Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Front page The transfer operator approach to “quantum chaos” Classical mechanics and the Laplace-Beltrami operator on PSL (Z) H 2 \ Tobias M¨uhlenbruch Joint work with D. Mayer and F. Str¨omberg Institut f¨ur Mathematik TU Clausthal [email protected] 22 January 2009, Fernuniversit¨at in Hagen Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Outline of the presentation 1 Modular surface 2 Spectral theory 3 Geodesics and coding 4 Transfer operator 5 Connection to spectral theory 6 Conclusions Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z The full modular group 2( ) Let PSL2(Z) be the full modular group PSL (Z)= S, T (ST )3 =1 = SL (Z) mod 1 2 h | i 2 {± } 0 1 1 1 1 0 with S = − , T = and 1 = . 1 0 0 1 0 1 a b az+b if z C, M¨obius transformations: z = cz+d ∈ c d a if z = . c ∞ Three orbits: the upper halfplane H = x + iy; y > 0 , the 1 { } projective real line PR and the lower half plane. z , z are PSL (Z)-equivalent if M PSL (Z) with 1 2 2 ∃ ∈ 2 M z1 = z2. The full modular group PSL2(Z) is generated by 1 translation T : z z + 1 and inversion S : z − . 7→ 7→ z (Closed) fundamental domain = z H; z 1, Re(z) 1 . F ∈ | |≥ | |≤ 2 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z H The modular surface 2( )\ The upper half plane H = x + iy; y > 0 can be viewed as a hyperbolic plane with constant{ negative} curvature 1. − d d 2 dx2+dy 2 Line element s s = y 2 d d dxdy Volume element A A = y 2 The M¨obius transformations are compatible with the hyperbolic metric. This way, it makes sense to speak of the modular surface PSL (Z) H. 2 \ Gluing along the identified edges, we obtain a realization of the modularF surface PSL (Z) H, a non-compact, finite surface with one 2 \ cusp at z i and two conic singularities at z = i and z = 1+i√3 . → ∞ 2 The hyperbolic plane H is also called the Poincar´ehalf plane. PSL (Z) H is arithmetic ( connection to number theory). 2 \ Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Hecke triangle groups Possible extension of PSL2(Z): Hecke triangle groups Gq G = S, T (ST )q =1 with q h q| q i 1 λ T = q , λ = 2cos π/q and q =3, 4, 5,.... 0 1 q Example: PSL2(Z)= G3. M¨obius transformation extends to Gq. Fundamental domain Re λq q = z H; z 1, (z) 2 . F n ∈ | |≥ | |≤ o Fundamental domain forms a (2, q, ) π π∞ hyperbolic triangle with angles 2 , q and 0. G H is finite and non-compact. q\ G H is non-arithmetic for q =3, 4, 6. q\ 6 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Maass cusp forms Maass cusp form u u : H C real-analytic function, → ∆u = s(1 s) u with ∆ = y 2 ∂2 + ∂2 , − − x y u(M z)= u(z) for all M PSL (Z), ∈ 2 u(x + iy)= y C as y for all C R. O → ∞ ∈ Maass cusp form at s = 1 + i13.7797 .... ∆ admits self-adjoined extension in 2 L2 PSL (Z) H . 2 \ s is called spectral parameter. s(1 s) > 1 , i.e., s 1 + iR⋆. − 4 ∈ 2 Discrete spectrum, eigenvalues have finite multiplicity, It is assumed that eigenvalues have multiplicity 1. Precise location of eigenvalues is unknown. 2πinx Maass cusp form at u(x + iy)= √y 0=n Z an Ks 1 (2π n y) e 6 ∈ − 2 | | s = 1 + i9.533 .... P 2 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The Laplace-Beltrami operator L2 PSL (Z) H are realized by functions h satisfying 2 \ h : H C measurable, → h(M z)= f (z) for all M PSL (Z) and ∈ 2 2 2 h(x, iy) y − dxdy < . | | ∞ RF Some properties of ∆ The Laplace-Beltrami operator on L2(M) is the self-adjoined extension of ∆. Spectrum of ∆ is discrete. Area(H) Eigenvalues λ = s(1 − s) obey Weyl’s law: ♯{λ ≤ Λ}∼ 4π . Conjectures about Eigenvalue statistics. Quantum unique ergodicity. Arithmetic properties: Hecke operators, associated L-series satisfy a GRH. