aa r X i v : . [ m a t h - ph ] O c t SMALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I
XIAOLONG HAN
Abstract.
In this series, we investigate quantum ergodicity at small scales for linear hyper-bolic maps of the torus (“cat maps”). In Part I of the series, we prove quantum ergodicity atvarious scales. Let N = 1 /h , in which h is the Planck constant. First, for all integers N ∈ N ,we show quantum ergodicity at logarithmical scales | log h | − α for some α >
0. Second, weshow quantum ergodicity at polynomial scales h α for some α >
0, in two special cases: N ∈ S ( N ) of a full density subset S ( N ) of integers and Hecke eigenbasis for all integers. Introduction
One of the main problems in
Quantum Chaos is to study the distribution of eigenstatesin the quantized system for which the classical dynamics is chaotic (i.e. hyperbolic). Inthis series, we consider the classical dynamics given by a hyperbolic linear map of the torus T = R / Z , commonly referred as a (classical) “cat map” due to Arnold [AA]. Such a mapis defined by a matrix M ∈ SL(2 , Z ) with | Tr M | >
2. Its iterations M t , t ∈ Z , induce adiscrete dynamical system that is hyperbolic [KH].The quantized system of a classical cat map, i.e. a quantum cat map, was proposed byHannay-Berry [HB]. In brief, this procedure restricts the Planck constant h to be the inverseof an integer: h = 1 /N for N ∈ N . The quantum cat map ˆ M acts on the N -dim Hilbert space H N of quantum states. There is an eigenbasis (orthonormal basis of eigenstates) { φ j } Nj =1 ofˆ M in H N . See Section 2 for the details of quantization in cat maps. The Quantum Ergodicity(QE) theorem [Sn, Ze1, CdV] in the context of cat maps is proved by [BDB, Ze2]. It assertsthat a full density (see (1.3)) eigenstates equidistribute.In this series, we investigate equidistribution of the eigenstates in quantum cat maps onballs B ( x, r ) ⊂ T at small scales r = r ( N ) → N → ∞ , i.e. small scale quantum ergod-icity . In Part I of the series, we first prove quantum ergodicity in cat maps at logarithmicalscales r = (log N ) − α for some α > N − α for some α > T is the phase space. Quantum states can then be described as distributions on R thatsatisfy perodic conditions in both position and momentum variables. We call such decrip-tion the position representation of the quantum states. Furthermore, due to the nature ofcat maps M ∈ SL(2 , Z ) on T , there is a rich arithmetic structure which can be used tostudy equidistribution of the eigenstates. It is explored by Degli Esposti-Graffi-Isola [DEGI], Mathematics Subject Classification.
Key words and phrases.
Hyperbolic linear maps of the torus, quantum ergodicity, small scale.
Kurlberg-Rudnick [KR1, KR2], Bourgain [Bo], etc. Quantum ergodicity at polynomial scales(Theorems 1.4 and 1.5) in this paper applies these results directly.The quantization precedure described above is rather restrictive to the fact that the phasespace for the quantum cat maps is T . In fact, quantization can be done for a much greaterclass of maps on general manifolds. That is, cat maps M ∈ SL(2 , Z ) on T are the sim-plest examples of symplectic maps on compact symplectic manifolds (as the phase spaces).Equipped with a natural complex structure, the phase space can be regarded as a K¨ahlermanifold. The quantum system can then be induced as the restriction of the classical systemto the holomorphic sections on the K¨ahler manifold. See e.g. Berezin [Bere] for a long historyin this framework of quantization.For a large class of symplectic maps, Zelditch [Ze2] defined their quantization as Toeplitzoperators. In such quantization approach, the quantum states are holomorphic sections onthe K¨ahler manifold. Under the same setting [Ze2], Chang-Zelditch [CZ] recently establishedquantum ergodicity at logarithmical scales for symplectic maps which satisfy appropriateconditions, including the cat maps. The holomorphic representation of quantum states [Ze2,CZ] is related to the position representation via Bargmann transform (see [Zw, Section 13]).We stress that our arguments and results are specifically adapted for cat maps and do notapply to the more general symplectic maps [Ze2, CZ]. Since our discussion of quantumergodicity is restricted to cat maps, we use the position representation of the quantum statesonly.In negatively curved manifolds, the classical dynamics given by the geodesic flow is knownto be hyperbolic [KH]. Equidistribution of the eigenstates at similar logarithmical scales (inthe physical space, see (1.4)) was proved by the author [Han1] and Hezari-Rivi`ere [HR1].However, unlike the case in negatively curved manifolds [Han1, HR1], in Part II of theseries, we show that the logarithmical scales for quantum ergodicity are optimal in cat maps.The optimality of logarithmical scales is related to the phenomenon of short periods of thelinear maps M ∈ SL(2 , Z ). That is, define the period P ( N ) as the smallest positive integersuch that M P ( N ) = Id mod N . Then we have that ˆ M P ( N ) = Id in H N . It is known that P ( N ) ≥ √ N for almost all N ∈ N . (See [KR2].) But there is a sequence { N k } ∞ k =1 ⊂ N suchthat P ( N k ) ∼ N k λ as k → ∞ , (1.1)for which we say ˆ M has short periods in H N k . Restricting to H N k , quantum ergodicity is notvalid beyond logarithmical scales.The phenomenon of short periods in cat maps also accounts for the scarring (i.e. non-equidistribution, see also the discussion below on Quantum Unique Ergodicity) of someeigenstates proved by Faure-Nonnenmacher-De Bi`evre [FNDB], optimal logarithmical rate ofquantum ergodicity proved by Schubert [Sc], and optimality proved by Faure-Nonnenmacher[FN] of the entropy bounds of semiclassical measures [A, AN]. In fact, the proof the opti-mality of logarithmical scales in quantum ergodicity applies the techniques in [FNDB, FN].See Part II of the series for details. We remark that such phenomenon is not shared by someother hyperbolic systems, in particular, the geodesic flows in negatively curved manifolds,for which the logarithmical scales in quantum ergodicity are unlikely to be optimal.In Part I, the second main result (Theorems 1.4) says that restricting to N ∈ N withlong periods (in particular, P ( N ) ≫ √ N e (log N ) δ for some δ >
