Dynamical Zeta Functions for Maps of the Interval

A dynamical zeta function Ç and a transfer operator Sf are associated with a piecewise monotone map / of the interval [0,1] and a weight function g . The analytic properties of f and the spectral properties of S? are related by a theorem of Baladi and Keller under an assumption of "generating partition". It is shown here how to remove this assumption and, in particular, extend the theorem of Baladi and Keller to the case when / has negative Schwarzian derivative. Let 0 = £z0 < o.\ < ■ ■ ■ < ajf = 1. We write X = [0, l]cl and assume that / is continuous X —> X and strictly monotone on the intervals J¡ = [a,_i, a¡]. Furthermore, let g:I-»C have bounded variation. A transfer operator ¿C acting on functions O : X —► C of bounded variation is defined by (-S* 0, then x — y), Baladi and Keller [1] have proved the following remarkable result (referred to as B-K theorem in what follows): The function m m—\