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
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 . Numerical example h2 ∈ × Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
The transfer operator for continued fractions
Recall
∞ h ST n + ∞ h ST n 2 2 ~ n=3 1 2s n=2 2 2s − s : V V ; s h(z)= n n . L → L P∞ h1 ST + P∞ h2 ST − n=2 2s n=3 2s P P
Theorem The transfer operator is nuclear of order 0. Ls The transfer operator s allows a meromorphic continuation into the complex s-plane. L
Corollary 2 For almost all ~h V the limit limn 1~h converges to the unique invariant measure.∈ →∞ L Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
The Selberg zeta-function and the transfer operator
Selberg zeta-function
∞ s m Z(s) := 1 el(ω) − − (Re(s) > 1) − Yω mY=0 h i { } where ω runs over distinct primitive periodic geodesics ω and l(ω) is{ its} length.
Properties of Z(s) Z(s) can be analytically continued to an entire function. The non-trivial zeros of Z(s) are located at s =1, 2s is Riemann zero or s is a spectral parameter. The trivial zeros of Z(s) are located at s = l, l =0, 1, 2,.... − − − Z(s) Γ(s 1 )ζ(2s 1) − 2 − Z(1 s) = Φ(s)Ψ(s) with scattering matrix Φ(s)= √π Γ(s)ζ(2s=) and− a (computable) function Ψ. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
The Selberg zeta-function and the transfer operator
Recall the Selberg zeta-function
∞ s m Z(s)= 1 el(ω) − − . − Yω mY=0 h i { }
Theorem det(1 )= Z(s) det(1 ) − Ls − Ks
where s is a simple operator with s det(1 s ) has no poles and K 2πik → − K simple zeros in sn,k = n + const , n Z 0, k Z. ∈ ≤ ∈ Corollary For 0 < Re(2s) < 1, 2s =1: 6 has eigenvalue 1 if and only if Z(s)=0 or s = s , . Ls n k has eigenvalue 1 only in spectral parameters s, ζ(2s)=0. Ls Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
Period functions and the transfer operator
Period function P
P : C′ := Cr ( , 0] C holomorphic, −∞ → 2s 1 P(z)= P(z +1)+(z + 1)− P z−+1 (three-term equation) and C 2C 2Re(s) Im(z) − 1+ z − if Re(z 0), | | | | ≤ P(z) Re z z ≪ 1 if ( ) > 0, 1 and 2Re(s) | |≤ z − if Re(z) > 0, z 1. | | | |≥ The period function P depends implicitly on s.
Theorem ([LZ01]) For Re(s) > 0: P is period function s is spectral parameter of a Maass cusp form. ⇐⇒ Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
Period functions and the transfer operator
h1 Let ~h = be an eigenfunction of s with eigenvalue 1. h2 L ω 1+√5 1 g C ( r 1, r), r = , s.th. g = h , g T − = h on [ 1, 1]. ∃ ∈ − − 2 1 2s 2 −
g satisfies the relation
n 1 n g g ∞ ST g ∞ T ST = 2s n=3 + 2s n=2 − − P P and on ( r, r) the 4-term equation −
g ST 2 g T 1 T 1ST 2 2s 1+ = 2s − + − − .
Theorem (R.W. Bruggeman, M)
For 0 < Re(2s) < 1, 2s =1 eigenfunctions of s with eigenvalue 1 correspond to Lewis-Zagier6 periodfunctions. L Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
A simplified overview
[LZ01] [BLZ]- Maasscuspforms periodfunctions 6 6 [BM09] ? ? [MMS] - zeros of Z(s) eigenfunctionsof s with eigenvalue 1 L
Advantages of the transfer operator approach:
Eigenfunctions of s can be calculated numerically, Z(s) is hard to calculate in general.L Extension to (non-arithmetic) Hecke triangle groups. Direct connection form eigenfunctions to period functions. Gives different approach to study problems in “quantum chaos”. Changes interpretation of s: from spectral value to a “weight-type” parameter. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions
Thank you!