0, see Theorem 5.2), quantum
MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 3 ergodicity holds at some polynomial scales r = r ( N ) = N − α , α >
0. Because ˆ M has long pe-riods in H N for almost all integers N ([KR2]), quantum ergodicity holds at polynomial scalesfor almost all N ∈ N . Similar argument concludes quantum ergodicity at finer polynomialscales for Hecke eigenbasis (Theorem 1.5), but for all integers.At this point, the background and setup of small scale quantum ergodicity are in order.The most studied classical dynamical system is given by the geodesic flow on a compactRiemannian manifold M . Let T ∗ M = { x = ( q, p ) : q ∈ M , p ∈ T ∗ q M } be the cotangentbundle of M . Then the geodesic flow is defined as the Hamiltonian flow (with Hamiltonian H ( q, p ) = | p | q ) in the phase space as T ∗ M . The eigenstates in the quantized system are theeigenfunctions of the (positive) Laplacian ∆ on M .If the geodesic flow is ergodic (a weaker condition than hyperbolicity), then the QuantumErgodicity theorem [Sn, Ze1, CdV] asserts that a full density eigenstates in any eigenbasis areequidistributed in the normalized phase space (that is, the cosphere bundle S ∗ M = { ( q, p ) ∈ T ∗ M : | p | q = 1 } .) More precisely, we associate a classical observable a ∈ C ∞ ( T ∗ M ) with aquantized operator Op( a ) acting on the L ( M ), the space of quantum states. Then for anyeigenbasis { φ j } ∞ j =1 , ∆ φ j = λ j φ j , there is a full density subset S ( N ) of integers such that for j → ∞ in S ( N ), h Op( a ) φ j , φ j i → Z S ∗ M a dµ for all a ∈ C ∞ ( T ∗ M ) , (1.2)in which µ is the normalized Liouville measure on S ∗ M . Here, S ( N ) has full density in theintegers N if lim N →∞ { j ∈ S ( N ) : 1 ≤ j ≤ N } N = 1 . Similarly, we say a subset S ( N ) of { , ..., N } has full density as N → ∞ iflim N →∞ { j ∈ S ( N ) } N = 1 . (1.3)See [Ze3, Sa] for the recent developments in Quantum Ergodicity.In small scale quantum ergodicity, we study (1.2) when the classical observable a hassupport in balls B ( x, r ) ⊂ T ∗ M with r = r ( λ j ) → j → ∞ . In particular, if a = χ B ( q,r ) as the indicator function of a geodesic ball B ( q, r ) in the physical space M , then h Op( χ B ( q,r ) ) φ j , φ j i = Z B ( q,r ) | φ j | d Vol . Here, d Vol is the Riemannian volume on M . Therefore, (1.2) is reduced to that (c.f. [Han1,Question 1.3]) for some full density S ( N ) in N , Z B ( q,r ( λ j )) | φ j | d Vol = Vol( B ( q, r ( λ j )))Vol( M ) + o (cid:0) r ( λ j ) d (cid:1) as j → ∞ in S ( N ) (1.4)for all q ∈ M . Here, d = dim M . If (1.4) holds, then we say that the eigenfunctions { φ j } for j ∈ S ( N ) are equidistributed at scale r = r ( λ ) in the physical space M . It should bedistinguished with quantum ergodicity at scale r = r ( λ ), i.e. equidistribution in the phasespace S ∗ M . See Definition 1.2 for the precise statement for quantum ergodicity at smallscales in quantum cat maps.In negatively curved manifolds, the geodesic flow is hyperbolic [KH]. Berry’s randomwave conjectures [Berr] suggest that the eigenfunctions of eigenvalue λ behave like randomwaves with frequency λ . Recent results about equidistribution at various polynomial scales XIAOLONG HAN of random waves on manifolds were proved in [Han2, HT, CI]. In comparison, we see thatthe logarithmical scales in [Han1, HR1] are at much weaker scales. i In the special case ofmodular surfaces and restricted to Hecke eigenbasis, (1.4) at smaller scales r = λ − α for some α > d -dim torus T d is integrable (sois not ergodic). ii But for any eigenbasis in T d , there is a full density subsequence that isequidistributed at r = λ − / ( d − o (1) by [HR2, LR, GW]. In T , the scale approaches thePlanck scale 1 /λ and in T d , d ≥
3, Bourgain [LR] showed that the scale is optimal.We now consider the classical cat map given by a matrix M ∈ SL(2 , Z ) : T → T with | Tr M | >
2. In this case, the torus T = { ( q, p ) : q, p ∈ T } is the phase space, in which q and p denote the position and momentum variables, respectively. We regard T as the physicalspace of the position variable q . Denote ˆ M the quantization of M . So ˆ M : H N → H N for N ∈ N . (See Section 2 for more background of cat maps.) Let B d ( x, r ) be a geodesicball in T d with radius r and center x . The first main theorem asserts equidistribution atlogarithmical scales in the physical space. Theorem 1.1 (Equidistribution at logarithmical scales) . For ≤ α < / and all N ∈ N ,any eigenbasis { φ j } Nj =1 of a quantum cat map ˆ M in H N contains a full density subset thatequidistributes at scale r = (log N ) − α in the physical space. That is, there is a full densitysubset S ( N ) of { , ..., N } such that for j ∈ S ( N ) , Z B ( q,r ) | φ j | d Vol = Vol( B ( q, r )) + o ( r ) as N → ∞ , (1.5) uniformly for all q ∈ T . To define small scale quantum ergodicity, i.e. equidistribution of eigenstates at small scalesin the phase space T , in (1.2), we can no longer choose the indicator function (not smooth) a = χ B ( x,r ) for x ∈ T and r = r ( N ) → N → ∞ . To remedy this, we use trigonometricpolynomials b ± x,r to approximate χ B ( x,r ) uniformly for all x ∈ T , i.e. b − x,r ≤ χ B ( x,r ) ≤ b + x,r and Z T b ± x,r d Vol = Vol( B ( x, r )) + o (cid:0) r (cid:1) uniformly for all x ∈ T . Such choices of approximation by trigonometric polynomials appear naturally in quantumcat maps. See Lemma 3.1 and (3.2) for their precise properties. With this understanding,we define small scale quantum ergodicity in quantum cat maps.
