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Mayer's Transfer Operator Approach to Selberg's Zeta Function

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Mayer's Transfer Operator Approach to Selberg's Zeta Function Algebra i analiz St. Petersburg Math. J. Tom 24 (2012), 4 Vol. 24 (2013), No. 4, Pages 529–553 S 1061-0022(2013)01252-0 Article electronically published on May 24, 2013 MAYER’S TRANSFER OPERATOR APPROACH TO SELBERG’S ZETA FUNCTION A. MOMENI AND A. B. VENKOV Abstract. These notes are based on three lectures given by the second author at Copenhagen University (October 2009) and at Aarhus University, Denmark (Decem- ber 2009). Mostly, a survey of the results of Dieter Mayer on relationships between Selberg and Smale–Ruelle dynamical zeta functions is presented. In a special sit- uation, the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions, and its matrix representation in a natural basis is given in terms of the Riemann zeta function and the Euler gamma function. Contents §1. General theory 529 §2. Mayer’s transfer operator for PSL(2, Z) 531 2.1. Matrix representation of Mayer’s transfer operator, its eigenvectors and eigenvalues 534 2.2. Nuclear spaces, nuclear operators, and Grothendieck’s theory 535 §3. Integral representation of Mayer’s transfer operator 538 §4. Calculation of the trace 540 4.1. Calculation of the trace via the integral representation 543 §5. Ruelle’s zeta function and transfer operator 544 §6. Selberg’s zeta function and transfer operator 546 §7. Number theoretic approach to the relationship between Selberg’s zeta function and Mayer’s transfer operator 549 References 552 §1. General theory We quote Ruelle [14, 15] to introduce his general notion of a transfer operator and a dynamical zeta function for a given dynamical system. First, we give the definition of a weighted dynamical system. Let Λ be a set weighted by a function g :Λ−→ C. Assume that Λ describes a system, then the dynamics of the 2010 Mathematics Subject Classification. Primary 11M36, 11M41. Key words and phrases. Mayer’s transfer operator, Selberg’s zeta function. The authors would like to thank Dieter Mayer for several important remarks and we would like to say also that all possible mistakes in the text belong to us but not to Mayer’s theory we presented in this paper. This work was supported by DAAD, the International Center of TU Clausthal, and the Danish National Research Foundation Center of Excellence, Center for Quantum Geometry of Moduli Spaces(QGM). c 2013 American Mathematical Society 529 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 530 A. MOMENI AND A. B. VENKOV system is given by a map F :Λ−→ Λ. The triplet D := (Λ,F,g) is called a weighted dynamical system or simply a dynamical system. The transfer operator method is applicable if the map F is not invertible, that is, for example, when its inverse is not unique. More precisely, the set of inverse branches of F must be finite or at least countable and discrete in a natural topology. For such a dynamical system, the action of the Ruelle transfer operator L on a function f :Λ−→ C is defined by (1.1) (Lf)(x)= g(y)f(y). y∈F −1{x} Let the set of transfer operators for all dynamical systems of the set Λ with respect to the product ◦ given by (L1 ◦L2)f = L1(L2f) be an algebra denoted by S.Atraceon this algebra is a linear functional Tr : S −→ C such that Tr(L1L2)=Tr(L2L1) for every L1 and L2 in S. For a given trace Tr, a determinant Det for the operators of the algebra can formally be defined by ∞ zm (1.2) Det(I − zL)=exp − Tr Lm . m m=1 On the other hand, a weighted dynamical system D =(Λ,F,g)isequippedwiththe so-called Ruelle dynamical zeta function defined by ∞ − zm m1 (1.3) ζ(z)=exp g(F kx) , m m=1 x∈Fix F m k=0 where Fix F m denotes the set of all fixed points of F m.ThesetFixF m is finite or countably infinite for all m>0. Like other zeta functions, the Ruelle dynamical zeta function has some sort of Euler product |P|−1 −1 |P | k (1.4) ζ(z)= 1 − z g(F xP ) , {P } k=0 where {P } denotes the set of periodic orbits of F with length |P | and xP is an arbitrary element of P . We shall assume that (1.2), (1.3), and (1.4) are absolutely convergent at least for z in a certain domain in C. In general, analytic properties of zeta functions give an important information about the corresponding