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

Abstract

The spectrum of the transfer operator ℒ 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 ℋ(υ) introduced in part I of this work. IfN denotes the operator ℒ−P ₁ withP ₁ the projector onto the eigenfunction to the dominant eigenvalueλ ₁ =1 of ℒ, then −N is au ₀-positive operator with respect to some cone and therefore has a dominant positive, simple eigenvalue −λ ₂. A minimax principle holds giving rigorous upper and lower bounds both forλ ₂ and the relaxation time of the mapT.