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Hurwitz continued fractions Definition (Hurwitz continued fractions (CF)) ⋆ We identify a sequence of integers, a0 ∈ Z, and a1, a2,... ∈ Z with a point a0 a1 a2 −1 x = T ST ST · · · 0= a0 + −1 =: [a0; a1, a2,...] a1 + −1 a2+ ... and say that it is a non-regular (formal) CF, [a0; a1, a2,...] in general. regular CF, [a0; a1, a2,...], if it does not contain “forbidden blocks”: no ±1 appear and if ai = ±2 then ai+1 ≶ 0. π = [3; 7, 16, 294, 3, 4, 5, 15,...] and e = [3;4, 2, 5, 2, 7, 2, 9,...] − − − − Equivalent points x and y are equivalent :⇔ there exist a g ∈ PSL2(Z) such that gx = y ⇔ the CF of x and y have the same tail or the CF of x and y have tail [ 3] and [ −3 ]. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Associated dynamical system The generating map f (x) Z denotes the nearest integer of x, i.e. x (x) 1 . ∈ | − |≤ 2 For I = 1 , 1 the generating map for the CF of x is − 2 2 1 1 f : I I ; x − − . → 7→ x − x The coefficients a0, a1,... computed by 1 a0 = (x) and xn+1 = f (xn)= − an+1 xn − satisfy x =[a0; a1,...] and the CF is regular. Natural extension of f The natural extension of f is 1 Ω Ω; (x, y) f (x), − → 7→ y + a1 with x = [0; a1,...]. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z 2( ) and geodesics Recall PSL (Z)= S, T . 2 h i We denote geodesics γ on H by its base points: γ = (γ ,γ+). − In the diagram we illustrate the closed geodesic 1 γ = ([0; 3, 4], [0; 4, 3]− ). − − − − Theorem Each geodesic γ′ is PSL2(Z)-equivalent with a geodesic 1 γ = (γ ,γ+) satisfying (γ ,γ+− ) Ω. − − ∈ If γ′ is closed then γ = [0; a1,..., an] is regular and 1 − γ+− = [0; an,..., a1] 1 1 If γ and υ satisfy (γ ,γ+− ), (υ ,υ+− ) Ω and no base point is − − ∈ PSL 3 √5 2(Z)-equivalent with [0; 3] = −2 then γ = υ. Nearest λ-multiple continued fractions Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The Ising model and transfer matrices Ernst Ising (1900 – 1998) discussed an 1 − d lattice spin model. Ising-Model N Config. space S = {±1} ; left-shift τ : S → S, (τξ)i := ξi+1. ∞ ∞ Total energy: E = −J i=1 ξi ξi+1 + B i=1 ξi with spin interaction J and magnetic field interaction B.) P P Ernst Ising ≈ 1925. Partition function Zm(A, s) m−1 τ k ξ −s k=0 A( ) 1 Zm(A, s)= e P with s = Temp. and A(ξ)= J ξ0ξ1 + Bξ0. ξ∈S; m−Xperiodic −1 free energy = limm→∞ m log Zm(A, s). Ising rewrote Zm as −s(Jξi ξj −Bξi ) Zm(A, s)= e(ξ1,ξ2) · · · e(ξm,ξ1) with e(ξi ,ξj )= e . ξ1,...,ξXm ∈{±1} Transfer matrix Ls e(+1, +1) e(+1, −1) L := satisfies Z (A, s) = trace Lm . s e(−1, +1) e(−1, −1) m s „ « ` ´ Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions General form of a transfer operator General form of a transfer operator Given a set Λ and maps f : Λ Λ and g : Λ C, a transfer operator acting on→ functions h :→ Λ C is defined by L → h (x)= g(y) h(y) L y Xf −1(x) ∈ Remarks: 1 Usually, take g = J − if the Jacobian J of f exists. | | 1 of the form h(x)= −1 f ′(y) − h(y) is also known as a L L y f (x) Perron-Frobenius OperatorP .∈ Relation of to the dynamical zeta-function: more L 1 ζ(z)= . detc (1 z ) − L Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The transfer operator for continued fractions Recall respectively define: 0 −1 1 1 PSL S = 1 0 and T = 0 1 ∈ 2(Z), ` a b ´ ` ´2 −s az+b h 2s c d (z) := (cz + d) h cz+d and `f :˛ −` 1 ,´´1 → −`1 , 1 ; x ´7→ −1 `(mod´ 1) = T −nS x for some n ∈ Z. ˛ 2 2 2 2 x ˆ ˜ ˆ ˜ The associated transfer operator is formally given by s h(x) := −1 f ′(y) − h(y) (x [ 1/2, 1/2]) Ls y f (x) ∈ − P ∈ ω Banach space (with sup-norm) V := C(D) C (D◦), D = z; z 1 . ∩ { | |≤ } Transfer operator for continued fractions The operator : : V V V V , Re(s) > 1, is defined as Ls × → × ∞ h ST n + ∞ h ST n ~ n=3 1 2s n=2 2 2s − s h(z)= n n L P∞ h1 ST + P∞ h2 ST − n=2 2s n=3 2s P P h for ~h = 1 V V .