Entropy and the localization of eigenfunctions By Nalini Anantharaman Abstract We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature { in fact, we only assume that the geodesic ow has the Anosov property. In the semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesics. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d 2, and assume that the geodesic ow (g t ) t2R , acting on the unit tangent bundle of M , has a chaotic" behaviour. This refers to the asymptotic properties of the ow when time t tends to in nity: ergodicity, mixing, hyperbolicity . : we assume here that the geodesic ow has the Anosov property, the main example being the case of negatively curved manifolds. The words quantum chaos" express the intuitive idea that the chaotic features of the geodesic ow should imply certain special features for the corresponding quantum dynamical system: that is, according to Schrodinger, the unitary ow exp(i~t 2 ) t2R acting on the Hilbert space L 2 (M ), where stands for the Laplacian on M and ~ is proportional to the Planck constant. Recall that the quantum ow converges, in a sense, to the classical ow (g t ) in the so-called semi-classical limit ~ ! 0; one can imagine that for small values of ~ the quantum system will inherit certain qualitative properties of the classical ow. One expects, for instance, a very di erent behaviour of eigenfunctions of the Laplacian, or the distribution of its eigenvalues, if the geodesic ow is Anosov or, in the other extreme, completely integrable (see [Sa95]). The convergence of the quantum ow to the classical ow is stated in the Egorov theorem. Consider one of the usual quantization procedures Op ~ , which associates an operator Op ~ (a) acting on L 2 (M ) to every smooth compactly supported function a 2 C 1 c (T M ) on the cotangent bundle T M . According to the Egorov theorem, we have for any xed t exp it ~ 2 Op ~ (a) exp it ~ 2 Op ~ (a g t ) L 2 (M) = O(~) ~!0 : 436 NALINI ANANTHARAMAN We study the behaviour of the eigenfunctions of the Laplacian, h 2 h = h in the limit h ! 0 (we simply use the notation h instead of ~, and now 1 h 2 ranges over the spectrum of the Laplacian). We consider an orthonormal basis of eigenfunctions in L 2 (M ) = L 2 (M; dVol) where Vol is the Riemannian volume. Each wave function h de nes a probability measure on M : j h (x)j 2 dVol(x); that can be lifted to the cotangent bundle by considering the microlocal lift", h : a 2 C 1 c (T M ) 7! hOp h (a) h ; h i L 2 (M) ; also called Wigner measure or Husimi measure (depending on the choice of the quantization Op ~ ) associated to the eigenfunction h . If the quantization procedure was chosen to be positive (see [Ze86, x3], or [Co85, 1.1]), then the distributions h s are in fact probability measures on T M : it is possible to extract converging subsequences of the family ( h ) h!0 . Re ecting the fact that we considered eigenfunctions of energy 1 of the semi-classical Hamiltonian h 2 , any limit 0 is a probability measure carried by the unit cotangent bundle S M T M . In addition, the Egorov theorem implies that 0 is invariant under the (classical) geodesic ow. We will call such a measure 0 a semi-classical invariant measure. The question of identifying all limits 0 arises naturally: the Snirelman theorem ([Sn74], [Ze87], [Co85], [HMR87]) shows that the Liouville measure is one of them, in fact it is a limit along a subsequence of density one of the family ( h ), as soon as the geodesic ow acts ergodically on S M with respect to the Liouville measure. It is a widely open question to ask if there can be exceptional subsequences converging to other invariant measures, like, for instance, measures carried by closed geodesics. The Quantum Unique Ergodicity conjecture [RS94] predicts that the whole sequence should actually converges to the Liouville measure, if M has negative sectional curvature. The problem was solved a few years ago by Lindenstrauss ([Li03]) in the case of an arithmetic surface of constant negative curvature, when the func-tions h are common eigenstates for the Laplacian and the Hecke operators; but little is known for other Riemann surfaces or for higher dimensions. In the setting of discrete time dynamical systems, and in the very particular case of linear Anosov di eomorphisms of the torus, Faure, Nonnenmacher and De Bievre found counterexamples to the conjecture: they constructed semi-classical invariant measures formed by a convex combination of the Lebesgue measure on the torus and of the measure carried by a closed orbit ([FNDB03]). However, it was shown in [BDB03] and [FN04], for the same toy model, that semi-classical invariant measures cannot be entirely carried on a closed orbit. ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS 437 1.1. Main results. We work in the general context of Anosov geodesic ows, for (compact) manifolds of arbitrary dimension, and we will focus our attention on the entropy of semi-classical invariant measures. The Kolmogorov-Sinai entropy, also called metric entropy, of a (g t )-invariant probability measure 0 is a nonnegative number h g ( 0 ) that measures, in some sense, the complex-ity of a 0 -generic orbit of the ow. For instance, a measure carried on a closed geodesic has zero entropy. An upper bound on entropy is given by the Ruelle inequality: since the geodesic ow has the Anosov property, the unit tangent bundle S 1 M is foliated into unstable manifolds of the ow, and for any invariant probability measure 0 one has (1.1.1) h g ( 0 ) Z S 1 M log J u (v)d 0 (v) ; where J u (v) is the unstable jacobian of the ow at v, de ned as the jacobian of g 1 restricted to the unstable manifold of g 1 v. In (1.1.1), equality holds if and only if 0 is the Liouville measure on S 1 M ([LY85]). Thus, proving Quantum Unique Ergodicity is equivalent to proving that h g ( 0) = j R S 1 M log J u d 0 j for any semi-classical invariant measure 0 . But already a lower bound on the entropy of 0 would be useful. Remember that one of the ingredients of Elon Lindenstrauss' work [Li03] in the arithmetic situation was an estimate on the entropy of semi-classical measures, proven previously by Bourgain and Linden-strauss [BLi03]. If the ( h ) form a common eigenbasis of the Laplacian and all the Hecke operators, they proved that all the ergodic components of 0 have pos-itive entropy (which implies, in particular, that 0 cannot put any weight on a closed geodesic). In the general case, our Theorems 1.1.1, 1.1.2 do not reach so far. They say that many of the ergodic components have positive entropy, but components of zero entropy, like closed geodesics, are still allowed { as in the counterexample built in [FNDB03] for linear hyperbolic toral automorphisms (called cat maps" thereafter). For the cat map, [BDB03] and [FN04] could prove directly { without using the notion of entropy { that a semi-classical measure cannot be entirely carried on closed orbits ([FN04] proves that if 0 has a pure point component then it must also have a Lebesgue component). Denote = sup v2S 1 M log J u (v) > 0: For instance, for a d-dimensional manifold of constant sectional curvature 1, we nd = d 1. Theorem 1.1.1. There exist a number > 0 and two continuous decreas-ing functions : [0; 1] ! [0; 1], # : (0; 1] ! R+ with (0) = 1, #(0) = +1, such that : If 0 is a semi-classical invariant measure, and 0 = Z