Variations on Quantum Ergodic Theorems

Transcription

1 Variations on Quantum Ergodic Theorems Michael Taylor Abstract We derive some quantum ergodic theorems, related to microlocal behavior of eigenfunctions of a positive, self-adjoint, elliptic pseudodifferential operator Λ on a compact Riemannian manifold M, emphasizing results that hold without the hypothesis that the Hamiltonian flow generated by the symbol of Λ be ergodic. Cases treated include both integrable Hamiltonians and some associated with soft chaos. 1 Introduction Let M be a compact Riemannian manifold and Λ OP S 1 (M) a first order, scalar, elliptic, positive self adjoint operator. Say Spec Λ = {λ k : k N}, (1.1) counted with multiplicity, with λ k. Let {ϕ k : k N} be an orthonormal basis of L 2 (M) consisting of eigenfunctions, Λϕ k = λ k ϕ k, or more generally we can let {ϕ k } be an orthonormal basis of L 2 (M) consisting of quasimodes, satisfying sup e s(λ λk) (Λ λ k )ϕ k L 2 = ε k 0, (1.2) 0 s 1 as k, though in examples we generally stick to the setting of actual eigenfunctions. Let S M T M denote the set where the principal symbol σ(λ) is equal to 1. The symplectic form on T M induces a volume form ds on 2010 Math Subject Classification: 35P20, 35S05, 81Q50. Key words. Quantum ergodic theorem, spectrum, eigenfunction, chaos. Work supported by NSF grant DMS

2 , which we normalize to have unit volume. The Hamiltonian vector field associated to σ(λ) generates a smooth flow G t on, preserving the volume form ds. Let P be the orthogonal projection of L 2 (, ds) onto We aim to prove the following. V = {b L 2 (, ds) : b G t b}. (1.3) Theorem 1.1 There is a set N N, of density zero, with the following property. Let A OP S 0 (M) and assume a = σ(a) satisfies P a = a := a ds. (1.4) Then lim (Aϕ k, ϕ k ) = a. (1.5) k / N,k If the flow {G t } is ergodic on, then V consists of constants, and (1.4) holds for all a. The ergodic case has been studied for some time, in [14], [5], [8], [17], [6], and other works. Theorem 1.1 applies to cases where such ergodicity does not hold. Such a formulation was mentioned in [13]. Here we intend to explore this formulation and its implications more thoroughly. More generally than taking A OP S 0 (M), which leads to a C (), we can take a C(), (1.6) and assign to a an operator A in the C -algebra Ψ(M) of operators on L 2 (M) generated by OP S 0 (M), with principal symbol a. See 3 for details. In this more general setting, still (1.4) (1.5). Going further, we obtain the following result, which we propose to call the Quantum Ergodic Theorem. Theorem 1.2 Take A Ψ(M), with principal symbol a C() and weaken the hypothesis (1.4) to P a C(). (1.7) If A p Ψ(M) has principal symbol P a, then, with N as in Theorem 1.1, lim (Aϕ k, ϕ k ) (A p ϕ k, ϕ k ) = 0. (1.8) k / N,k 2

3 As a corollary of (1.8), we get lim sup a (Aϕ k, ϕ k ) sup P a a. (1.9) k / N,k The proof of Theorem 1.1 is parallel to previous proofs (cf. [5], [8], [6], [18], Chap. 15, and [9], pp ), done in the context where {G t } is ergodic. We record the details to verify that the result holds in the more general setting put forward here. In 2 we show that a Weyl formula, standard if {ϕ k } are eigenfunctions of Λ, holds under the more general hypothesis (1.2). In 3 we discuss a special class of quantization procedures, known as Friedrichs quantization, which gives that, if A = op F (a), (Aϕ k, ϕ k ) = a, µ k (1.10) defines µ k as a probability measure on, not just a distribution. In 4 we apply Egorov s theorem to show that, given a C(), (a a G t ) dµ k 0 as k, (1.11) for each t (under the hypothesis (1.2)). In 5 we use a standard Mean Ergodic Theorem argument to complete the proof of Theorem 1.1. In 6 we prove the Quantum Ergodic Theorem stated above. In 7 we give examples of cases where {G t } is not ergodic on but (1.4) holds for interesting classes of operators A, and (1.7) holds for a general class. The examples treated there are the following. 1. M = T n, Λ =. Here (1.4) holds for a = a(x), and (1.7) holds for all a C(). 2. M = S n, Λ =. Here (1.7) holds for all a C(), and an explicit formula reveals many cases where (1.4) holds. 3. M = T 2, a certain non-flat torus in R 3, Λ =. The analysis of P a is more complicated here, due partly to the existence of a hyperbolic closed geodesic on T 2. Again, (1.7) holds for all a C(). However, unlike Cases 1 and 2, in this case we do not have P : C () C (). In fact, members of the image of P can have less than Hölder regularity. This illustrates the value of having results for A Ψ(M), rather than just for A OP S 0 (M). 4. M = S 2, Λ = Λ c = (Y Y cy 2 3 ), Y j generate rotation about 3

4 the x j -axis. We take c > 0, c 1. There is a uniform description for the dynamics on and behavior of the projection P, despite the fact that the spectral behavior of Λ c depends delicately on c. Thus, certain implications of the quantum ergodic theorem that look natural, for irrational c, seem a bit more surprising for rational c ( 1). The curious phenomenon arising in Example 4 is related to an issue that arises when two self-adjoint operators with discrete spectra commute. Namely, must all eigenfunctions of one operator be joint eignfunctions of both? As seen in Example 4, sometimes they do and sometimes they do not. In 8, we look into this phenomenon in a more general setting. The examples arising in 7 are all integrable systems. In 9 we take a look at situations that might yield soft chaos, involving a mixture of integrability and chaos. This analysis makes use of some basic results in KAM theory. In 10 we discuss the issue of the existence of soft chaos, describing both known results and open problems. We bring in some examples expected to exhibit soft chaos, and describe how Theorem 1.1 applies to some of them. We plan to investigate in future work some problems raised in Sections 9 and 10. We end with two appendices. In Appendix A, we establish some results on conditional expectation of use in 7. In Appendix B we establish a general result on invariance properties of commuting, measure-preserving flows, which contains Proposition 8.2, the classical version of the quantum result, Proposition Weyl law Our first goal is to prove the following. This will play a role in the proof of Theorem 1.1 in 5. Proposition 2.1 Let A OP S 0 (M), with principal symbol σ(a) C (T M\ 0), homogeneous of degree zero. Then lim N 1 N N (Aϕ k, ϕ k ) = k=1 σ(a) ds. (2.1) Note that if the limit on the left side of (2.1) were known to exist for all A OP S 0 (M), it must depend only on σ(a), since if à OP S0 (M) had 4

