Transfer operators, the Selberg zeta function and the Lewis-Zagier theory of periodic functions (Q2872742)

Two approaches are known for analytic continuation of Selberg zeta functions. One is to use the Selberg trace formula discovered by Selberg in 1950's, and the other is to express the zeta function as a determinant of transfer operators. The latter method was first developed by Ruelle, Smale and Fried for compact manifolds in 1970's and 80's, and it was a long standing open problem to generalize their results to noncompact manifolds such as the modular surface.NEWLINENEWLINEThe author of this article is the one who discovered an unexpected expression of the Selberg zeta function for the modular group in [Bull. Am. Math. Soc., New Ser. 25, No. 1, 55--60 (1991; Zbl 0729.58041)] in the context of the transfer operators. It was a surprising result, in the sense that he created a completely new idea not using the traditional transfer operator based on the geodesic flow but using the arithmetic property of the modular group concerning the Gauss map for continued fractions. After this discovery, many mathematicians developed the theory to get generalizations.NEWLINENEWLINEThis article offers a survey of this theory with its developments. In particular, the author focuses on a generalization to a subgroup of the full modular group of finite index. This is a nice introduction for those who need to study the whole picture of the theory. The only thing that the reviewer thinks the author should have referred to is the work of \textit{T. Morita} [Ergodic Theory Dyn. Syst. 17, No. 5, 1147--1181 (1997; Zbl 0893.60052)], who generalized the author's determinant expression to cofinite Fuchsian groups.NEWLINENEWLINEFor the entire collection see [Zbl 1237.51005].