Continued fractions and Hecke triangle groups€¦ · Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions Front page Continued

$Page 1: Continued fractions and Hecke triangle groups€¦ · Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions Front page Continued$
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions

Front page

Continued fractions and Hecke triangle groups

Tobias MuhlenbruchJoint work with Dieter Mayer and Fredrik Stromberg

Lehrgebiet StochastikFernUniversitat Hagen

[email protected]

19 May 2010EU-Young and Mobile Workshop: Dynamical Systems and Number

Theory

http://www.fernuni-hagen.de/

http://www.dynamik.tu-clausthal.de/institute/

http://www3.mathematik.tu-darmstadt.de/hp/algebra-geometry-and-functional-analysis/stroemberg-fredrik/research.html

http://www.fernuni-hagen.de/stochastik/

http://www.fernuni-hagen.de

mailto:[email protected]

http://www.icms.org.uk/workshops/euyam

http://www.icms.org.uk/workshops/euyam


Outline of the presentation

1 Hecke triangle groups

2 Continued fractions

3 Coding of geodesics

4 Transfer operator

5 Model system

6 Conclusions


Hecke triangle groups

Hecke triangle groups Gq

Gq =⟨S ,Tq|(STq)q = 1

⟩⊂ PSL2(R) with S =

(0 −11 0

),

Tq =

(1 λq

0 1

), λq = 2 cos

(π/q

)and q = 3, 4, 5, . . ..

Mobius transformations:

(a bc d

)z =

az+bcz+d if z ∈ C,ac if z =∞.

Translation Tq : z 7→ z + λq and rotation S : z 7→ −1z .

Gq is a Fuchsian group of the first kind.

Gq\H is finite and non-compact.

limit set Gq∞ ⊂ Q(λq) ⊂ P1R is dense.

Gq\H is non-arithmetic for q 6= 3, 4, 6.

(Closed) fundamental domain Fq =z ∈ H; |z | ≥ 1, |Re(z)| ≤ λq

2

.


Nearest λ-multiple continued fractions

Recall Gq =⟨

S, Tq |(STq )q = 1⟩

with S =

(0 −11 0

), Tq =

(1 λq0 1

), λq = 2 cos

(π/q

)and q = 3, 4, 5, . . ..

Nearest λ-multiple continued fractions (λCF)

We identify a sequence of integers, a0 ∈ Z, and a1, a2, . . . ∈ Z? with a point

x = T a0q ST a1

q ST a2q · · · 0 = a0λ+

−1

a1λ+ −1

a2λ+−1...

=: [a0; a1, a2, . . .]q

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”.

Forbidden blocks are blocks of the form h =

q2− 1 if 2 | q,

q−32

if 2 - q.

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], [(−1)h,−2, (−1)h,−m].

for all m ∈ N. Hecke continued fractions


Nearest λ-multiple continued fractions

Exampleq π e −λq/23 [3;−7, 16, 294, 3, 4, 5, 15, . . .]3 [3; 4, 2,−5,−2, 7, 2,−9,−2, . . .]3 [0; 2]3

4 [2;−2, 2, 8,−4,−5,−1, 2, 3, . . .]4 [2; 6,−1, 3, 1,−2,−1, 1,−7, . . .]4 [0; 1]4

5 [2; 7, 1, 2,−1, 1,−1, 1,−3, . . .]5 [2; 1,−2,−14, 5, 1, 9,−2,−1, . . .]5 [0; 1, 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 [0; 1, 1]6

Equivalent points

x and y are Gq-equivalent :⇐⇒ ∃ g ∈ Gq such that g x = y⇐⇒

the CF of x and y have the same tail or

the CF of x and y have tail

q = 3 [ 3 ]3 or [−3 ]3,

even q [ 1h−1, 2 ]q or [ (−1)h−1,−2 ]q, or

odd q ≥ 5 [ 1h, 2, 1h−1, 2 ]q or [ (−1)h,−2, (−1)h−1,−2 ]q.

λCF’s admit natural lexicographic order ≺.