5 the same principal symbol, then K = A Ã would be compact on L2 (M). Since ϕ k 0 weakly in L 2 (M), we then have Kϕ k 0 in L 2 -norm, so (Aϕ k, ϕ k ) (Ãϕ k, ϕ k ) 0. (2.2) Proof of Proposition 2.1. First note that e tλ ϕ k e tλ k ϕ k = e tλ k = e tλ k t 0 t 0 d ds e s(λ λ k) ϕ k ds e s(λ λ k) (Λ λ k )ϕ k ds, (2.3) so, for t (0, 1], e tλ ϕ k e tλ k ϕ k L 2 ε k t e tλ k, (2.4) with ε k as in (1.2). Hence e tλ k (Aϕ k, ϕ k ) = k k (Aϕ k, e tλ ϕ k ) + r(t), (2.5) with r(t) t k ε k e tλ k = o(tr e tλ ), as t 0. (2.6) Hence, as t 0, e tλ k (Aϕ k, ϕ k ) (Aϕ k, e tλ ϕ k ) k k = Tr Ae tλ ( ) σ(a) ds Tr e tλ ( = ) σ(a) ds e tλ k. k (2.7) The third line holds via a standard parametrix construction for e tλ. The result (2.1) follows from this, via Karamata s Tauberian theorem. 5

6 3 Friedrichs quantization To proceed, we discuss the notion of a quantization, which is a continuous linear map op : C () OP S 0 1,0(M) (3.1) with the property that, given a C (), A = op(a) has principal symbol a (mod OP S1,0 1 (M)). We insist that op(1) = I, the identity map. The existence of quantizations follows via local coordinate charts and partitions of unity from ψdo calculus on Euclidean space. There are many different quantizations. Each one gives rise to a sequence of elements µ k D (), defined by a, µ k = (op(a)ϕ k, ϕ k ). (3.2) It follows from (2.2) that if õp is another quantization, yielding µ k D (), then for each a C (), a, µ k a, µ k 0 as k. Basic examples are Kohn-Nirenberg quantizations: op KN : C () OP S 0 (M) OP S 0 1,0(M). (3.3) However, for the analysis to follow, it is useful to bring in the existence of a Friedrichs quantization, having the property op F : C () OP S 0 1,0(M), (3.4) a 0 = op F (a) 0. (3.5) This is constructed on the Euclidean space level from op KN (a) via Friedrichs symmetrization. See [16], Chapter 7. It is the case that a C () = op F (a) op KN (a) OP S1,0 1 (M). (3.6) This result is not trivial. In fact, it is the main technical result in the Friedrichs approach to the proof of the sharp Gårding inequality. From (3.5) it follows that op F (a) L(L 2 ) sup a, (3.7) and hence (3.4) has a unique extension to op F : C() L(L 2 (M)), (3.8) 6

7 with (3.5) holding for all a C(). From here on we will take a Friedrichs quantization, and set A = op F (a). In such a case, the distributions µ k D () defined by (3.2) satisfy a 0 = a, µ k 0. (3.9) Also 1, µ k = (ϕ k, ϕ k ) = 1. Consequently, for each k, µ k is a probability measure on. (3.10) We write (Aϕ k, ϕ k ) = a dµ k. (3.11) Remark. The image of C() in (3.8) is contained in the C -algebra of operators on L 2 (M) generated by OP S 0 (M), which we denote Ψ(M). If we compose the map op F : C() Ψ(M) (3.12) with taking the quotient by K(L 2 (M)), the space of compact operators on L 2 (M), we get an isomorphism of C -algebras: C() Ψ(M)/K(L 2 (M)). (3.13) 4 Application of Egorov s theorem The following result will play a useful role in 5. Proposition 4.1 Given a C(), we have (a a G t ) dµ k 0, as k, (4.1) locally uniformly in t. Proof. It suffices to prove (4.1) for a C (). Set A = op F (a) and By Egorov s theorem (and (3.6)) A t = e itλ Ae itλ. (4.2) A t op F (a G t ) OP S1,0 1 (M), (4.3) 7

8 and this holds locally uniformly in t, so a G t dµ k (Ae itλ ϕ k, e itλ ϕ k ) 0, as k, (4.4) locally uniformly in t. Now e itλ ϕ k e itλ k ϕ k = e itλ k = e itλ k t 0 t 0 d ds eis(λ λ k) ϕ k ds e is(λ λ k) (Λ λ k )ϕ k ds, (4.5) so, with ε k as in (1.2), Hence e itλ ϕ k e itλ k ϕ k L 2 ε k t. (4.6) (Ae itλ ϕ k, e itλ ϕ k ) = (Aϕ k, ϕ k ) + r k (t), (4.7) with r k (t) 0 as k, locally uniformly in t. This plus (4.4) gives (4.1), for a C (), and the general result follows, via (3.10). 5 Proof of Theorem 1.1 To proceed, given a C(), set a T = 1 T T 0 a G t dt, a = a ds. (5.1) The Mean Ergodic Theorem implies that, as T, a T P a in L 2 -norm, (5.2) where P is the orthogonal projection of L 2 () onto V = {b L 2 () : b G t b}. (5.3) If the flow {G t } is ergodic on, then V consists of constants, and then P a = a. (5.4) 8

9 Rather than assuming {G t } is ergodic, we will make (5.4) an hypothesis. Under this hypothesis, we have a T a ds 0 as T. (5.5) Thus, for ε (0, 1], there exists T ε < such that T T ε = a T a ds ε. (5.6) Now, Proposition 4.1 gives, for all a C(), T <, (a T a) dµ k 0 (5.7) as k, hence (a T a) dµ k as k. Furthermore, Proposition 2.1 implies lim N 1 N N k=1 (a a) dµ k 0, (5.8) b dµ k = b ds, (5.9) for each b C (), and (3.10) then gives this result for all b C(). Taking b = a T a gives lim N 1 N N k=1 a T a dµ k = for each T <. Comparison with (5.6) gives T T ε = lim N 1 N N k=1 a T a ds, (5.10) a T a dµ k ε, (5.11) if a satisfies (5.4). It follows that there exists a set N ε (a) N, of density zero, such that T = T ε = lim sup a T a dµ k 2ε. (5.12) k / N ε (a),k 9

