On the relaxation time of Gauss' continued-fraction map. II. The Banach space approach (transfer operator method)

The spectrum of the transfer operator for the mapTx=1/x–1/x] when restricted to a certain Banach space of holomorphic functions is shown to coincide with the spectrum of the adjointU* of Koopman's isometric operatorUf(x)=f·T(x) when the former is restricted to the Hilbert space () introduced in part I of this work. IfN denotes the operator –P ₁ withP ₁ the projector onto the eigenfunction to the dominant eigenvalue ₁ =1 of , then –N is au ₀-positive operator with respect to some cone and therefore has a dominant positive, simple eigenvalue – ₂. A minimax principle holds giving rigorous upper and lower bounds both for ₂ and the relaxation time of the mapT.