Associated dynamical system

The generating map fq

(x)q ∈ Z denotes the nearest λ-multiple of x :

|x − (x)qλ| ≤ λ2

.

For Iq =[−λ

2, λ

2

]the generating map is

f : Iq → Iq; x 7→ −1x−(−1

x

)qλ.

If we set x0 = − 1x

then the CF x = [a0; a1, . . .]q arecomputed by

an = (xn)q and xn+1 = f (xn) = −1xn− anλ.

Natural extension of fq

The natural extension of fq is (x = [0; a1, . . .]q)

Ωq → Ωq; (x , y) 7→(f (x), −1

y+a1λ

).


λCF and coding of geodesics

Recall Gq = 〈S ,Tq〉.

We denote geodesics γ on H by their basepoints: γ = (γ−, γ+).

In the diagram we illustrate the closedgeodesic γ = ([0;−3,−4], [0;−4,−3]−1).

γ? is closed ⇐⇒ γ− = [0; . . . , a1, . . . , an]q is regular andγ−1

+ = [0; . . . , an, . . . , a1]q is dual regular.

Equivalent geodesics

γ and γ′ are Gq-equivalent :⇐⇒∃ g ∈ Gq such that (g γ−, g γ+) = (γ′−, γ

′+)



Theorem

Each geodesic γ′ in H is Gq-equivalent with a geodesic γ = (γ−, γ+)satisfying (γ−, γ

−1+ ) ∈ Ω.

If γ and υ satisfy (γ−, γ−1+ ), (υ−, υ

−1+ ) ∈ Ω and no base point is

Gq-equivalent with

q = 3 [0; 3 ]3 or [0;−3 ]3,

even q [0; 1h−1, 2 ]q or [0; (−1)h−1,−2 ]q, or

odd q ≥ 5 [0; 1h, 2, 1h−1, 2 ]q or [0; (−1)h,−2, (−1)h−1,−2 ]q.

then γ = υ.

Closed geodesics γ have at most 2 Gq-equivalent geodesics “in Ω”.



TheoremThe following statements are equivalent:

γ? is a closed geodesic on Gq\H.

γ = (γ−, γ+) with γ− = [0; a1, . . . , an]q is regular andγ−1

+ = [0; an, . . . , a1]q is dual regular.

The map γ? 7→ γ is bijective except if [a1, . . . , an] is Gq-equivalent to

q = 3 [ 3 ]3 or [−3 ]3,

even q [ 1h−1, 2 ]q or [ (−1)h−1,−2 ]q, or

odd q ≥ 5 [ 1h, 2, 1h−1, 2 ]q or [ (−1)h,−2, (−1)h−1,−2 ]q.


General form of a transfer operator


Given a set Λ and maps f : Λ→ Λ and g : Λ→ C,a transfer operator L acting on functions h : Λ→ C is defined by(

Lh)(x) =

∑y∈f−1(x)

g(y) h(y)

Remarks:

Usually, take g = |J|−1 if the Jacobian J of f exists.

L of the form Lh(x) =∑

y∈f−1(x)

(f ′(y)

)−1h(y) is also known as a

Perron-Frobenius Operator.

Relation of L to the dynamical zeta-function: more

ζ(z) =1

detc (1− zL).

Related to the Ising 1d spin model. Ising Model


The transfer operator for continued fractions

Recall respectively define:

S =(0 −1

1 0

)and T =

(1 10 1

)∈ G3 = PSL2(Z),(

h∣∣

2s

(a bc d

))(z) :=

((cz + d)2

)−sh(

az+bcz+d

)and

f :[− 1

2 ,12

]→[− 1

2 ,12

]; x 7→ −1

x (mod 1) = T−nS x for some n ∈ Z.

The associated transfer operator is formally given by

Lsh(x) :=∑

y∈f−1(x)

(f ′(y)

)−sh(y) (x ∈ [−1/2, 1/2])

Banach space (with sup-norm) V := C(D) ∩ Cω(D), D = z ; |z | ≤ 1.