systems in question. For example, a Tauberian theorem yields the classical prime number theorem from the positions of poles and zeros of the Riemann zeta function in the critical strip. In the same way, we are interested in the analytic properties of the dynamical zeta function to get more information about the corresponding dynamical system. An important method to study the analytic properties of dynamical zeta functions is the transfer operator method. In this method, the analytic properties of the zeta function are related to the spectral properties of a transfer operator through a relationship between the Fredholm determinant of the transfer operator and the dynamical zeta function. An interesting realization of the general program described above is the Mayer trans- fer operator acting on some Banach space of holomorphic functions on a disk [9]. This operator is assigned to the dynamical system related to the geodesic flow on the hyper- bolic plane modulo an arithmetical cofinite discrete group Γ. In this case the Fredholm determinant of the transfer operator is equal to the Selberg zeta function for the corre- sponding discrete group, which is one of the most important aspects of Mayer’s transfer operator theory. Indeed this identity provides us a new insight to the theory of quantum chaos. It turns out that the Mayer transfer operator, which is a purely classical object, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MAYER’S TRANSFER OPERATOR 531 surprisingly contains all information we can obtain from the corresponding Schr¨odinger operator. §2. Mayer’s transfer operator for PSL(2, Z) We start with introducing some notation and definitions. The hyperbolic plane H is the upper half-plane {x + iy ∈ C |y>0} equipped with the Poincar´emetricds2 = y−2(dx2 + dy2)andthemeasuredμ(z)=y−2 dx dy. Thus, geodesics on H are the semicircles with centra and the end points on the real axis. The group of all orientation preserving isometries of the hyperbolic plane H is identified with the group (2.1) PSL(2, R)=SL(2, R)/{±I} acting on H by linear fractional transformations defined by az + b ab z → gz = ,g= ,z∈ H. cz + d cd The modular group PSL(2, Z) is a discrete subgroup of PSL(2, R) defined by ab (2.2) PSL(2, Z)= ad − bc =1, a,b,c,d∈ Z /{±I}. cd This is a noncocompact Fuchsian group of the first kind. Let M =PSL(2, Z)\H be the quotient space of the hyperbolic plane H mod PSL(2, Z). This is a surface with one cusp and two conical singularities. Consider the continuous dynamics given by the geodesic flow ϕt on T1M, the unit tangent bundle of M.Froma physics point of view, the tangent bundle said to be unit if the geodesic flow describes the motion of a free particle on M with unit magnitude of velocity. As has already been mentioned, a transfer operator can be defined if the corresponding dynamical map has a finite or countable set of inverse branches, while the geodesic flow is continuous and determines an invertible map on T1M. Thus, we first discretize the geodesic flow by constructing a Poincar´e map of ϕt. It is known that, by a suitable choice of the Poincar´e section in T1M, the dynamics of ϕt reduces to the Poincar´e map given by (see [3]) P :[0, 1] × [0, 1] × Z2 → [0, 1] × [0, 1] × Z2, (2.3) 1 − P (x1,x2,)= TGx1, 1 , , [ ]+x2 x1 where [x] denotes the integral part of x and 1 ∈ x mod 1 if x (0, 1], (2.4) TGx = 0ifx =0 is the Gauss map. Remark 2.1. The group PSL(2, Z) does not contain the reflection relative to the y-axis. Consequently, for every geodesic on M there exists a unique geodesic on M such that their representatives on the upper half-plane are located symmetrially relative to the y-axis.ThesameistruefortheorbitsonT1M. This fact is reflected by two possible values of the parameter . We are interested in the expanding part of the map P , which reflects the ergodic aspects of the geodesic flow ϕt, Pex :[0, 1] × Z2 → [0, 1] × Z2, (2.5) Pex(x, )=(TGx, −). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 532 A. MOMENI AND A. B. VENKOV It remains to select a suitable weight function. Mayer chose the following weight function: s 2s (2.6) g(x, )=(TG) (x)=x , where s is a complex parameter. In fact, in accordance with Sina˘ı’s paper [18], the ergodic properties of ϕt are related to this weight function.
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