Quantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

Transcription

1 Quantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces Etienne Le Masson (Joint work with Tuomas Sahlsten) School of Mathematics University of Bristol, UK August 26, 2016

2 Hyperbolic Plane Upper half plane model H = {z = x + iy C y > 0} ds 2 = dx 2 + dy 2 y 2 dµ(z) = dx dy y 2 Isometry group: SL(2, R) acting via Möbius transformations Poincaré disk model H = {z = x + iy C z < 1} ds 2 = 4 dx 2 + dy 2 (1 z 2 ) 2 dµ(z) = 4 dx dy (1 z 2 ) 2 Isometry group: SU(1, 1) acting via Möbius transformations

3 Hyperbolic Surfaces Hyperbolic surface: Quotient Γ\H of the plane by a discrete subgroup of isometries. Surface of constant negative curvature 1. Example in the disk model

4 Spectrum of the Laplacian Let M = Γ\H be a compact hyperbolic surface. the Laplacian. In the upper-half plane model = y 2 ( 2 x + 2 y ). There exists an orthonormal basis (ψ j ) j N of eigenfunctions in L 2 (M). ψ j = λ j ψ j λ 0 λ 1... λ j +

5 Quantum ergodicity (quantitative version) Theorem (Zelditch 94) On compact hyperbolic surfaces, for any function a C (M), lim λ + 1 N(λ) ψ j, a ψ j a dµ λ j λ where N(λ) is the number of eigenvalues λ. 2 = O(log(λ) 1 ),

6 Quantum ergodicity (quantitative version) Theorem (Zelditch 94) On compact hyperbolic surfaces, for any function a C (M), lim λ + 1 N(λ) ψ j, a ψ j a dµ λ j λ where N(λ) is the number of eigenvalues λ. More generally: for any a C (M S 1 ) lim λ + 1 N(λ) ψ j, Op(a) ψ j a(z, θ) dµ(z)dθ λ j λ where Op(a) is a pseudo-differential operator of order 0. 2 = O(log(λ) 1 ), 2 = O(log(λ) 1 ),

7 Quantum ergodicity (an alternative approach) ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Question Equidistribution theory of eigenfunctions for bounded eigenvalues instead of large eigenvalues λ j

8 Quantum ergodicity (an alternative approach) ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Question Equidistribution theory of eigenfunctions for bounded eigenvalues instead of large eigenvalues λ j Fix any bounded interval I (1/4, + ) (fixed spectral window)

9 Quantum ergodicity (an alternative approach) ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Question Equidistribution theory of eigenfunctions for bounded eigenvalues instead of large eigenvalues λ j Fix any bounded interval I (1/4, + ) Consider just ψ j with λ j I. (fixed spectral window)

10 Quantum ergodicity (an alternative approach) ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Question Equidistribution theory of eigenfunctions for bounded eigenvalues instead of large eigenvalues λ j Fix any bounded interval I (1/4, + ) Consider just ψ j with λ j I. Now vary the surface M (e.g. volume, genus, injectivity radius etc.) (fixed spectral window)

11 Quantum ergodicity (an alternative approach) ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij. Question Equidistribution theory of eigenfunctions for bounded eigenvalues instead of large eigenvalues λ j Fix any bounded interval I (1/4, + ) Consider just ψ j with λ j I. Now vary the surface M (e.g. volume, genus, injectivity radius etc.) (fixed spectral window) What happens to ψ j 2 with λ j I as we vary the geometry of M?

12 Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1.

13 Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1. Pointwise injectivity radius of M at x: InjRad(M, x) is the largest R > 0 such that B(x, R) M is isometric to a ball in H.

14 Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1. Pointwise injectivity radius of M at x: InjRad(M, x) is the largest R > 0 such that B(x, R) M is isometric to a ball in H.

15 Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1. Pointwise injectivity radius of M at x: InjRad(M, x) is the largest R > 0 such that B(x, R) M is isometric to a ball in H.

16 Example: injectivity radius Let M be a hyperbolic surface, i.e. constant negative curvature 1. Pointwise injectivity radius of M at x: InjRad(M, x) is the largest R > 0 such that B(x, R) M is isometric to a ball in H. Injectivity radius of M is then: InjRad(M) := inf InjRad(M, x) x M

17 Example: injectivity radius Increase InjRad(M) = More eigenvalues λ j I.

18 Example: injectivity radius Increase InjRad(M) = More eigenvalues λ j I.

19 Example: injectivity radius Increase InjRad(M) = More eigenvalues λ j I.

20 Example: injectivity radius Increase InjRad(M) = More eigenvalues λ j I.

21 Equidistribution for large injectivity radius ψ j = λ j ψ j, 0 λ 1 λ 2..., ψ i, ψ j = δ ij, I (1/4, ). Do most ψ j 2, λ j I, equidistribute when InjRad(M) is large? Problem (Colin de Verdière 2010s) Fix a bounded interval I (1/4, ). Is it true that 1 {λ j I } j:λ j I ψ j, a ψ j a dµ 2 0 as InjRad(M)? (assuming a uniform spectral gap condition)

22 Motivation Quantum Ergodicity on graphs

23 Motivation Quantum Ergodicity on graphs QUE in the level aspect

24 Arithmetic example: the level aspect Coverings X q of the modular surface SL(2, Z)\H X q = Γ 0 (q)\h where {( ) a b Γ 0 (q) = SL c d 2 (Z) : c 0 } mod q Theorem (Nelson 2012, Nelson-Pitale-Saha 2014) Holomorphic forms of fixed weight and increasing level q equidistribute. (QUE) The question is open for Maass forms, i.e. eigenfunctions of the Laplacian.