Definition 1.2 (Small scale quantum ergodicity in quantum cat maps) . Let ˆ M be a quantumcat map and G ⊂ N . We say quantum ergodicity at scale r = r ( N ) holds for N ∈ G if forany eigenbasis { φ j } Nj =1 of ˆ M in H N , there is a full density subset S ( N ) of { , ..., N } suchthat for j ∈ S ( N ) , (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) = Vol( B ( x, r )) + o ( r ) as N → ∞ in G, (1.6) uniformly for all x ∈ T . i In fact, it was only shown in [Han1, HR1] that the two sides of (1.4) are comparable uniformly for q ∈ M .This is a weaker statement than the uniform equidistribution at small scales in (1.4). ii We remark that the cat map is not the time 1 map of some Hamiltonian flow on the torus, thereforediffers with the geodesic flow on the torus [HR2, LR, GW] signicantly.
MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 5
Then we have that
Theorem 1.3 (Quantum ergodicity at logarithmical scales) . For ≤ α < / and all N ∈ N ,quantum ergodicity at scale r = (log N ) − α holds. The second main result in this paper treats two cases for which the logarithmical scalesin Theorems 1.1 and 1.3 can be significantly improved to polynomials scales. In these twocases, we apply the results of Kurlberg-Rudnick [KR1, KR2], where the problem of QuantumUnique Ergodicity (QUE) in cat maps is studied.If QUE holds, then equidistribution in (1.2) is valid for the whole sequence of any eigenba-sis. While the QUE conjecture in negatively curved manifolds proposed by Rudnick-Sarnak[RS] is still open, some (positive and negative) results are known in different dynamicalsystems. Hassell [Has] showed that in generic Bunimovich stadia, QUE does not hold. Inarithmatic hyperbolic surfaces and restricting to Hecke eigenbasis, QUE has been verified[L, SV, HS, BL]. The Hecke eigenbasis is the joint eigenbasis of a family of commutativegroup of operators including the Laplacian. (Note that Brooks-Lindenstauss [BL] provedQUE for the joint eigenbasis of the Laplacian and one
Hecke operator.)In the context of cat maps, if QUE holds for a subset G ⊂ N , then h Op N ( a ) φ j , φ j i → Z T a dµ as N → ∞ in G, for any eigenbasis { φ j } ∞ j =1 in H N . Faure-Nonnenmacher-De Bi`evre [FNDB] proved that QUEdoes not hold for G = N . That is, along the sequence { N k } ∞ k =1 with short periods (1.1), thereare eigenstates in H N k that fail to be equidistributed, which are called “scarred” eigenstates.On the positive side of QUE in cat maps, Kurlberg-Rudnick [KR2] proved that there is afull density subset G ⊂ N such that QUE holds. (Earlier QUE result for a sparse subset of N was proved in [DEGI], assuming the Generalized Riemann Hypothesis.) The cat map in H N for N ∈ G has sufficiently long periods and our quantum ergodicity at polynomial scalesis also in this case: Theorem 1.4 (Quantum ergodicity at polynomial scales) . There is a full density subset S ( N ) of integers such that for N ∈ S ( N ) , • any eigenbasis { φ j } Nj =1 of the quantum cat map ˆ M in H N contains a full density subsetthat equidistributes at scale r = N − α , ≤ α < / , in the physical space, • quantum ergodicity at scale r = N − α , ≤ α < / , holds. With suitable additional assumptions of the cat map M , Kurlberg-Rudnick [KR1] intro-duced the Hecke theory associated with the cat map M . It is the analogue of the Hecke theoryin arithmetic hyperbolic surfaces. That is, in H N for each integer N ∈ N , they define a familyof commutative group of unitary operators including the quantum cat map ˆ M . Then thereis a joint eigenbasis for all these operators, similarly called the Hecke eigenbasis. Kurlberg-Rudnick [KR1] then proved QUE for the Hecke eigenbasis for all integers. Using [KR1], ourquantum ergodicity follows at a better polynomial scales than the ones in Theorem 1.4. Theorem 1.5 (Quantum ergodicity at polynomial scales for Hecke eigenbasis) . For all N ∈ N and Hecke eigenbasis { φ j } Nj =1 in H N , • { φ j } Nj =1 contains a full density subset that equidistributes at scale r = N − α , ≤ α < / , in the physical space, • quantum ergodicity for { φ j } Nj =1 at scale r = N − α , ≤ α < / , holds. XIAOLONG HAN
Remark . We shall remark the crucial difference of the Hecke theory in cat maps and theone in arithmetic hyperbolic surfaces. In arithmetic hyperbolic surfaces, all eigenbases are conjectured to be Hecke eigenbasis [Sa], which means that QUE for Hecke eigenbasis shouldimply QUE.However, in cat maps, not all eigenbases are Hecke eigenbasis. The variety of eigenbasesdisplay very different distribution properties. That is, the Hecke eigenbasis satisfies QUE[KR1] but some other eigenbasis fails QUE [FNDB]. Similarly in the small scale quantumergodicity, the Hecke eigenbasis satisfies quantum ergodicity at polynomial scales (Theorem1.5), but some other eigenbasis can only equidistribute up to the logarithmical scale (see PartII).We shall also remark that the polynomial scales in Theorems 1.4 and 1.5 are unlikelyoptimal.