10 Hence, by (5.8), for all ε > 0, Now we can produce such that, for all l N, lim sup k / N ε(a),k a Then (5.13) gives, for all ε (0, 1], lim sup a a dµ k 2ε. (5.13) N (a) N, of density zero, (5.14) N 2 l(a) \ N (a) is finite. (5.15) k / N (a),k so, if a C() satisfies (5.4), To proceed, let lim k / N (a),k a a dµ k 2ε, (5.16) a dµ k = 0. (5.17) I = {a C() : P a = a}, (5.18) which is a closed, linear subspace of C() (equal to C() if {G t } is ergodic). We can take a countable set {a ν }, dense in I, and produce such that, for all ν, Then N N, of density zero, (5.19) N (a ν ) \ N is finite. (5.20) lim k / N,k a a dµ k = 0, (5.21) whenever a = a ν, and hence, by a limiting argument, using (3.10), for all a I. We record the conclusion. Theorem 5.1 Let A OP S 0 (M) and assume a = σ(a) satisfies (5.4). Then, with N as in (5.19) (5.20), lim (Aϕ k, ϕ k ) = a ds. (5.22) k / N,k This also holds whenever a C() satisfies (5.4) and A = op F (a). 10

11 6 Proof of the quantum ergodic theorem Here we will weaken the hypothesis (5.4). Instead, we will assume A Ψ(M) has principal symbol a L C(), (6.1) where L is a closed linear subspace of C() satisfying P : L C(). (6.2) Under the hypotheses (6.1) (6.2), we can apply Theorem 5.1 to Note that P b = 0 = b, so Theorem 5.1 implies where b = a P a. (6.3) lim ((A A p)ϕ k, ϕ k ) = 0, (6.4) k / N,k A p = op F (P a). (6.5) This gives the Quantum Ergodic Theorem, which we restate here. Theorem 6.1 Assume (6.1) (6.2) hold. There is a set N N, of density 0, independent of the choice of such a L, with the property that (6.4) holds, with A p given by (6.5). If P a = a, then A p ϕ = aϕ, so (6.4) contains Theorem 5.1. Note also that a (Aϕ k, ϕ k ) = ((a A p )ϕ k, ϕ k ) + ((A p A)ϕ k, ϕ k ), (6.6) and, by (3.7), a A p L(L 2 ) sup This together with (6.4) yields the following. a P a. (6.7) Corollary 6.2 Assume (6.1) (6.2) hold. Take A Ψ(M) with principal symbol a. Then, with N as in Theorem 6.1, lim sup a (Aϕ k, ϕ k ) sup P a a. (6.8) k / N,k 11

12 7 Examples To begin, let M = T n = R n /(2πZ n ), the flat torus. Take Λ =. In such a case, the geodesic flow is integrable. However, it is elementary that More generally, a(x, ξ) = a(x) = P a = a. (7.1) P a(x, ξ) = 1 (2π) n a(y, ξ) dy. (7.2) T n Hence, if {ϕ k } is an orthonormal basis of L 2 (T n ), satisfying (1.2), then there exists a sparse set N N such that, for all a C (T n ), lim (a(x)ϕ k, ϕ k ) = a. (7.3) k / N,k If ϕ k consists of the standard complex exponentials, (a(x)ϕ k, ϕ k ) = a(x) ϕ k (x) 2 dx a, (7.4) T n and (7.3) is trivial. However, in this setting eigenspaces of have high multiplicities, and other forms of {ϕ k } can be produced. For them, (7.3) seems not to be trivial. For the second example, we consider M = S n, the unit sphere in R n+1, with its standard round metric. Again take Λ =. The geodesic flow is again integrable. In this case, G t is periodic of period 2π, and = S S n is foliated into circles, orbits of {G t }. We have P a(x, ξ) = 1 2π a(g t (x, ξ)) dt. (7.5) 2π 0 In case a C(S n ), i.e., a = a(x), P a = a provided a(x) = a + b(x), b( x) = b(x). (7.6) (For general a = a(x), P a need not be a function of x alone.) Again in this setting, eigenspaces of have high multiplicity, but here each eigenspace consists of functions that are either all even or all odd. Consequently, the results of Theorem 1.1 are trivial for a(x, ξ) = a(x) in this case. However, there are many functions a C (S S n ), lacking any symmetry, whose 12

13 averages over the various closed orbits of {G t } all coincide, hence satisfying P a = a. Theorem 1.1 applied to these operators has a nontrivial conclusion. The formulas (7.2) and (7.5) show that, for both families of examples mentioned above, with = S T n and = S S n, respectively, we have and also P : C() C(), (7.7) P : C () C (). (7.8) In general, there is a formula for P a as a conditional expectation, P a = E(a F), (7.9) where F is the σ-algebra of Borel sets in that are G t -invariant. In case (7.2), = T n S n 1 and F = {T n B : B S n 1 Borel}. (7.10) In case (7.5), F is the σ-algebra generated by { } G t (B) : B compact. (7.11) 0 t 2π For the third example, we consider a non-flat torus, an inner tube T 2, the image of T 2 under the map Φ : T 2 R 3, Φ(θ, ω) = ( (2 + cos θ) cos ω, (2 + cos θ) sin ω, sin θ ). (7.12) We use (θ, ω) as coordinates on T 2. Then = S T 2 = T 2 S 1, and on the factor S 1 we use the coordinate ϕ, the angle a tangent vector makes with the vector θ Φ(θ, ω) (and use the metric tensor to identify tangent vectors and cotangent vectors). Due to rotational symmetry about the x 3 -axis, T 2 also has an integrable geodesic flow. In this case, we have (7.9) with F = {Ω 1 (B) : B R Borel}, (7.13) where Ω : R is (the restriction to = S T 2 of) the angular momentum, i.e., the principal symbol of i / ω. Writing = {(θ, ϕ, ω) : θ, ϕ, ω T 1 }, (7.14) we see that Ω is independent of ω, say Ω(θ, ϕ, ω) = Ω(θ, ϕ), and F = { Ω 1 (B) T 1 : B R Borel}. (7.15) 13

