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

It is shown thatU ^*, the adjoint of Koopman's isometric operatorUf(x) =f(Tx) corresponding to the mapTx=x ^–1 (mod 1) of the unit interval, is isomorphic to a symmetric integral operator when restricted to a Hilbert space of holomorphic functionsf This result, also obtained by Babenko in a different setting, allows us to derive new trace formulas. Using generalized Temple's 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 mapT.