Organization of the paper.
We organize this paper as follows. In Sections 2, we reviewclassical and quantum cat maps. In Section 3, we gather some results that are used to proveequidistribution in the physical space and quantum ergodicity at small scales. In Section4, we prove equidistribution in the physical space and quantum ergodicity at logarithmicalscales, i.e. Theorems 1.1 and 1.3. In Section 5, we prove equidistribution in the physicalspace and quantum ergodicity at polynomial scales, i.e. Theorems 1.4 and 1.5.2.
Classical dynamics and quantum dynamics in cat maps
In this section, we review the background on classical and quantum cat maps. See [HB,BDB, KR1, FNDB] for more details. Here, we mainly follow [BDB, Section 6].2.1.
Classical cat maps.
Consider the quadratic Hamiltonian on the plane R H ( q, p ) = 12 αq + 12 βp + γqp. (2.1)It generates the Hamiltonian flow M ( t ) : x (0) = ( q (0) , p (0)) → x ( t ) = ( q ( t ) , p ( t ))such that dq ( t ) dt = ∂Hdp = βp + γq and dp ( t ) dt = − ∂Hdq = − αq − γp. So explicitly M ( t ) = exp (cid:26) t (cid:18) γ β − α − γ (cid:19)(cid:27) . If γ > αβ , then the flow M ( t ) is hyperbolic with Lyapunov exponent λ = p γ − αβ . Denote M := M (1) = exp (cid:26)(cid:18) γ β − α − γ (cid:19)(cid:27) = (cid:18) A BC D (cid:19) . Then M ∈ SL(2 , R ) : R → R is a hyperbolic map with eigenvalues ± e λ . Notice thatthroughout the paper, we use M to denote both the hyperbolic map and the matrix thatdefines it. Remark . We remark that M ∈ SL(2 , R ) preserves the Liouville measure dµ = dqdp on R .Moreover, define the symplectic product on R u ∧ v = u v − u v for u = ( u , u ) , v = ( v , v ) ∈ R . (2.2) MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 7
Then uM ∧ vM = u ∧ v. (2.3)That is, M preserves the symplectic product. Definition (Classical cat maps) . Let M ∈ SL(2 , R ) : R → R be a hyperbolic map.Suppose further that M ∈ SL(2 , Z ), i.e. A, B, C, D ∈ Z . Since( x + n ) M = xM + nM = xM mod 1 for x ∈ R and n ∈ Z ,M induces a map on T that is hyperbolic, by which we refer as a classical cat map. Example (Arnold cat map) . The Arnold cat map is defined by M = (cid:18) (cid:19) . The Lyapunov exponent λ = 3 + √ . As mentioned in the introduction, our main theorems of quantum ergodicity in quantumcat maps are closely related to the periods of classical cat maps.
Definition (Periods of cat maps) . Let M ∈ SL(2 , Z ) be a classical cat map. Define P ( N ) asthe period (or order) of M module N , that is, the smallest positive integer k ≥ M k = Id mod N .The following proposition provides estimates of periods for cat maps [K, KR1, KR2]. Proposition 2.1 (Estimates of the periods of cat maps) . Let M be a classical cat map.(i). There is C > depending only on M such that λ log N − C ≤ P ( N ) ≤ N for all N ∈ N . (ii). There is a full density subset S ( N ) of integers such that P ( N ) ≥ √ N for all N ∈ S ( N ) . (iii). There is a sequence of integers { N k } ∞ k =1 such that P ( N k ) = 2 log N k λ + O (1) as k → ∞ . Quantum cat maps.
We first recall the quantization on the real line R , in which casethe phase space is R . For a detailed discussion, see [Zw, Chapter 4].Let h be the Planck constant and we are interested in the semiclassical limit that h → a → Op h ( a ), we assign a quantum observableOp h ( a ) on L ( R ) to a classical observable a ∈ C ∞ ( R ). Then a is called a symbol of Op h ( a ).Write x = ( q, p ) ∈ R , in which q and p denote position and momentum variables respec-tively. Define the position and momentum self-adjoint operators ˆ q = Op h ( q ) and ˆ p = Op h ( p ):ˆ qψ ( q ) := qψ ( q ) and ˆ pψ := h πi dψ ( q ) dq for ψ ∈ C ∞ ( R ) . So we have that [ˆ q, ˆ p ] := ˆ q ˆ p − ˆ p ˆ q = ih π Id . Here, Id is the identity map that Id ψ = ψ . XIAOLONG HAN
The Weyl quantization of the Hamiltonian in (2.1) isˆ H = Op h ( H ) = 12 α ˆ q + 12 β ˆ p + γ q ˆ p + ˆ p ˆ q ) . It generates the Schr¨odinger flow such that for a quantum state ψ (0) ∈ L ( R ), ψ (0) → ψ ( t ) = e − πit ˆ H/h ψ (0) . So ψ ( t ) solves the Schr¨odinger equation ih π ∂ψ ( t ) ∂t = ˆ Hψ ( t ) . The quantization of the hyperbolic map M on R is the Schr¨odinger flow at t = 1:ˆ M = e − πi ˆ H/h . (2.4)Consider v = ( v , v ) ∈ R . Define the phase space translation operatorˆ T v := exp (cid:18) − πih ( v ˆ p − v ˆ q ) (cid:19) . It readily follows that ˆ T ⋆v = ˆ T − v . Moreover,ˆ M ˆ T v ˆ M − = ˆ T vM . (2.5)Notice also that for u = ( u , u ) and v = ( v , v ),ˆ T u ˆ T v = e πi ( u ∧ v )2 h ˆ T u + v , (2.6)in which u ∧ v is the symplectic product of u and v defined in (2.2).The function ψ on R that defines a quantum state on T should be periodic in positionand in momentum. That is, ψ is invariant under the phase translations ˆ T n for n ∈ Z . Inparticular, ˆ T (1 , ψ = e πiκ ψ and ˆ T (0 , ψ = e πiκ ψ. Here, we allow the phase shifts e πiκ and e πiκ for some κ = ( κ , κ ) ∈ T , because undersuch phase shifts the function defines the same quantum state. It then follows from suchperiodicity that ˆ T (1 , ˆ T (0 , = ˆ T (0 , ˆ T (1 , restricted to the quantum states on T . But in the view of (2.6), since (1 , ∧ (0 ,