14 and Thus, given a C(), say a = a(θ, ϕ, ω), we have P a = P b a b, a b (θ, ϕ) = 1 a(θ, ϕ, ω) dω, (7.16) 2π T 1 where P b a b (θ, ω) = E(a b F b )(θ, ω), (7.17) F b = { Ω 1 (B) : B R Borel}, F b T 2. (7.18) Regarding the level sets of Ω, we see that Ω has four critical points: max at θ = 0, ϕ = π 2, min at θ = 0, ϕ = π 2, saddles at θ = π, ϕ = ± π 2. (7.19) The max and min correspond to closed geodesics on the outer equator of T 2, going counterclockwise or clockwise, and the saddles correspond to closed geodesics along the inner equator of T 2, going counterclockwise or clockwise. There are four curves (on each of which Ω is constant) that function as separatrices, two for each saddle, which separate T 2 = {(θ, ϕ)} into four regions. Two regions correspond to geodesics on T 2 that cross the inner equator infinitely often. The other two correspond to geodesics that never cross the inner equator. Now, given a C(), we have a b C(T 2 ). Formula (7.17) and the analysis above of F b guarantee that P b a b is continuous on the interior of each of the four regions described above, and extends to be continuous on the closure of each one of these. Furthermore, the limit on each separating curve from each side is the value of a b at the corresponding saddle point. Hence P b a b does not jump across these separating curves, so again (7.7) holds. On the other hand, (7.8) typically fails in this case. Instead, we have a C () = P a C ω (), (7.20) with ω(h) = 1, log h for h << 1. (7.21) See Appendix A for details. Remark. With a b given by (7.16), we see that Thus Theorem 5.1 implies the following. P (a a b ) = 0. (7.22) 14

15 Proposition 7.1 Given A Ψ(T 2 ) with symbol a, let A b Ψ(T 2 ) have symbol a b. If {ϕ k } is an orthonormal basis of L 2 (T 2 ) consisting of eigenfunctions of, or more generally satisfying (1.2), then there is a set N N, of density 0, such that, for each a C(), lim (Aϕ k, ϕ k ) (A b ϕ k, ϕ k ) = 0. (7.23) k / N,K Remark. Both T 2 and S 2 are invariant under rotation about the x 3 -axis. However, if one replaces T 2 by S 2 in Proposition 7.1, the conclusion does not hold. This is because the σ-algebra of G t -invariant Borel subsets of S S 2 in much richer than that for S T, and (7.22) fails. For the fourth example (or class of examples), we return to M = S 2, and consider other elliptic operators. Let Y j be vector fields generating 2π-periodic rotation about the x j -axis. For c (0, ) take Λ c = (Y1 2 + Y cy 3 2 ). (7.24) Then Λ 1 = is what we considered in Example 2. Now, commutes with each Y j. Hence Λ c commutes with Y 3 for each c R +, but if c 1, Λ c does not commute with Y 1 or Y 2. Let c T S 2 denote the set where σ(λ c ) = 1, G c,t the flow on c generated by the Hamiltonian vector field associated to σ(λ c ). There is a natural probability measure on c, invariant under {G c,t : t R}. Let P c denote the orthogonal projection of L 2 ( c ) onto the space of G c,t -invariant functions. Let {ϕ k : k N} be an orthonormal basis of L 2 (Λ c ), consisting of eigenfunctions of Λ c. (This time, we abuse notation, and do not call these functions ϕ c,k.) By Theorem 6.1, there is a set N c N of density 0 such that if A Ψ(S 2 ) has principal symbol a C( c ), and if P c a C( c ), forming then the principal symbol of A p,c Ψ(S 2 ), then lim (Aϕ k, ϕ k ) (A p,c ϕ k, ϕ k ) = 0. (7.25) k / N c,k When c 1, the structure of P c is quite a bit different from that of P 1, given by (7.5). By (7.9), we have P c a = E(a F c ), (7.26) where F c is the σ-algebra of Borel sets in c that are G c,t -invariant. F 1 is given by (7.11), but F c = {Ω 1 c (B) : B R Borel}, if c 1, (7.27) 15

16 where Ω c is the (restriction to c of) the principal symbol of iy 3. The flow on S 2 generated by Y 3 has a natural lift to a flow H t on T (S 2 ) (generated by the Hamiltonian vector field associated to σ(iy 3 )). The flow H t leaves c invariant. Furthermore, for c 1, each Borel set in F c is H t -invariant. It follows that if a C( c ), P c a = P c a b, a b = 1 2π 2π 0 a H t dt. (7.28) Note the parallel to (7.16). The argument going from (7.22) to Proposition 7.1 applies here, to give (for c 1) lim (Aϕ k, ϕ k ) (A b ϕ k, ϕ k ) = 0, (7.29) k / N c,k where A b Ψ(S 2 ) has principal symbol a b, given by (7.28). Since Λ c and Y3 2 commute, one can pick an orthonormal basis of L2 (S 2 ) consisting of joint eigenfunctions for these two operators. In such a case, (7.29) seems relatively natural, at least when Af(x) = a(x)f(x). One is tempted to wonder whether an eigenfunction of Λ c must also be an eigenfunction of Y3 2. As we will see, this holds for some values of c but fails in a big way for some other values of c. Since Λ 2 c = (c 1)Y3 2, (7.30) we can read off Spec Λ c from the joint spectrum of and Y 3. We recall that and L 2 (S 2 ) = V k, = k 2 + k on V k, (7.31) k=0 Spec iy 3 Vk = {j Z : j k}, Spec( Y 2 3 ) Vk = {j 2 : 0 j k}. (7.32) Hence Spec Λ 2 c Vk = {λ cjk : 0 j k} λ cjk = k 2 + k + (c 1)j 2. (7.33) If c is irrational, then the number λ cjk uniquely determines j and k, given that they both are in N {0}. In view of the fact that each joint eigenspace of and iy 3 is one dimensional, we have: Proposition 7.2 If c R + is irrational, each eigenspace of Λ c has dimension 1 or 2, and each eigenfunction of Λ c is also an eigenfunction of Y

