On the relaxation time of Gauss' continued-fraction map. I: The Hilbert space approach (Koopmanism) (Q1111602)

It is shown that \(U^*\), the adjoint of Koopman's isometric operator \(Uf(x)=f(Tx)\) corresponding to the map \(Tx=1/x (mod 1)\) of the unit interval, is isomorphic to a symmetric integral operator when restricted to a Hilbert space of holomorphic functions f. This result, also obtained by K. I. Babenko in a different setting, allows us to derive new trace formulas. Using generalized Temple inequalities, we determine the relaxation time of the above system with great accuracy. In contrast to a widespread belief, it appears to be unrelated to the entropy of the map T.