1) = − e πi/h = 1. Hence, N := 1 h ∈ N . We always assume this condition throughout the paper. Under such condition, the space ofquantum states H N,κ on T is an N -dim space that consists of distributions of the form ψ ( q ) = X k ∈ Z Ψ( k ) δ ( q − ( k + κ ) /N ) , in which Ψ( k + N ) = e − πκ Ψ( k ) . (2.7)So H N,κ is a Hilbert space equipped with the inner product h ψ, φ i = 1 N N X k =1 Ψ( k )Φ( k ) . MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 9
Definition (Quantum cat maps) . Let M be a classical cat map and ˆ M be defined in (2.4).Then for any N ∈ N , there exists i κ ∈ T such that ˆ M : H N,κ → H
N,κ . We fix suchchoice of κ (that depends on M and N ) and simply denote the Hilbert space of quantumstates as H N . There is then an eigenbasis { φ j } Nj =1 ⊂ H N such that ˆ M φ j = e iθ j φ j for0 ≤ θ ≤ θ ≤ · · · ≤ θ N < π . Remark (Hecke eigenbases) . Assume in addition that M = Id mod 4. Then one can intro-duce the Hecke theory associated with ˆ M . That is, there are a group of operators, called theHecke operators, which commute with ˆ M acting on H N . There is therefore a joint eigenbasisin H N , i.e. Hecke eigenbasis, of all the Hecke operators and ˆ M . The Hecke theory in cat mapswas introduced by Kurlberg-Rudnick and we refer to [KR1] for the precise construction.Any phase translation ˆ T v acts on H N only if ˆ T v commutes with ˆ T n for all n ∈ Z . Applying(2.6) again, e πi ( v ∧ n ) /h = 1 for all n ∈ Z . So v ∈ Z /N . For notational convenience, we writeˆ T N ( n ) := ˆ T n/N . Let a ∈ C ∞ ( T ) be a classical observable. Define its Weyl quantization as an operator on H N : Op N ( a ) = X n ∈ Z ˜ a ( n ) ˆ T N ( n ) . (2.8)Here, ˜ a ( n ) is the Fourier coefficients of a that a ( x ) = X n ∈ Z ˜ a ( n ) e πi ( n ∧ x ) . In the quantum cat system that M is linear, we have the following exact Egorov’s theorem.The proof is straightforward from its linear nature and we provide it here. Theorem 2.2 (Egorov’s theorem) . Let a ∈ C ∞ ( T ) . Then ˆ M − t ◦ Op N ( a ) ◦ ˆ M t = Op N ( a ◦ M t ) for all t ∈ Z . Proof.
It suffices to show the case when Op N ( a ) = ˆ T N ( n ) and t = 1. Observe thatˆ T N ( n ) = Op N (cid:0) e πi ( n ∧ x ) (cid:1) . By (2.5), we have that ˆ M − ˆ T N ( n ) ˆ M = ˆ T nM − /N = Op N (cid:16) e πi ( nM − /N ∧ x ) (cid:17) = Op N (cid:0) e πi ( n/N ∧ xM ) (cid:1) = ˆ T N ( n ) ◦ ˆ M .
Here, we used the fact that the symplectic map M preserves the symplectic product so( nM − /N ) ∧ x = ( n/N ) ∧ ( xM ). (cid:3) Notice that from (2.7) (see [KR1, Lemma 4] for a short proof)Tr (cid:16) ˆ T N ( n ) (cid:17) = ( N if n = 0 mod N, . (2.9) i See the detailed discussion in [BDB, Section 6].
We then derive the following trace formula in H N . Theorem 2.3 (Trace formula) . Let a ∈ C ∞ ( T ) . Then Tr (Op N ( a )) = X j ∈ Z ˜ a ( N j ) . Equidistribution and quantum ergodicity at small scales
Recall that N = 1 /h ∈ N as h →
0. Distribution of the eigenstates of { φ j } Nj =1 is studiedthrough h Op N ( a ) φ j , φ j i for appropriate classical observables a . • For equidistribution at small scale r = r ( N ) in the physical space T , we choose a = χ B ( q,r ) for B ( q, r ) ⊂ T so h Op N ( χ B ( q,r ) ) φ j , φ j i = Z B ( q,r ) | φ j | d Vol . • For equidistribution at small scale r = r ( N ) in the phase space T , i.e. small scalequantum ergodicity, we can only choose smooth functions a ≈ χ B ( x,r ) for B ( x, r ) ⊂ T .In both cases, notice that the quantization of a in cat maps (2.8) is via Fourier series of a . It is natural to approximate indicator functions of balls in T d , d = 1 ,
2, by trigonometricpolynomials.Roughly speaking, to approximate χ B ( q,r ) or χ B ( x,r ) , we need trigonometric polynomials ofdegree D = D ( r ) such that 1 /D = o ( r ). These trigonometric polynomials are the appropriateversions of Beurling-Selberg polynomials, which are well studied [Har, Ho, HV]. Here we recall[LR, Lemma 2.5] that is explicit for our purpose. Lemma 3.1.
