The Milnor-Thurston determinant and the Ruelle transfer operator (Q261599)

The kneading matrix for a continuous piecewise monotone interval map was introduced by \textit{J. Milnor} and \textit{W. Thurston} [Lect. Notes Math. 1342, 465--563 (1988; Zbl 0664.58015)]. For continuous piecewise monotone maps it was observed that the ``Milnor-Thurston'' determinant and a dynamical determinant associated to the Ruelle transfer operator have the same leading and peripheral zeros. This phenomenon was studied by \textit{V. Baladi} and \textit{D. Ruelle} [Ergodic Theory Dyn. Syst. 14, No. 4, 621--632 (1994; Zbl 0818.58013)] and \textit{S. Gouëzel} [Manuscr. Math. 106, No. 3, 365--403 (2001; Zbl 1048.37019)]. The main result of the paper reviewed here is that the ``Milnor-Thurston'' determinant matches a dynamical determinant associated to the Ruelle transfer operator when the transfer operator is restricted to suitable small subspace of bounded variation functions, thus providing an explanation for the matching zeros.