Fredholm determinant for piecewise linear transformations (Q916815)

Let \(I\) be a finite union of intervals, \(F: I\to I\) piecewise linear, and suppose that \(\xi\), the lower Lyapunov number of \(F\), is strictly positive. Denote by \(\text{Spec}(F)\) the spectrum of the Perron-Frobenius operator of \(F\) (acting on the space of functions of bounded variation on \(I\)). The main result of the paper states: Let \(\lambda\in {\mathbb C}\), \(| \lambda | >e^{-\xi}.\) Then \(\lambda\in \text{Spec}(F)\) if and only if \(\lambda\) is a zero of a function \(D(z)\) which is defined in terms of the symbolic dynamics of \(F\) and analytic in \(\{| z| <e^{-\xi}\}.\) (\(D(z)\) is called the Fredholm determinant of \(F\).) \(D(z)\) coincides up to a factor which has no zeros and poles in \(\{| z| <e^{-\xi}\}\) with the Ruelle zeta function for \(F\). This result is essentially contained in \textit{F. Hofbauer} and the reviewer [J. Reine Angew. Math. 352, 100--113 (1984; Zbl 0533.28011)], except that a formally different kind of symbolic dynamics is used there to link the Perron-Frobenius operator with the zeta function. For piecewise holomorphic systems the same result was proved by the reviewer [Trans. Am. Math. Soc. 314, No. 2, 433--497 (1989; Zbl 0686.58027)], and for quite arbitrary piecewise monotone interval maps by \textit{V. Baladi} and the reviewer [Commun. Math. Phys. 127, No. 3, 459--478 (1990; Zbl 0703.58048)].