Transfer operator for continued fractions (q = 3)

The operator Ls : : V × V → V × V , Re(s) > 1, is defined as

Ls~h(z) =

(∑∞n=3 h1

∣∣2sST n +

∑∞n=2 h2

∣∣2sST−n∑∞

n=2 h1

∣∣2sST n +

∑∞n=3 h2

∣∣2sST−n

)for ~h =

(h1h2

)∈ V × V . Numerical example


The transfer operator for continued fractions

Recall

Ls : V 2 → V 2; Ls~h(z) =

(∑∞n=3 h1

∣∣2sST n +

∑∞n=2 h2

∣∣2sST−n∑∞

n=2 h1

∣∣2sST n +

∑∞n=3 h2

∣∣2sST−n

).

TheoremThe transfer operator Ls is nuclear of order 0.

The transfer operator Ls allows a meromorphic continuation into thecomplex s-plane.

Corollary

For almost all ~h ∈ V 2 the limit limn→∞ L1~h converges to the unique

invariant measure.


Setup

Model system

1 particle freely moving on Gq\H

Quantum-mechanical interpretation Maass cusp formsMaass cusp forms

Classical mechanical picture (closed) geodesics

Selberg trace formula:Quantum mechanical system ←→ Classical mechanical system∑

test function over spectral values ←→∑

test function over closed geodesics

∃ Maass cusp form ←→ Zeros of Selberg zeta-function


Geodesics on Gq\H

We denote geodesics γ on H by their basepoints: γ = (γ−, γ+).

The diagram illustrates a closed geodesic γ

Classical mechanics =⇒ geodesics on Gq\H.

The classical system is chaotic =⇒ closed geodesics are dense.

Each geodesic γ induces an geodesic γ? on Gq\H.

Each geodesic γ? on Gq\H induces class of geodesics Gq γ on H.

Coding of geodesics.(γ? on Gq\H is closed ⇐⇒ γ− = [0; a1, . . . , an]q , γ−1

+ = [0; an, . . . , a1]q


The Selberg zeta-function and the transfer operator

Selberg zeta-function

Z (s) :=∏ω

∞∏m=0

(1−

[e l(ω)

]−s−m)

(Re(s) > 1)

where ω runs over distinct primitive periodic geodesics ω andl(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 Riemannzero or s is a spectral parameter.

The trivial zeros of Z (s) are located at s = −l , l = 0,−1,−2, . . ..

Z(s)Z(1−s) = Φ(s)Ψ(s) with scattering matrix Φ(s) =

√π

Γ(s− 12 )ζ(2s−1)

Γ(s)ζ(2s)

(for q = 3) and a (computable) function Ψ.


The Selberg zeta-function and the transfer operator

Recall the Selberg zeta-function

Z (s) =∏ω

∞∏m=0

(1−

[e l(ω)

]−s−m).

Theorem

det(1− Ls) = Z (s) det(1−Ks)

where Ks is a simple operator with s → det(1−Ks) has no poles andsimple zeros in sn,k = n + 2πik

const , n ∈ Z≤0, k ∈ Z.

Corollary

For 0 < Re(s) < 1, 2s 6= 1:

Ls has eigenvalue 1 if and only if Z (s) = 0 or s = sn,k .

(for q = 3) Ls has eigenvalue 1 only in spectral parameters s,ζ(2s) = 0.

Remark: Connection between 1-eigenfunctions of Ls and period functions (for q = 3)

Period functions


A simplified overview

Maass cusp forms period functions

zeros of Z (s) eigenfunctions of Ls

with eigenvalue 1

- [LZ01] [BLZ]

- [MMS]

6

?

[Mar03, §14]

6

?

[BM09]

Advantages of the transfer operator approach:

Eigenfunctions of Ls can be calculated numerically, Z (s) is hard tocalculate in general.

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.


Thank you!

Thank you for your time!

References Abstract 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. Muhlenbruch,

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 Grossen,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.

http://dx.doi.org/10.1016/j.jnt.2008.08.003

http://gepris.dfg.de/gepris/OCTOPUS/;?module=gepris&task=showDetail&context=projekt&id=5443346

http://www.ams.org/mathscinet-getitem?mr=1106989

http://dx.doi.org/10.1007/BF02391885

http://dx.doi.org/10.1090/S0273-0979-06-01115-3


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” (SchlossReisensburg, Gunzburg, 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 Gunzburg, Germany, 4-11 October 2003.