17 We now observe how very different the spectrum of Λ c is when c = 2. By (7.33), Spec Λ 2 2 Vk = {k 2 + k + j 2 : 0 j k}. (7.34) Now If we set n = k 2 + k + j 2 = 4n + 1 = (2k + 1) 2 + (2j) 2. (7.35) S N = {4n + 1 : j, k such that 0 j k, 4n + 1 = (2k + 1) 2 + (2j) 2, n N}, T N = {(j, k) : 0 j k, (2k + 1) 2 + (2j) 2 S N }, we see that the number of elements of T N satisfies (7.36) #(T N ) π N, as N. (7.37) 8 On the other hand, it is a classical result of lattice point counting that These two results give: #(S N ) N 0 as N. (7.38) Proposition 7.3 If the eigenvalues of Λ 2 are with corresponding eigenspaces Ṽk, then 0 = ω 1 < ω 2 < < ω k, (7.39) lim n 1 n n dim Ṽk =. (7.40) k=1 The argument proving Proposition 7.3 extends readily to treat In such a case, c 1 = a2, a, b N. (7.41) b2 λ = k 2 + k + a2 b 2 j2 = (4λ + 1)b 2 = ((2k + 1)b) 2 + (2ja) 2, (7.42) and considerations parallel to (7.36) (7.38) apply. We have: 17

18 Proposition 7.4 The conclusion of Proposition 7.3 holds for Λ c whenever c 1 Q. (7.43) We briefly mention a further generalization of (7.24), namely Λ c = (c 1 Y1 2 + c 2Y2 2 + c 3Y3 2), c = (c 1, c 2, c 3 ), c j > 0. (7.44) Since commutes with each Y j, we see that Λ c and commute, c (R + ) 3. (7.45) Thus their symbols σ(λ c ) and σ( ) Poisson commute, and H σ(λc) is integrable. One could also look into how Λ c operates on the spaces V k introduced in (7.31), using basic representation theory of SO(3), but we will not take this up here. 8 Joint eigenfunctions of commuting operators Let Λ OP S 1 (M) be as in 1. Assume we have Y = Y OP S 1 (M), ΛY = Y Λ. (8.1) Then L 2 (M) has an orthonormal basis consisting of joint eigenfunctions of Λ and Y, but a random orthonormal basis consisting of eigenfunctions of Λ might not also be eigenfunctions of Y, as the example M = S 2, Λ = considered in 7 shows. The following result generalizes Proposition 7.2. To state it, note that there is an interval I R, containing 0, such that α I = Λ + αy OP S 1 (M) is elliptic. (8.2) Proposition 8.1 There is a countable set C R such that, for all α I \C, if u L 2 (M) is an eigenfunction of Λ+αY, then u is an eigenfunction both of Λ and of Y. Proof. We have L 2 (M) = V k, Λ = λ k on V k, 0 λ 0 < λ 1 < λ 2 <. (8.3) k=0 By (8.1), Y : V k V k. Say Spec Y Vk = {ω jk : 1 j d k }, (8.4) 18

19 where the numbers ω jk are distinct, for each fixed k. Then Spec(Λ + αy ) Vk = {λ k + αω jk : 1 j d k }. (8.5) To prove Proposition 8.1, we need to construct C R such that α I \ C, λ k + αω jk = λ l + αω ml (8.6) implies λ k = λ l. Note that the hypotheses of (8.6) imply and if λ k λ l, then also ω jk ω ml, so So to construct C, first consider α(ω jk ω ml ) = λ l λ k, (8.7) α = λ l λ k ω jk ω ml. (8.8) S = {λ k : k Z + } {ω jk : k Z +, 1 j d k }, (8.9) which is countable. Then let C R be the field generated by S, which is also countable. Alternatively, C could just be the set of quotients that appear on the right side of (8.8), with ω jk ω ml. This spectral result suggests a dynamical counterpart. To state it, let α = {(x, ξ) T M : σ(λ + αy )(x, ξ) = 1}, (8.10) which gets a natural probability measure, invariant under the flow generated by H σ(λ+αy ). Also, α is invariant under the flows generated by H σ(λ) and H σ(y ). Proposition 8.2 There exists a countable set C I such that, for α I \C, the following holds. Given b L 2 ( α ), H σ(λ+αy ) b = 0 = H σ(λ) b = 0 and H σ(y ) b = 0. (8.11) For such α, a Borel set S α is invariant under the flow genreated by H σ(λ+αy ) if and only if S is simultaneously invariant under the flow generated by H σ(λ) and that generated by H σ(y ). Note that, by homogeneity, we can simply work on = 0, rather than on α, in Proposition 8.2. In Appendix B, we establish a more general result, on commuting, measure-preserving flows, which implies Proposition

20 9 Soft chaos Here, we take M, Λ, and as in 1, but require dim M = 2, (9.1) so dim = 3. Assume the flow G t has an elliptic periodic orbit, γ. Let T 0 denote its minimal period, so p γ G T0 p = p. We describe how soft chaos (a term used in [7]) can arise in this setting, as a consequence of KAM theory. Pick p γ, and let Σ be a 2-dimensional surface, transversal to γ, containing p. Let R : Σ Σ be the Poincaré first return map. Thus R(p) = p. The symplectic form on T M pulls back to a nondegenerate, closed, 2-form on Σ, invariant under R. Thus Σ has an area element, and R is area-preserving. That γ is elliptic means DR(p) : T p Σ T p Σ has eigenvalues of the form {e iα, e iα }. In such a case, R is called an α-twist. It is a classical result of G. Birkhoff that (under some extra hypotheses) R has a normal form. The formulation below is from [1], p Proposition 9.1 If α is not 0 or an integral multiple of π/2 or 2π/3, then (after perhaps shrinking Σ) there exist symplectic coordinates u = u 1 + iu 2 on Σ such that u(p) = 0 and R(u) = u e i(α+β u 2) + O( u 4 ). (9.2) One says the first return map R is an elementary twist map if α is as in Proposition 9.1 and (9.2) holds with β 0. We next recall a stability theorem, proved by J. Moser, in [11]. We say a cycle in Σ is a simple, closed, C 1 curve in Σ that encloses p. Theorem 9.2 Assume R : Σ Σ is an elementary twist map. Then there is a collection {σ I} of invariant cycles, contained in Σ, such that (i) For each σ I, R σ has irrational rotation number, and each orbit of R σ is dense in σ. (ii) σ I σ Σ is closed. (iii) For each ε > 0, there exists a neighborhood U of p in Σ such that the union {σ I : σ U} has 2D measure (1 ε) Area(U). 20

