Dynamics of the continued fraction map and the spectral theory of \(\text{SL}(2,\mathbb{Z})\)
The paper answers a question raised in \textit{D. Mayer} [Bull. Am. Math. Soc., New Ser. 25, 55-60 (1991; Zbl 0729.58041)]. Consider the map of \([0,1)\) given by \(\{1/x\}\), \(x\neq 0\), \(T(0)=0\). The periodic points of the dynamical system defined by iterating \(T\) are periodic continued fractions. Mayer has developed a way of obtaining a meromorphic continuation of the Ruelle zeta-function of \(T\). The Ruelle zeta-function \(\zeta_ 1(s)\) associated to \(T\) is given by \[ \zeta_ 1(x)= \text{det} (1+ L_{s+1})/ \text{det} (1-L_ s). \] Here \(L_ s\) is the family of operators defined on the Banach space of functions \(f\) analytic in the disc \(| z-1|< 3/2\) and continuous on the boundary by \[ (L_ s f)(z)= \sum_{m=1}^ \infty (z+m)^{-2s} f \bigl( \textstyle {{1\over {z+m}}} \bigr), \qquad \text{for Re } s>1/2. \] Here the author considers a variant of this zeta-function \[ \zeta_ 2(s)= \text{det}(1+ L^ 2_{s+1})/ \text{det} (1-L_ s^ 2). \] He shows that the reciprocal of Selberg's zeta-function \(Z(s)\) of \(\text{PSL} (2,\mathbb{Z})\) factors as an infinite product of integer translates of \(\zeta_ 2(s)\). Mayer's question is: ``what aspects of the spectral theory of \(L^ 2 (\text{PSL} (2,\mathbb{Z}) \setminus H)\) (if any) are reflected by the following factorization of Selberg's zeta-function: \[ Z(s)= \text{det} (1-L_ s^ 2)= \text{det} (1-L_ s) \text{det}(1+ L_ s)? \text{''} \] The main Theorem of the present paper gives explicit factorizations of \(\text{det} (1\pm L_ s)\). The theorem implies the following Corollary. \(\text{Det} (1-L_ s)=0\) if and only if \(s(1-s)\) is an eigenvalue of \(\Delta\) for an even eigenfunction on \(L^ 2 (\text{PSL} (2,\mathbb{Z})\setminus H)\) or if \(\pi^{-s} \Gamma(s) \zeta(2s)=0\); \(\text{det} (1+L_ s)=0\) iff \(s(1-s)\) is an eigenvalue of \(\Delta\) for an odd eigenfunction on \(L^ 2 (\text{PSL} (2,\mathbb{Z}) \setminus H)\). The proof involves making precise the relations between continued fractions, quadratic forms and fundamental units in real quadratic number fields.
scientific article; zbMATH DE number 549646
scientific article