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Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T, g) extends meromorphically to {∣z∣-1} (where θ=lim ∥g°Tn-1...g°Tg∥∞1/n) and thatz is a pole of ζ if and only ifz−1 is an eigenvalue of the corresponding transfer operator L. We do not assume that L leaves a reference measure invariant.

Zeta functions and transfer operators for piecewise monotone transformations