21 In addition to the proof in [11], there is a treatment, in the real analytic case, in [15], See also discussions in [12], Chapter 2, 4, and in [1]. The following result is perhaps implicit in these works, and is certainly implicit in illustrative figures. We make it explicit. Proposition 9.3 Given two invariant cycles σ σ, in I, either σ encloses σ or σ encloses σ. Proof. The invariant cycles in I are obtained as small perturbations of the cycles u = small const., which are invariant under R 0 (u) = ue i(α+β u 2). Thus if σ, σ I, either one encloses the other or they have a nonempty intersection. Then, part (i) of Theorem 9.2 implies this intersection must be dense in σ, and in σ, which requires σ = σ. At this point, we pick a base invariant cycle σ 0, and consider only those σ enclosed by (or equal to) σ 0. Given an invariant cycle σ I, let T σ = {G t (x) : x σ, t R} (9.3) be the tube swept out by σ under the flow G t. Each such T σ is homeomorphic to T 2. Results in Theorem 9.2 and Proposition 9.3 imply the following. (I) Each T σ has zero 3D measure. (II) The union of all such T σ for σ I is closed. (III) Given two invariant tubes T σ T σ, either T σ encloses T σ or T σ encloses T σ. Now, let Given x, let T # σ = closed, solid tube enclosed by T σ. (9.4) T x = T σ, (9.5) and T # x = σ I,x/ T # σ σ I,x/ T # σ T # σ. (9.6) 21

22 That is, T x is the union of all 2D tubes T σ that neither enclose x nor contain x, and T x # is the union of all closed solid tubes T σ # that do not contain x. Note that, given x, y, possibly and if not, either If (9.7) holds, we write and if it fails, we write T x = T y, (9.7) T x T y, or T y T x. (9.8) x y, (9.9) x y, or y x, (9.10) in the two cases in (9.8). We also have natural notions of x y or y x. Now, using the invariant volume element on arising in 1, we define a function Φ : R by Φ(x) = Vol T x. (9.11) Note that T x = T Gt x, for all t R, x, so Φ = Φ G t. We view the following as a key result. Proposition 9.4 The function Φ is continuous on. Proof. Let x k, x, x k x. We need to show that Φ(x k ) Φ(x). It suffices to show this in each of the following three cases: x k x, x x k, x x k, k, (9.12) and x k x. Consider the first case, x k x, k. Reordering the sequence (x k ) if necessary, we can assume In such a case, hence x 1 x 2 x 3. (9.13) T x1 T x2 T x3, (9.14) Φ(x 1 ) Φ(x 2 ) Φ(x 3 ) Φ(x). (9.15) Furthermore, the monotone convergence theorem implies ( Φ(x k ) Vol T xk ). (9.16) 22 k 1

23 Of course, T xk T x. k 1 Recall that T x is the union of all 2D tubes T σ that neither enclose x nor contain it. If T σ is such a tube, x has a positive distance from T σ. Since x k x, we deduce that for all sufficiently large k, T σ will neither enclose nor contain x k. Thus T σ T xk for all sufficiently large k, so in fact T xk = T x. (9.17) k 1 This proves Proposition 9.4 for the first case (and also the last case) of (9.12). It remains to check the case x x k, k. This time, we can reorder (x k ) to achieve x 3 x 2 x 1. (9.18) In such a case, hence T x1 T x2 T x3, (9.19) Φ(x 1 ) Φ(x 2 ) Φ(x 3 ) Φ(x). (9.20) This time, the monotone convergence theorem implies ( Φ(x k ) Vol T xk ). (9.21) Clearly k 1 T xk T x. k 1 Recall that T xk is the union of all the 2D tubes T σ that neither enclose nor contain x k. Now suppose you take T σ such that T σ is not a subset of T x. (9.22) It follows that T σ either contains x or encloses x. If T σ encloses x, then x has a positive distance from T σ, hence for all sufficiently large k, T σ must enclose x k, so T σ is not a subset of T xk for large k. We deduce that, for each σ for which (9.22) holds, T σ k 1 T xk = x T σ. (9.23) 23

24 Now condition (III) implies there can be at most one such σ. We conclude that either T xk = T x, or T xk = T x T σ1, (9.24) k 1 k 1 for an invariant cycle σ 1. Then condition (I) yields Φ(x k ) Φ(x), (9.25) and the proof of Proposition 9.4 is complete. Proposition 9.4 gives a nontrivial function Φ C() that is invariant under G t for all t R. Of course, for each continuous β : R R, β Φ belongs to C() and is invariant under all G t. Thus the range of P contains lots of elements of C(), where P is the orthogonal projection arising in Theorem 1.1. We are interested in the following complementary situation. Problem. Determine when one can take a C() and guarantee that P a C(). Our quantum ergodic theorem points to the usefulness of obtaining results on this problem. We intend to address this in future work. 10 On the existence of soft chaos As stated in [7], p. 118, the term soft chaos is somewhat lacking in mathematical precision. A number of different varieties of soft chaos come to mind. To set definitions, let H be a Hamiltonian vector field on T M, a constant energy surface, assumed to be compact, G t the flow on generated by H. Here is a weak notion of soft chaos. The flow G t is neither integrable nor ergodic. (10.1) In this case, it has been proven in [10] that, in the generic case, (10.1) holds. Tools to show that, in certain cases, G t is not ergodic include KAM theory. Tools to show that, in certain cases, G t is not integrable include the detection of homoclinic tangles. 24

25 A stronger version of soft chaos is the following. There is a partition = 0 1 with the following properties. 0 is closed and the union of invariant Lagrangian tori. 1 = α α, with G t acting ergodically on each α. Finally, 0 and 1 both have positive measure. (10.2) It is widely believed (and this belief seems to be well supported by numerical evidence) that (10.2) holds generically, but proving this (or even proving some examples exist) has been a problem for some time, a problem mentioned in [12], p. 109, in 1973, and in [4] in Of course, KAM can establish that 0 has positive measure, in certain cases. The problem is to show that \ 0 has positive measure. In this context, it should be mentioned that Smale horseshoes that arise from homoclinic tangles have measure zero. The remarks above regarding (10.2) apply to cases where M has no boundary, which is the setting of this paper. There do exist bounded domains in Euclidean space, with piecewise smooth boundary, whose associated billiard ball maps have been proven to have property (10.2). More precisely, what is called the mushroom domain in R 2 has been shown in [3] to have the property that = 0 1 with the billiard ball map integrable on 0 and ergodic on 1, and both 0 and 1 have positive measure. Study of the mushroom domain and variants has produced a growing literature. As an example, we mention [2]. We now discuss some examples where one can expect to find soft chaos, and it is likely one can prove that (10.1) holds, though we do not propose proofs here. It also seems likely that (10.2) holds for these examples, but proofs of this will certainly have to wait! Our first class of examples arise from geodesic flows on certain perturbations of the inner tube T 2 R 3, given by (7.17). To begin, we consider T 2 a R 3, the image of T 2 under the maps Φ a : T 2 R 3, given by Φ a (θ, ω) = ( (2 + cos θ) cos ω, a(2 + cos θ) sin ω, sin θ ). (10.3) In other words, T 2 a is the image of T 2 under the map M a : R 3 R 3, M a (x, y, z) = (x, ay, z). (10.4) For each a > 0, Ta 2 has four closed geodesics in the plane {z = 0}. Two (related by time reversal) are outer equatorial geodesics, and the other two (also related by time reversal) are inner equatorial geodesics. The outer 25

