Fredholm determinant for piecewise monotonic transformations
Let \(I\) be a finite union of bounded intervals. For a piecewise monotone and piecewise \(C^ 2\) map \(F: I\to I\) let \[ \xi = \liminf_{n\to \infty} \text{ ess }\inf_{x\in I}\textstyle{1\over n} \log| (F^ n)'(x)| \] and \[ \zeta(x) = \text{exp}\left[\sum^ \infty_{n=1} \textstyle{1\over n}\sum_{x = F^ n x}1/|(F^ n)'(x)|\right]. \] The main result of the paper is: The reciprocal \(1/\zeta(z)\) of the Ruelle-Artin-Mazur zeta function has an analytic extension to the complex domain \(\{z: | z| < e^ \xi\}\), and the zeros of \(1/\zeta\) are the reciprocals of the isolated eigenvalues of the Perron-Frobenius operator of \(F\) acting on the space of functions of bounded variation. The main tools for the proof are signed symbolic dynamics, a renewal equation and the approximation of general \(F\) by ``formal piecewise linear'' ones. [For related results see \textit{V. Baladi} and \textit{G. Keller}, Commun. Math. Phys. 127, No. 3, 459-477 (1990; Zbl 0703.58048) and \textit{G. Keller} and \textit{T. Nowicki}, Commun. Math. Phys. 149, No. 1, 31-69 (1992; Zbl 0763.58024)].
scientific article; zbMATH DE number 219890