Thank you for your time! References Additional things
References
[BLZ]: R.W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology, in preparation. [BM09]: R.W. Bruggeman and T. M¨uhlenbruch, Eigenfunctions of transfer operators and cohomology, J. Number Theory 129 (2009) 158–181. [DFG]: DFG Research Project “Transfer operators and non arithmetic quantum chaos” (Ma 633/16-1). [He83]: D.H. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol.2, Lecture Notes in Mathematics 1001, Springer-Verlag, 1983. [He92]: D.H. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992). [Hu89]: A. Hurwitz, Uber¨ eine besondere Art der Kettenbruch-Entwicklung reeller Gr¨ossen, Acta Mathematica 12 (1889), 367–405. [KU07]: S. Katok,and I. Ugarcovic, Symbolic dynamics for the modular surface and beyond, Bulletin of the American Mathematical Society 44 (2007), 87–132. References Additional things
References
[LZ01]: J. Lewis and D. Zagier, Period functions for Maass wave forms. I , Annals of Mathematics 153 (2001), 191–258. [Mar03]: J. Marklof, Selberg’s trace formula: an introduction, Proceedings of the International School “Quantum Chaos on Hyperbolic Manifolds” (Schloss Reisensburg, G¨unzburg, Germany, 4-11 October 2003). [Mar06]: J. Marklof, Arithmetic quantum chaos, Encyclopedia of Mathematical Physics, editors J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford, Elsevier, 2006, Volume 1, pp. 212–220. [Ma03]: D. Mayer, Transfer operators, the Selberg-zeta function and Lewis-Zagier theory of period functions, Lecture notes of a course given in G¨unzburg, Germany, 4-11 October 2003. [MS08]: D. Mayer and F. Str¨omberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, Journal of Modern Dynamics 2 (2008), 581–627. [MMS]: D. Mayer, T. M¨uhlenbruch and F. Str¨omberg, Nearest λ-multiple fractions, In preparation. References Additional things
References
[Na95]: H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, Fractals, and Dynamics, Edited by T. Takahashi, Plenum Press, New York, 1995. [Ro54]: D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Mathematical Journal 21 (1954), 549–563. [Ru78]: D. Ruelle, Thermodynamic formalism, 2nd edition, Cambridge University Press, 2004. [Ru02]: D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc. 49 (2002), no. 8, 887–895. [SS95]: T.A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Mathematische Zeitschrift 220 (1995), 369–397. [Str]: F. Str¨omberg, Computation of Selberg Zeta Functions on Hecke Triangle Groups, Preprint. References Additional things
Nearest λ-multiple continued fractions and the interval map fq
Forbidden block are blocks of the form q = 3 [1], [−1], [2, m] and [−2, −m], even q [1h+1], [(−1)h+1], [1h, m] and [(−1)h, −m], odd q ≥ 5 [1h+1], [(−1)h+1], [1h, 2, 1h, m] and [(−1)h, −2, (−1)h, −m].
q 2 − 1 if 2 | q, for all m ∈ N and h = hq = q−3 ( 2 if 2 ∤ q.
Nearest λ-multiple continued fractions (λCF) ⋆ We identify a sequence of integers, a0 ∈ Z, and a1, a2,... ∈ Z with a point
a0 a1 a2 −1 x = TQ STq STq · · · 0= a0λ + −1 =: [a0; a1, a2,...]q a1λ + −1 a2λ+ ... and say that it is a
regular λCF, [a0; a1, a2,...]q , if it does not contain “forbidden blocks”, and
dual regular λCF, [a0; a1, a2,...]q , if it does not contain reversed “forbidden blocks”. References Additional things
Nearest λ-multiple continued fractions and the interval map fq
Example q π e 3 [3; −7, 16, 294, 3, 4, 5, 15,...]3 [3; 4, 2, −5, −2, 7, 2, −9, −2,...]3 4 [2; −2, 2, 8, −4, −5, −1, 2, 3,...]4 [2; 6, −1, 3, 1, −2, −1, 1, −7,...]4 5 [2; 7, 1, 2, −1, 1, −1, 1, −3,...]5 [2; 1, −2, −14, 5, 1, 9, −2, −1,...]5 6 [2; 2, 2, 1, 2, 1, −1, −3, −1,...]6 [2; 1, 1, −1, −1, 3, 5, 1, 1, −7,...]6
Equivalent points x and y are equivalent : g G such that g x = y ⇐⇒ ∃ ∈ q ⇐⇒ the CF of x and y have the same tail or the CF of x and y have tail q =3 [ 3] or [ 3 ], − even q [ 1h 1, 2] or [ ( 1)h 1, 2 ], or − − − − odd q 5 [ 1h 1, 2, 1h, 2] or [ ( 1)h 1, 2, ( 1)h, 2 ]. ≥ − − − − − − References Additional things
Nearest λ-multiple continued fractions and the interval map fq
The generating map fq
λ (x)q ∈ Z denotes the nearest λ-multiple of x, i.e. |x − (x)qλ|≤ 2 . The generating map for the regular CF of x is −1 −1 λ f : Iq → Iq ; x 7→ x − x q . 1 ` ´ If we set x0 = − x then the CF x = [a0; a1,...]q are computed by −1 an =(xn)q and xn+1 = f (xn)= − anλ. xn
The natural extension of fq is
−1 Ω → Ω ; (x, y) 7→ f (x), q q y + a λ „ 1 «
with x = [0; a1,...]q .
[0; a , a ,...] , [0; a , a ,...]⋆ ([0; a ,...] , [0; a , a , a ,...]⋆). ⇒ “ 1 2 q 0 −1 q ” 7→ 2 q 1 0 −1 3 ⋆ where [0; a0,...]3 denotes dual regularCFs (no reversed forbidden block appear). Bijection to a sofic symbolic left shift on two sided sequences. ⇒ Purely periodic nearest integer CFs ⇐⇒ equivalence classes of closed geodesics
PSL Z 2( ) and geodesics References Additional things
The dynamical zeta function and the the counting determinant
Recall the transfer operator LΦ(x)= y∈f −1(x) g(y) h(y). P counting trace and counting determinant c Define the counting trace trace (L)= x∈Fix(f ) g(x) and m c 1 ∞ zm c the counting determinant det − zLP= exp − m=1 m trace (L) . ` ´ “ P “ ” ” Dynamical zeta function The dynamical zeta function for a dynamical system (S, f ) is
∞ m−1 z m ζ(z) = exp g f k (x) . 0 m 1 m=1 x∈Fix(f m) k=0 X X X ` ´ @ A
1 ζ(z)= detc 1 − yL ` ´ Reference: [Ru02] Transfer operators References Additional things
Spectrum of the transfer operator
Spectrum of the transfer operator s for Hecke triangle group G5 and s = 1 + iR, R [6, 14]: L 2 ∈
Transfer operators Movie