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Zeta functions and transfer operators for piecewise monotone transformations. (English) Zbl 0703.58048

Given a piecewise monotone transformation T of the interval and a piecewise continuous complex weight function g of bounded variation, the authors prove that the Ruelle zeta function \(\zeta\) (z) of (T,g) extends meromorphically to \(\{| z| <\theta^{-1}\}\) (where \(\theta =\lim_{n\to \infty}\| g\circ T^{n-1}\cdot...\cdot g\circ T\cdot g\|_{\infty}^{1/n})\) and that z is a pole of \(\zeta\) if and only if \(z^{-1}\) is an eigenvalue of the corresponding transfer operator \({\mathcal L}\). They do not assume that \({\mathcal L}\) leaves a reference measure invariant.
Reviewer: G.M.Rassias

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C80 Symmetries, equivariant dynamical systems (MSC2010)
30D50 Blaschke products, etc. (MSC2000)
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