[MM]: D. Mayer, T. Muhlenbruch,

Nearest λq-multiple fractions.

[MMS]: D. Mayer, T. Muhlenbruch, F. Stromberg,

The transfer operator for the Hecke triangle groups.

[MS08]: D. Mayer and F. Stromberg,

Symbolic dynamics for the geodesic flow on Hecke surfaces,Journal of Modern Dynamics 2 (2008), 581–627.

http://projecteuclid.org/euclid.annm/1026916788

http://arxiv.org/abs/math/0407288

http://www.maths.bris.ac.uk/~majm/bib/arithmetic.pdf

http://www.dynamik.tu-clausthal.de/research/preprints/Mayer.001.ps

http://arxiv.org/abs/0902.3953

http://arxiv.org/abs/0912.2236

http://dx.doi.org/10.3934/jmd.2008.2.581


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. Stromberg,

Computation of Selberg Zeta Functions on Hecke Triangle Groups,Preprint.

http://www.ams.org/mathscinet-getitem?mr=1402490

http://dx.doi.org/10.1215/S0012-7094-54-02154-7

http://www.ams.org/notices/200208/fea-ruelle.pdf

http://dx.doi.org/10.1007/BF02572621

http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg/research/pub/ComputingSelbergZ.pdf


Abstract of the talk

Classical mechanics and Maass cusp forms

I present the Nakada continuous fraction expansion which is of the form

λa0 +−1

λa1 + −1λa2+ −1

...

, λ = 2 cos

(π

q

), q = 3, 4, 5, . . . .

The coefficients ai are non-zero integers, satisfying certain conditions.We present some properties of these continued fractions and show thatthey can be used to code closed geodesics on the surface H/Gq, H beingthe upper complex plane and Gq the qth Hecke triangle group.As an application we demonstrate the construction of a transfer operator.It turns out this transfer operator can be used to express the Selberg zetafunction associated to H/Gq.


Maass cusp forms

Maass cusp form u

u : H→ C real-analytic function,

∆u = s(1− s) u with ∆ = −y 2(∂2

x + ∂2y

),

u(M z) = u(z) for all M ∈ Gq,

u(x + iy) = O(yC)

as y →∞ for all C ∈ R.

∆ admits self-adjoined extension in L2(Gq\H

).

s is called spectral parameter.

s(1− s) ≤ 14 , i.e., s ∈ 1

2 + iR?.

Discrete spectrum.(For G3 multiplicity 1 conjectured.)

Precise locations of eigenvalues are “not known”.

u(x + iy) =√y∑

n∈Z 6=0

an Ks− 12(2π |n| y) e

2πinxλq

Maass cusp form at

s = 12 + i13.78 . . . for G3.

Maass cusp form at

s = 12 + i9.533 . . . for G3.


Maass cusp forms

L2(Gq\H

)are realized by functions h satisfying

h : H→ C measurable,

h(M z) = f (z) for all M ∈ Gq and∫F |h(x + iy)|2 y−2dxdy <∞.

(s = 12 + i12.173 . . ., G3)

Some properties of ∆

The Laplace-Beltrami operator on L2(M) is the self-adjoined extension of ∆.

∆ can be viewed as a time-independent Schrodinger operator.

Spectrum of ∆ is discrete.

Eigenvalues λ = s(1− s) obey Weyl’s law: ]λ ≤ Λ ∼ Area(H)4π

Λ.

Conjectures about Eigenvalue statistics.

Quantum unique ergodicity.

Arithmetic properties for G3, G4 and G6:Hecke operators, associated L-series satisfy a GRH.

Model system


Hurwitz continued fractions

Hurwitz continued fractions (HCF)

We identify a sequence of integers, a0 ∈ Z, and a1, a2, . . . ∈ Z? with a point

x = T a0 ST a1 ST a2 · · · 0 = a0 +−1

a1 + −1

a2+−1...