Let B d (0 , r ) ⊂ T d and D = D ( r ) such that rD ≥ . There exist trigonometricpolynomials a ± r such that(i). a − r ( y ) ≤ χ B d (0 ,r ) ( y ) ≤ a + r ( y ) for all y ∈ T d ,(ii). f a ± r ( n ) = 0 if | n | ≥ D ,(iii). f a ± r (0) = Vol( B d (0 , r )) + O ( r d − /D ) ,(iv). (cid:12)(cid:12)(cid:12)f a ± r ( n ) (cid:12)(cid:12)(cid:12) ≤ cr d for all n ∈ Z , in which c depends only on d .In particular, if D = D ( r ) such that /D = o ( r ) , then (iii) becomes f a ± r (0) = Vol( B d (0 , r )) + o (cid:0) r d (cid:1) . Here, a + r and a − r above are called a majorant and a minorant of the indicator function χ B d (0 ,r ) . For any x ∈ T d , the trigonometric polynomials b ± x,r ( · ) := a ± ( · − x ) (3.1)majorize and minorize χ B d ( x,r ) . Since f b ± x,r ( n ) = e − πi ( n ∧ x ) f a ± ( n ), b ± x,r satisfy the estimates inLemma 3.1 independent of x ∈ T d . In particular, with D = D ( r ) such that 1 /D = o ( r ), f b ± x,r (0) = Vol( B d ( x, r )) + o (cid:0) r d (cid:1) and (cid:12)(cid:12)(cid:12) f b ± x,r ( n ) (cid:12)(cid:12)(cid:12) ≤ cr d . (3.2)Define the p -moment MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 11
Definition ( p -moment) . Let 1 ≤ p < ∞ and a ∈ C ∞ ( T ). Define the p -moment V p ( N, Op N ( a )) := 1 N N X j =1 |h Op N ( a ) φ j , φ j i − µ ( a ) | p . For notational simplicity, we also write V p ( N, a ) = V p ( N, Op N ( a )).Inspired by [LR], we prove the following crucial lemma. From this lemma, equidistributionat small scales in the phase space (Theorems 1.3, 1.4, and 1.5) is derived. Lemma 3.2.
Let
L > and b ± x,r be defined in (3.1) for d = 2 . Define S ± ( N, L ) := ( ≤ j ≤ N : sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) µ (cid:0) b ± x,r (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ L ) . Denote p ′ = p/ ( p − . Assume that D = D ( r ) such that /D = o ( r ) as r → . Then S ± ( N, L ) N ≤ cD pp ′ L p X ≤| n |≤ D V p (cid:16) N, ˆ T N ( n ) (cid:17) , in which c depends on p . Remark . In particular, when p = 1, the above inequality reads S ± ( N, L ) N ≤ cL X ≤| n |≤ D V (cid:16) N, ˆ T N ( n ) (cid:17) . It can be viewed as a variation in quantum cat maps of [LR, § Proof.
By the uniform control of the Fourier coefficients of b ± x,r in (3.2),sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) µ (cid:0) b ± x,r (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P n ∈ Z f b ± x,r ( n ) D ˆ T N ( n ) φ j , φ j Ef b ± x,r (0) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ≤| n |≤ D f b ± x,r ( n ) D ˆ T N ( n ) φ j , φ j Ef b ± x,r (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ sup x ∈ T X ≤| n |≤ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f b ± x,r ( n ) f b ± x,r (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ pp ′ X ≤| n |≤ D (cid:12)(cid:12)(cid:12)D ˆ T N ( n ) φ j , φ j E(cid:12)(cid:12)(cid:12) p ≤ cD pp ′ X ≤| n |≤ D (cid:12)(cid:12)(cid:12)D ˆ T N ( n ) φ j , φ j E(cid:12)(cid:12)(cid:12) p . Here, we used H¨older’s inequality with exponents p and p ′ . Hence, using Chebyshev’s in-equality, S ± ( N, L ) N ≤ N L p N X j =1 sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) µ (cid:0) b ± x,r (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ cD pp ′ N L p N X j =1 X ≤| n |≤ D (cid:12)(cid:12)(cid:12)D ˆ T N ( n ) φ j , φ j E(cid:12)(cid:12)(cid:12) p ≤ cD pp ′ L p X ≤| n |≤ D V p (cid:16) N, ˆ T N ( n ) (cid:17) . (cid:3) Notice that when a ∈ C ∞ ( T ), i.e. a depends only on the position variable q , a ( q ) = X m ∈ Z ˜ a ( m ) e πimq . In this case, the quantization of a on T isOp N ( a ) = X m ∈ Z ˜ a ( m ) ˆ T N ( m, . We prove the following lemma, from which equidistribution at small scales in the physicalspace (Theorems 1.1, 1.4, and 1.5) is derived. The proof is similar as in Lemma 3.2 so weomit it here.
Lemma 3.3.
Let
L > and b ± x,r be defined in (3.1) for d = 1 . Then S ± ( N, L ) N ≤ cD pp ′ L p X ≤| m |≤ D V p (cid:16) N, ˆ T N ( m, (cid:17) , in which c depends on p . Logarithmical scales
To prove equidistribution at small scales using Lemma 3.2, we need to estimate the p -moments of basic Fourier modes ˆ T N ( n ). Denote the Ehrenfest time T E := log Nλ .
The following proposition provides the estimate of 2-moments of ˆ T N ( n ) by T E . Proposition 4.1.
Let { φ j } Nj =1 be an eigenbasis of a quantum cat map ˆ M in H N . Supposethat ≤ | n | < N and < δ < − log | n | log N .
Then V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ δT E . Proof.
