Eigenfunctions of transfer operators and cohomology (Q2378050)

The general method of transfer operators (or generalized Perron-Frobenius operators) were introduced by Ruelle [cf. e.g. \textit{D.~Ruelle}, Thermodynamic Formalism (Encycl. Math. Appl. 5, Addison-Wesley, Reading Mass.) (1978; Zbl 0401.28016)] and an overview of the general theory can be found in e.g. [\textit{D.~Ruelle}, Notices Am. Math. Soc. 49, No. 8, 887--895 (2002; Zbl 1126.37305)]. In the paper under review, there are two transfer operators under consideration and they both correspond to interval maps given by generating maps for different kind of continued fractions. The first transfer operator of this type was constructed by \textit{D. Mayer} [Commun. Math. Phys. 130, No. 2, 311--333 (1990; Zbl 0714.58018)] and corresponds to the generating map for the simple or Gauss continued fractions, \(x\mapsto\frac{1}{x}\mod1\). This transfer operator ``Mayer's operator'' is denoted by \(\mathcal{L}_{s}^{\text{Ma}}\) in this paper. The other transfer operator, denoted by \(\mathcal{L}_{s}\) corresponds to the generating function of the nearest integer continued fractions, \(x\mapsto-\frac{1}{x}-\left[x\right]\) where \(\left[x\right]\) is the nearest integer of \(x\) (with some fixed choice at the half integers). There is another interpretation of these maps, which is as factor maps of Poincaré maps of particular Poincaré sections for the geodesic flow on the modular surface \(\mathcal{M}=\text{PSL}\left(2,\mathbb{Z}\right)\backslash\mathbb{H}\) (or the corresponding billiard system). See e.g.~\textit{D.~Mayer} [Bull. Am. Math. Soc., New Ser. 25, No. 1, 55--60 (1991; Zbl 0729.58041)]. Starting from the geodesic flow, there are many possible choices involved in the construction of the Poincaré section and different choices give rise to different maps. Some examples of Poincaré sections and corresponding maps can be found in [\textit{S.~Katok} and \textit{I.~Ugarcovici}, Bull.~Am.~Math.~Soc., New Ser.~44, No.~1, 87--132 (2007; Zbl 1131.37034)]. Using for instance properties of the (simple or nearest integer) continued fractions one can express the Selberg zeta function for the modular group, \(Z\left(s\right)\), in terms of Fredholm determinants of the corresponding transfer operators. In fact \(Z\left(s\right)=\det\left(1-\left(\mathcal{L}_{s}^{\text{Ma}}\right)^{2}\right)\) and \(Z\left(s\right)=K\left(s\right)\det\left(1-\mathcal{L}_{s}\right)\) (where \(K(s)\) is non-zero). The Selberg zeta function is an object which is closely related to the corresponding quantum mechanical free particle system on \(\mathcal{M}\). In particular it has zeros on the values of \(s\) where the hyperbolic Laplacian \(\Delta\) (which is essentially the time-independent Schrödinger operator) on \(\mathcal{M}\) has discrete eigenvalues. This furnish a relationship between the set of values of \(s\) for which \(1\) is in the spectrum of the transfer operator in question (together with \(-1\) in the case of \(\mathcal{L}_{s}^{\text{Ma}}\)) and the set of discrete eigenvalues of \(\Delta\). In the case of \(\mathcal{L}_{s}^{\text{Ma}}\) the relation to the corresponding quantum mechanical system was made much more explicit in [\textit{J.~Lewis} and \textit{D.~Zagier}, Ann.~Math.~(2) 153, No.~1, 191--258 (2001; Zbl 1061.11021)] where it was showed that there is a direct correspondence between eigenfunctions of \(\mathcal{L}_{s}^{\text{Ma}}\) with eigenvalue \(\pm1\) (in a certain space, satisfying some extra conditions) and Maass cusp forms, that is \(L^{2}\)-eigenfunctions of \(\Delta\). Until now it has seemed like it was more or less pure luck that the first transfer operator that was constructed (\(\mathcal{L}_{s}^{\text{Ma}}\)) exhibited this kind of explicit relation to Maass waveforms. Although this type of correspondence is believed to be quite general no one has been able to prove it for single transfer operator except \(\mathcal{L}_{s}^{\text{Ma}}\) until the work in the present paper. In the paper under review, the authors show that there is a one-to-one correspondence between eigenfunctions of \(\mathcal{L}_{s}\) with eigenvalue \(1\) and Maass waveforms. By an elaborate construction, using non-trivial facts about various cohomology groups for the modular group they show that there is a relation between eigenfunctions of \(\mathcal{L}_{s}^{\text{Ma}}\) with eigenvalue \(\pm1\) and eigenfunctions of \(\mathcal{L}_{s}\) with eigenvalue \(1\) (in certain spaces, satisfying some particular conditions). The method they use to prove this also includes a proof that both these spaces are in a one-to-one correspondence with the space of Maass cusp forms. The bijection is unfortunately not explicit and there is still much work to be done in explicitly relating eigenfunctions of one operator to those of the other. The importance of the present work is of course that it shows that in this case there is a one-to-one correspondence. The generality of the approach; i.e. relating eigenfunctions of transfer operators to certain cohomology groups and the fact that Maass cusp forms are more or less directly related to other types of cohomology groups seems to indicate that the method is valid in a much greater context.