26 equatorial geodesics are elliptic and the inner equatorial geodesics are hyperbolic. For (almost all?) a > 0, we expect the results of 9 to apply near the elliptic closed geodesics, yielding some nontrivial G t -invariant functions in C(), = S Ta 2. The results of 9 do not imply that there are stochastic regions in a small tubular neighborhood of the outer equatorial geodesics, though we expect this to be the case. Another potential source of chaos is associated with the inner equatorial geodesics, which are hyperbolic. Identify an inner equatorial geodesic with the periodic G t -orbit γ. Pick p γ, take a surface Σ, through p, transversal to γ, and let R : Σ Σ be the Poincaré first return map. Then R(p) = p, and p is a hyperbolic fixed point. Thus there are invariant curves through p, one the stable manifold (near p) and one the unstable manifold. If a = 1, then, globally, these two coincide, and we have a homoclinic invariant curve. One would achieve chaos if, as a is moved from 1, these did not coincide, but intersected transversally, yielding a homoclinic tangle. However, for M a as in (10.3) (10.4), this does not happen. In fact, for each a > 0, in this setup, the stable and unstable manifolds coincide. This can be established using the symmetries of Ta 2. Each surface Ta 2 is invariant under three reflections, namely reflection across {x = 0}, across {y = 0}, and across {z = 0}. Hence Ta 2 is invariant under rotation by 180 about the x-axis (and also the y-axis, and for that matter, also the z-axis). Consider various geodesics in Ta 2 starting at the point e 1 = (1, 0, 0) Ta 2, where θ = ω = 0. If the velocity vector is close to (0, 0, 1), the geodesic winds aronnd Ta 2, crossing the inner equatorial geodesic infinitely often. If the velocity vector is close to (0, 1, 0), the geodesic never crosses the inner equatorial geodesic. There are 4 critical velocity directions leading to geodesics through e 1 that tend toward the inner equatorial geodesic as t +. Now the rotational symmetry mentioned above implies these critical directions occur in pairs, one velocity being the negative of the other. It follows that if a geodesic through e 1 tends toward the inner equatorial geodesic as t +, it also tends toward same as t. To address this phenomenon, we modify T 2 a, obtaining surfaces T 2 a,b, for which the reflection symmetries across {x = 0} and {y = 0} are destroyed, though we retain the reflection symmetry across {z = 0}. We take T 2 a,b R3 to be the image of T 2 under the map Φ a,b : T 2 R 3, with We take Φ a,b (θ, ω) = Φ a (θ, ω) + bψ(ω), Ψ(ω) = (sin 3ω, cos 5ω, 0). (10.5) 0 < b << a. (10.6) 26

27 In such a case, Ta,b 2 still has inner and outer equatorial geodesics, in {z = 0}. Under such conditions, it is reasonable to expect that homoclinic tangles arise near the inner equatorial geodesic. We turn to another class of examples. Let M 0 be a compact 2-dimensional Riemannian manifold for which the geodesic flow is known to be ergodic on S M 0. Take the standard unit sphere S 2. Cut a small open geodesic disk D from S 2, cut a small open geodesic disk D 0 from M 0, and join what remains by a neck N, obtaining M = (S 2 \ D) N (M 0 \ D 0 ). (10.7) Endow M with a smooth metric tensor, agreeing with the original metric tensors on S 2 \ D and on M 0 \ D 0. Let D S 2 be the image of D under the antipodal map of S 2. The set of points in S S 2 whose image under the geodesic flow lies over S 2 \ (D D ) for all t has positive measure. This gives rise to a subset Y of S M of positive measure (in fact, with nonempty interior) on which the geodesic flow is integrable. It is tempting to conjecture that S M \ Y has a subset of positive measure on which the geodesic flow is ergodic. Needless to say, we do not have a proof of this. We conclude with a description of a class of symbols to which Theorem 1.1 applies. With M as above, and β R, take a C(S M) such that a averages to β on each closed orbit in Y, a = β on S M \ Y. (10.8) Then P a β, and Theorem 1.1 applies. A Conditional expectation near a hyperbolic critical point In (7.17) we have the conditional expectation of a function on T 2, with respect to the σ-algebra of sets of the form Ω 1 (B), for Borel B R, where Ω C (T 2 ) has two hyperbolic critical points. We state there that this conditional expectation operator maps C(T 2 ) to C(T 2 ) and C (T 2 ) to C ω (T 2 ), with ω given by (7.21). Here we show how to establish these results. To simplify the calculations, we look at a model case, with one hyperbolic critical point. Namely, we analyze P f = E(f F), on Q = [ 1, 1] [ 1, 1] R 2, with Lebesgue measure, where F is the σ-algebra of the form g 1 (B), for Borel B R, with g(x, y) = y 2 x 2, which has a critical point at (0, 0). 27

28 We make a few preliminary observations. Clarly P : L (Q) L (Q) has norm 1, so to show P : C(Q) C(Q), (A.1) it suffices to show P : C (Q) C(Q). (A.2) Also, clearly f C(Q) (resp., f C (Q)) implies P f is continuous (resp., P f is smooth) on Q \, where = {(x, y) Q : x = ±y}. (A.3) Consequently, the following result is useful. Lemma A.1 If f C 1 (Q), then z k Q \, z k z 0 = P f(z k ) f(0, 0). (A.4) Proof. The set Q \ has four connected components. It will suffice to consider the case where each z k lies in the upper quarter, where y > x, as similar arguments apply in the other cases. We may as well drop the subscripts, and take y = ε + x 2, ε > 0. (A.5) Then P f(x, y) depends only on ε, and we have P f(x, y) = 1 1 f(s, ε + s 2 ) 1 ds ds, A(ε) = A(ε). 1 ε + s 2 1 ε + s 2 (A.6) Remark. Actually, the domain of integration should be s [ 1 ε, 1 ε]. For notational convenience, we ignore this, here and below. For the denominator A(ε), we have A(ε) = 1/ ε 1/ ε du 1 + u 2 = 2 sinh 1 1 ε 1 ε 1 ) = 2 log( + ε 1 ( = log 1 ) (1 ) + O(ε). ε (A.7) 28