Since ˆ
M φ j = e iθ j , we have that D ˆ T N ( n ) φ j , φ j E = D ˆ T N ( n ) e − itθ j φ j , e − itθ j φ j E = D ˆ T N ( n ) ˆ M − t φ j , ˆ M − t φ j E = D ˆ M t ◦ ˆ T N ( n ) ◦ ˆ M − t φ j , φ j E MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 13 = D ˆ T N (cid:0) nM t (cid:1) φ j , φ j E , by Egorov’s theorem in Theorem 2.2. Then compute that V (cid:16) N, ˆ T N ( n ) (cid:17) = 1 N N X j =1 (cid:12)(cid:12)(cid:12)D ˆ T N ( n ) φ j , φ j E(cid:12)(cid:12)(cid:12) = 1 N N X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* T T − X t =0 ˆ M t ◦ ˆ T N ( n ) ◦ ˆ M − t φ j , φ j +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 N N X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* T T − X t =0 ˆ T N (cid:0) nM t (cid:1) φ j , φ j +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N T N X j =1 * T − X t =0 ˆ T N (cid:0) nM t (cid:1)! ⋆ T − X s =0 ˆ T N ( nM s ) ! φ j , φ j + by Cauchy-Schwarz inequality= 1 N T T − X t,s =0 N X j =1 D ˆ T N (cid:0) nM t (cid:1) ⋆ ˆ T N ( nM s ) φ j , φ j E = 1 N T T − X t,s =0 Tr (cid:16) ˆ T N (cid:0) nM t (cid:1) ⋆ ˆ T N ( nM s ) (cid:17) . (4.1)For 1 ≤ | n | < N , since 0 < δ < − log | n | log N , there is δ such that 0 < δ < δ < − log | n | log N .
We then have that | n | < N − δ . Set T = δT E = δ log Nλ .
Now if 0 ≤ t, s ≤ T −
1, then | nM t − nM s | ≤ | nM t | + | nM s | ≤ | e λT n | = 2 N δ | n | < N δ N − δ = 2 N δ − δ < N. It implies that for 0 ≤ t, s ≤ T − nM t = nM s mod N only if nM t = nM s , i.e. t = s. (4.2)Notice that from (2.9), Tr ( T N ( k ) ⋆ T N ( j )) = ( N if j = k mod N, . Therefore, in the view of (4.2), we have thatTr (cid:16) ˆ T N (cid:0) nM t (cid:1) ⋆ ˆ T N ( nM s ) (cid:17) = ( N if t = s, . Hence, (4.1) continues as V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ N T T − X t,s =1 Tr (cid:16) ˆ T N (cid:0) nM t (cid:1) ⋆ ˆ T N ( nM s ) (cid:17) = 1 T = 1 δT E . (cid:3) Now we prove equidistribution at logarithmical scales in the physical space.
Proof of Theorem 1.1.
For 0 ≤ α < /
2, let D = (log N ) β with some β ∈ ( α, / /D = o ( r ) as r → r = (log N ) − α . By Proposition 4.1, we can choose12 < δ < − log D log N such that V (cid:16) N, ˆ T N ( m, (cid:17) ≤ δT E for all 1 ≤ | m | ≤ D. Applying Lemma 3.3 for d = 1 and p = 2, we immediately have that S ± ( N, L ) N ≤ cD pp ′ L X ≤| m |≤ D V (cid:16) N, ˆ T N ( m, (cid:17) ≤ CDL X ≤| m |≤ D δT E ≤ CD L log N .
Since 0 < β < /
2, let γ = 1 − β > L = 1(log N ) γ . Then CD L log N = C (log N ) − β − γ = C (log N ) γ → N → ∞ . Denote S ( N ) = { , ..., N } \ {S + ( N, L ) ∪ S − ( N, L ) } . It is evident that S ( N ) has full density in { , ..., N } as N → ∞ .If j
6∈ S + ( N, L ), then sup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b + q,r ) φ j , φ j (cid:11) µ (cid:0) b + q,r (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L = 1(log N ) γ . This means that (cid:10) Op N ( b + q,r ) φ j , φ j (cid:11) ≤ (1 + L ) µ (cid:0) b + q,r (cid:1) if j
6∈ S ( N, L ) , MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 15 uniformly for all q ∈ T . In the view of (3.2), Z B ( q,r ) | φ j | d Vol ≤ Vol( B ( q, r )) + o ( r ) if j
6∈ S + ( N, L ) . A similar analysis implies that the above inequality holds for j
6∈ S − ( N, L ) with inequalityreversed. Hence, Z B ( q,r ) | φ j | d Vol = Vol( B ( q, r )) + o ( r ) if j ∈ S ( N ) , uniformly for all q ∈ T . (cid:3) We then prove equidistribution at logarithmical scales in the phase space, i.e. small scalequantum ergodicity.
Proof of Theorem 1.3.
For 0 ≤ α < /
4, let D = (log N ) β with some β ∈ ( α, / /D = o ( r ) as r → r = (log N ) − α . By Proposition 4.1, we can choose12 < δ < − log D log N such that V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ δT E for all 1 ≤ | n | ≤ D. Applying Lemma 3.2 for d = 2 and p = 2, we immediately have that S ± ( N, L ) N ≤ cD pp ′ L X ≤| n |≤ D V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ CD L X ≤| n |≤ D δT E ≤ CD L log N .
Since 0 < β < /
4, let γ = 1 − β > L = 1(log N ) γ . Then CD L log N = C (log N ) − β − γ = C (log N ) γ → N → ∞ . Denote S ( N ) = { , ..., N } \ {S + ( N, L ) ∪ S − ( N, L ) } . It is evident that S ( N ) has full density in { , ..., N } as N → ∞ . If j ∈ S ( N ), we deduce thatsup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) µ (cid:0) b + x,r (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L = 1(log N ) γ . In the view of (3.2), this means thatlim N →∞ (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) = Vol( B ( x, r )) + o ( r ) if j ∈ S ( N ) , uniformly for all x ∈ T . (cid:3) Polynomial scales
Polynomial scales for full density integers.