=: [a0; a1, a2, . . .]

and say that it is a

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 exists 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 ].


Associated dynamical system

The generating map f

(x) ∈ Z denotes the nearest integer of x , i.e., |x − (x)| ≤ 12.

For I =[− 1

2, 1

2

]the generating map for the CF of x is

f : I → I ; x 7→ −1

x−(−1

x

).

The coefficients a0, a1, . . . computed by

a0 = (x) and xn+1 = f (xn) = −1xn− an+1

satisfy x = [a0; a1, . . .] and the CF is regular.

Natural extension of fThe natural extension of f is

Ω→ Ω; (x , y) 7→(f (x),

−1

y + a1

)with x = [0; a1, . . .].

Reference: [Hu89] Nearest λ-multiple continued fractions


The Ising model and transfer matrices

Ernst Ising (1900 – 1998) discussed an 1d lattice spin model.

Ising-Model

Config. space S = ±1N; left-shift τ : S → S , (τξ)i := ξi+1.Total energy: E = −J

∑∞i=1 ξiξi+1 + B

∑∞i=1 ξi

with spin interaction J and magnetic field interaction B.)Ernst Ising ≈ 1925.

Partition function Zm(A, s)

Zm(A, s) =∑ξ∈S ;

m−periodic

e−s∑m−1

k=0A(τkξ) with s = 1

Temp.and A(ξ) = J ξ0ξ1 + Bξ0.

free energy = limm→∞m−1 logZm(A, s).

Ising rewrote Zm as

Zm(A, s) =∑

ξ1,...,ξm∈±1

e(ξ1, ξ2) · · · e(ξm, ξ1) with e(ξi , ξj ) = e−s(Jξiξj−Bξi ).

Transfer matrix Ls

Ls :=

(e(+1,+1) e(+1,−1)e(−1,+1) e(−1,−1)

)satisfies Zm(A, s) = trace

(Lm

s

).



The dynamical zeta function and the the counting determinant

Recall the transfer operator LΦ(x) =∑

y∈f−1(x) g(y) h(y).

Define the counting trace tracec (L) =∑

x∈Fix(f ) g(x) and

the counting determinant detc(1− zL

)= exp

(−∑∞

m=1zm

m

(tracec (L)

)m).

Dynamical zeta function

The dynamical zeta function for a dynamical system (S , f ) is

ζ(z) = exp

∞∑m=1

zm

m

∑x∈Fix(f m)

m−1∑k=0

g(f k (x)

) .

ζ(z) =1

detc(1− yL

)Reference: [Ru02] Transfer operators


Spectrum of the transfer operator

Spectrum of the transfer operator Ls for Hecke triangle group G5 ands = 1

2 + iR, R ∈ [6, 14]:

Movie

Transfer operators


Period functions and the transfer operator

Period function P

P : C′ := Cr (−∞, 0]→ C holomorphic,

P(z) = P(z + 1) + (z + 1)−2sP(−1z+1

)(three-term equation) and

P(z)

|Im(z)|−C

(1 + |z |2C−2Re(s)

)if Re(z ≤ 0),

1 if Re(z) > 0, |z | ≤ 1 and

|z |−2Re(s) 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.


Period functions and the transfer operator

Let ~h =(

h1

h2

)be an eigenfunction of Ls with eigenvalue 1.

∃g ∈ Cω(−r − 1, r), r = 1+√

52 , s.th. g = h1, g

∣∣2sT−1 = h2 on [−1, 1].

g satisfies the relation

g = g∣∣2s

∑∞n=3 ST

n + g∣∣2s

∑∞n=2 T

−1ST−n

and on (−r , r) the 4-term equation

g∣∣2s

(1 + ST 2

)= g

∣∣2s

(T−1 + T−1ST−2

).

Theorem ([BM09])

Assume q = 3, 0 < Re(s) < 1 and 2s 6= 1.Eigenfunctions of Ls with eigenvalue 1 correspond to (Lewis-Zagier)period functions.

Selberg zeta-function and transfer operator

Documents

Continued fractions and Hecke triangle groups€¦ · Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions Front page Continued