29 We can rewrite (A.6) as P f(x, y) = 1 1 f ε (s) A(ε) ds, 1 ε + s 2 (A.8) where f ε (s) = f(s, ε + s 2 ). (A.9) Note that, if f C 1 (Q), then the family f ε is uniformly Lipschitz on [ 1, 1], for ε (0, 1/2]. We then have P f(x, y) = f ε (0) A(ε) = f ε (0) + 1 A(ε) f ε (s) f ε (0) ε + s 2 ds s ε + s 2 g ε(s) ds, (A.10) where Since g ε (s) = f ε(s) f ε (0) s = f(s, ε + s 2 ) f(0, ε). (A.11) s ε + s 2 ε = 1 2 s 2 0 dy 1 s 2 ε + y 2 0 dy y = s, (A.12) and f is C 1, hence Lipschitz, we have g ε (s) L <, ε. (A.13) Thus we have 1 1 hence, for (x, y) as in (A.5), f C 1 (Q), s g ε(s) ds 2L, ε, (A.14) ε + s 2 P f(x, y) f(0, ε) 2L A(ε). (A.15) This establishes Lemma A.1. It hence yields (A.1), and also P : C 1 (Q) C ω (Q), (A.16) with ω(h) = 1, 0 < h << 1. (A.17) log h 29

30 Remark. The estimate (A.13) and the Lebesgue dominated convergence theorem yield lim ε s ε + s 2 g ε(s) ds = = f(s, s ) f(0, 0) (sgn s) s f(s, s ) f(0, 0) s ds. ds (A.18) Since there are functions f C 1 (Q) for which this last integral is 0, this implies that (A.16) (A.17) is sharp, as far as the range is concerned. B Invariance properties of commuting, measurepreserving flows Our goal here is to establish a general result that contains Proposition 8.2. To set things up, let (, µ) be a probability space. Assume L 2 (, µ) is separable. For j = 1, 2, t R, let Fj t : be a 1-parameter group of measure-preserving transformations. Assume these transformations commute, i.e., F1 tf 2 s = F 2 sf 1 t, for all s, t R. Also assume the following continuity: f L 2 (, µ) = t f F t j is continuous from R to L 2 (, µ). (B.1) For α R, set The following result contains Proposition 8.2. G t α = F t 1 F αt 2. (B.2) Proposition B.1 Under the hypotheses stated above, there exists a countable set C R such that, for each α R \ C, f L 2 (, µ), f G t α = f, t R = f F t j = f, t R, j. (B.3) Proposition B.1 follows from the next proposition, an abstract result about commuting unitary groups. To state it, let {Uj t : t R} be strongly continuous unitary groups on a separable Hilbert space H, and assume they commute, i.e., U1 tu 2 s = U 2 su 1 t, for all s, t R. For α R, set V t α = U t 1U αt 2. (B.4) 30

31 Proposition B.2 Given Uj t as above, there exists a countable set C R such that, for each α R \ C, f H, V t αf = f, t R = U t jf = f, t R, j. (B.5) To prove Proposition B.2, we use the following version of the Spectral Theorem. There exists a σ-finite measure space (Y, ν), a unitary map W : H L 2 (Y, ν), and ν-measurable functions a j : Y R such that Note that and W (U t jf)(y) = e ita j(y) W f(y), f H, t R. (B.6) U t jf = f t a j W f = 0 (ν-a.e.), (B.7) V t αf = f t (a 1 + αa 2 )W f = 0 (ν-a.e.). (B.8) Hence Proposition B.2 is a consequence of the following. Lemma B.3 Given ν-measurable functions a j : Y R as above, there exists a countable set C R such that for each α R \ C, g L 2 (Y, ν), (a 1 + αa 2 )g = 0 (ν-a.e.) = a 1 g = a 2 g = 0 (ν-a.e.). (B.9) Proof. Consider the following subsets of Y : S α = {y Y : a 1 (y) = αa 2 (y)}, S = {y Y : a 1 (y) = a 2 (y) = 0}. (B.10) Clearly S S α, for each α. To prove Lemma B.3, it suffices to show that there exists a countable set C R such that α R \ C = ν(s α \ S) = 0. (B.11) Note that S α \ S = T α = {y Y : a 1 (y) = αa 2 (y), a 2 (y) 0}. (B.12) On the other hand, clearly α α T α T α =. Hence ν(t α ) 0 for at most countably many α R, and we are done. We note the following n-dimensional version of Lemma B.3. 31

32 Lemma B.4 Given a ν-measurable a : Y R n, a = (a 1,..., a n ), there exists E R n of Lebesgue measure 0 such that, for each ω R n \ E, g L 2 (Y, ν), (ω a)g = 0 (ν-a.e.) = a 1 g = = a n g = 0 (ν-a.e.). (B.13) This follows by induction on n, starting at n = 2, by Lemma B.3. This result leads to the following n-dimensional version of Proposition B.2. Proposition B.5 Let U be a strongly continuous unitary representation of R n on a separable Hilbert space H. Then there exists a subset E R n, of Lebesgue measure 0, such that, for each ω R n \ E, f H, U(tω)f = f, t R = U(ξ)f = f, ξ R n. (B.14) This in turn leads to an n-dimensional version of Proposition B.1, which we leave to the reader. References [1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, Reading MA, [2] A. Barnett and T. Betche, Quantum mushroom billiards, Chaos 17 (2007), [3] L. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11 (2001), [4] L. Bunimovich, Relative volume of Kolmogorov-Arnold-Moser tori and uniform distribution, stickiness and nonstickiness in Hamiltonian systems, Nonlinearity 21 (2008), T13 T17. [5] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacian, Comm. Math. Phys. 102 (1985), [6] H. Donnelly, Quantum unique ergodicity, Proc. AMS 131 (2003), [7] M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer- Verlag, New York,

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