25 Benjamini-Schramm convergence A sequence of hyperbolic surfaces (M n ) Benjamini-Schramm converges to H if for any R > 0, Vol({x M n : InjRad(M n, x) < R}) lim = 0 n Vol(M n )

26 Benjamini-Schramm convergence A sequence of hyperbolic surfaces (M n ) Benjamini-Schramm converges to H if for any R > 0, Vol({x M n : InjRad(M n, x) < R}) lim = 0 n Vol(M n ) Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault, Samet: On the growth of L 2 -invariants for sequences of lattices in Lie groups. Preprint. arxiv: , 98 pp

27 To simplify, we state the theorem for large injectivity radius, but the ideas adapt to Benjamini-Schramm convergence Theorem (LM-Sahlsten 2016) Fix a closed interval I (1/4, + ) and let a L 2 (M), ψ j, a ψ j a dµ λ j I where ρ(β) is a function of the spectral gap β. 2 I a 2 2 ρ(β) InjRad(M),

28 To simplify, we state the theorem for large injectivity radius, but the ideas adapt to Benjamini-Schramm convergence Theorem (LM-Sahlsten 2016) Fix a closed interval I (1/4, + ) and let a L 2 (M), ψ j, a ψ j a dµ λ j I where ρ(β) is a function of the spectral gap β. 2 I a 2 2 ρ(β) InjRad(M), For a sequence M n with uniform spectral gap and such that InjRad(M n ) + we have 1 {λ j I } ψ 2 j, a n ψ j a n dµ 0 λ j I for any sequence of test functions a n L 2 (M n ).

29 Examples Compact arithmetic surfaces of congruence type when the level tends to infinity. (Abert et al. On the growth of L 2 -invariants for sequences of lattices in Lie groups. Preprint. arxiv: ) Large random Riemann surfaces for a natural model satisfy the conditions almost surely (R. Brooks and E. Makover: Random Construction of Riemann Surfaces, J. Differential Geom. 68(1), 2004.)

30 Reduction to a Hilbert-Schmidt estimate Replace the wave propagator with renormalized averages over discs P t u(z) = e t/2 u(w)dµ(w) D(z,t)

31 Reduction to a Hilbert-Schmidt estimate Replace the wave propagator with renormalized averages over discs P t u(z) = e t/2 u(w)dµ(w) D(z,t) Proposition (Spectral side) If a dµ = 0 then by estimating the spherical transform of the kernel of P t we have for any T λ j I ψ j, a ψ j 2 1 T I T 0 P t a P t dt 2 HS

32 Geometric side Kernel of P t a P t : [P t a P t ](z, z ) = e t D(z,t) D(z,t) a(w) dµ(w)

33 Geometric side Kernel of P t a P t : [P t a P t ](z, z ) = e t D(z,t) D(z,t) a(w) dµ(w) We want to compute 1 T T 0 P t a P t dt 2 HS = 1 T T 0 2 [P t a P t ](z, z ) dµ(z) dµ(z )

34 Nevo s mean ergodic theorem Gorodnik, Nevo: The ergodic theory of lattice subgroups, Annals of Mathematics Studies 172, Princeton University Press, 2010 Theorem (Nevo) Let (F t ) be a family of measurable sets of positive measure and π(f t ) the averaging operator over F t (convolution with the characteristic function of the set), then π(f t) f M f dµ 2 1 µ(f t ) ρ(β) f 2.

35 Sketch of proof We propagate only up to the radius of injectivity T InjRad(M) Almost orthogonality 1 T T 0 P t a P t dt 2 HS = 1 T T 2 P t a P t 2 HS dt 0

36 Sketch of proof We propagate only up to the radius of injectivity T InjRad(M) Almost orthogonality 1 T T 0 P t a P t dt 2 HS = 1 T T 2 P t a P t 2 HS dt 0 Using Nevo s theorem P t a P t 2 HS a 2 2

37 Sketch of proof We propagate only up to the radius of injectivity T InjRad(M) Almost orthogonality 1 T T 0 P t a P t dt 2 HS = 1 T T 2 P t a P t 2 HS dt 0 Using Nevo s theorem P t a P t 2 HS a 2 2 Finally 1 T T 0 P t a P t dt 2 HS 1 T T 2 a 2 2 dt a T

38 Perspectives Phase space version

39 Perspectives Phase space version More general manifolds, variable curvature, Maass forms

40 Perspectives Phase space version More general manifolds, variable curvature, Maass forms Quantum Unique Ergodicity: arithmetic QUE, entropy results