To prove Theorem 1.4, we need theresults in Kurlberg-Rudnick [KR2, Proposition 8 and Theorem 17] which provide the controlof the 4-moment of ˆ T N ( n ). These results are used to prove QUE in [KR2, Theorems 1 and2] for a full density subset of integers; they were improved by Bourgain [Bo, Theorem 3] toinclude a larger set of integers (still full density). But the improvement does not providesmaller scale in quantum ergodicity so we use [KR2] here. Theorem 5.1.
Let { φ j } Nj =1 be an eigenbasis of a quantum cat map ˆ M in H N . There are δ > and a full density subset S ( N ) of integers such that for all ε > and N ∈ S ( N ) wehave that V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ C | n | ε N e (log N ) δ for | n | > , in which C depends only on M and δ . In particular, if N ∈ S ( N ) here, then P ( N ) ≫ √ N e (log N ) δ . W now prove equidistribution in the physical space and quantum ergodicity at small scalesin Theorem 1.4.
Proof of Theorem 1.4.
We first use Lemma 3.2 for d = 2 and p = 4 to prove quantumergodicity at small scales. S ± ( N, L ) N ≤ cD pp ′ L X ≤| n |≤ D V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ CD L X ≤| n |≤ D | n | ε N e (log N ) δ ≤ CD L N e (log N ) δ X ≤| n |≤ D | n | ε ≤ CD ε L N e (log N ) δ . For 0 ≤ α < /
16, let D = N β for some β ∈ ( α, / /D = o ( r ) since r = N − α .Choose ε > − (16 + ε ) β >
0. Let γ = 1 − (16 + ε ) β > L = 1 N γ . Compute that CD ε L N e (log N ) δ = CN (16+ ε ) β N − γ N e (log N ) δ = CN − (16+ ε ) β − γ e (log N ) δ = Ce (log N ) δ → N → ∞ . It follows that S ± ( N, L ) both have zero density in { , ..., N } as N → ∞ . Denote S ( N ) = { , ..., N } \ {S + ( N, L ) ∪ S − ( N, L ) } . It is evident that S ( N ) has full density in { , ..., N } as N → ∞ . If j ∈ S ( N ), we deduce thatsup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11)R T b ± x,r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L = 1 N γ . MALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I 17
In the view of (3.2), this means thatlim N →∞ (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11) = Vol( B ( x, r )) + o ( r ) if j ∈ S ( N ) , uniformly for all x ∈ T .We then use Lemma 3.3 for d = 1 and p = 4 to prove equidistribution at small scales inthe physical space. S ± ( N, L ) N ≤ cD pp ′ L X ≤| m |≤ D V (cid:16) N, ˆ T N ( m, (cid:17) ≤ CD L X ≤| m |≤ D | m | ε N e (log N ) δ ≤ CD L N e (log N ) δ X ≤| m |≤ D | m | ε ≤ CD ε L N e (log N ) δ . For 0 ≤ α < /
12, let D = N β for some β ∈ ( α, / /D = o ( r ) since r = N − α .Choose ε > − (12 + ε ) β >
0. Let γ = 1 − (12 + ε ) β > L = 1 N γ . Compute that CD ε L N e (log N ) δ = CN (12+ ε ) α N − γ N e (log N ) δ = CN − (12+ ε ) β − γ e (log N ) δ = Ce (log N ) δ → N → ∞ . It follows that S ± ( N, L ) both have zero density in { , ..., N } as N → ∞ . Denote S ( N ) = { , ..., N } \ {S + ( N, L ) ∪ S − ( N, L ) } . It is evident that S ( N ) has full density in { , ..., N } as N → ∞ . If j ∈ S ( N ), we deduce thatsup x ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Op N ( b ± x,r ) φ j , φ j (cid:11)R T b ± x,r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L = 1 N γ . In the view of (3.2), similarly as in the proof of Theorem 1.1, this means that Z B ( q,r ) | φ j | d Vol = Vol( B ( q, r )) + o ( r ) if j ∈ S ( N ) , uniformly for all q ∈ T . (cid:3) Polynomial scales for Hecke eigenbasis.
To prove Theorem 1.5, we need an estimateof the 4-moments for Hecke eigenbasis [KR1, Theorem 10].
Theorem 5.2.
Let { φ j } Nj =1 be a Hecke eigenbasis of a quantum cat map ˆ M in H N . Thenfor all ε > we have that V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ C | n | N − ε for | n | > , in which C depends only on M and ε . Proof of Theorem 1.5.
We first use Lemma 3.2 for d = 2 and p = 4 to prove quantumergodicity at small scales. S ± ( N, L ) N ≤ cD pp ′ L X ≤| n |≤ D V (cid:16) N, ˆ T N ( n ) (cid:17) ≤ CD L X ≤| n |≤ D | n | N − ε ≤ CD L N − ε X ≤| n |≤ D | n | ≤ CD L N − ε . Similar argument as in the proof of Theorem 1.4 shows quantum ergodicity at scales N − α ,0 ≤ α < /
12, for Hecke eigenbasis.We then use Lemma 3.3 for d = 1 and p = 4 to prove quantum ergodicity at small scales. S ± ( N, L ) N ≤ cD pp ′ L X ≤| m |≤ D V (cid:16) N, ˆ T N ( m, (cid:17) ≤ CD L X ≤| m |≤ D | m | N − ε ≤ CD L N − ε X ≤| m |≤ D | m | ≤ CD L N − ε . Similar argument as in the proof of Theorem 1.4 shows equidistribution in the physical spaceat scales N − α , 0 ≤ α < /
10, for Hecke eigenbasis. (cid:3)
Acknowledgements
I benefited from discussions with Fr´ed´eric Faure, Andrew Hassell, Ze´ev Rudnick, StephaneNonnenmacher, and Steve Zelditch. This work began when I participated the Special YearProgram in Analysis at Australian National University in 2018. I thank the institute for thehospitality.
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