow invariance, we need to compare the quantum time evolution with the classical one. Denote by the propagator generated by the Hamiltonian : it solves the time-dependent Schrödinger equation, and thus provides the quantum evolution. Let us state Egorov’s theorem, which is a rigorous form of quantum-classical correspondence in terms of observables:
Since is an eigenstate of , we directly get
These properties of semiclassical measures naturally lead to the following question:
Among all flow-invariant probability measures supported on , which ones appear as semiclassical measures associated with eigenstates of ?
To start answering this question, we will investigate the Kolmogorov-Sinai entropy of semiclassical measures. We will show that, in the case of an Anosov flow, the requirement of being a semiclassical measure implies a nontrivial lower bound on the entropy.
3. From classical to quantum entropies
3.1. Entropies and pressures [KatHas95]
3.1.1. Kolmogorov-Sinai entropy of an invariant measure
In this paper we will deal with several types of entropies. All of them are defined in terms of certain discrete probability distributions, that is finite sets of real numbers satisfying
The entropy associated with such a set is the real number
Our first example is the entropy associated with a -invariant probability measure on the energy shell and a finite measurable partition of . That entropy is given by
One can then use the flow in order to refine the partition . For each integer we define the -th refinement as the partition composed of the sets
where can be any sequence of length with symbols . In general many of the sets may be empty, but we will nonetheless sum over all sequences of a given length . More generally, for any , we consider the partition made of the sets
From this refined partition we obtain the entropy .
From the concavity of the logarithm, one easily gets
If the measure is -invariant, this has for consequence the subadditivity property:
It thus makes sense to consider the limit
the Kolmogorov-Sinai entropy of the invariant measure , associated with the partition . The KS entropy per se is defined by maximizing over the initial (finite) partition :
For an Anosov flow, this supremum is actually reached as soon as the partition has a sufficiently small diameter (that is, its elements have uniformly small diameters).
3.1.2. Pressures associated with invariant measures
Let us come back to our probability distribution . We may associate to it a set of weights, that is of positive real numbers , making up a weighted probability distribution. The pressure associated with this weighted distribution is the real number555The factor appearing in front of the second term is convenient for our future aims.
For instance, in the case of a flow-invariant measure on and a partition , we can select weights on each component , and define the pressure
We want to refine this pressure using the flow. The weights corresponding to the -th refinement can be simply defined as