Sharp Determinants and Kneading Operators for Holomorphic Maps

In an earlier paper 1] two of us studied generalized transfer operators M associated to maps of intervals of R. In particular the analyticity of a sharp determinant Det # (1?zM) was analyzed on the basis of spectral properties of M and by relating Det # (1 ? zM) to the determinant of 1 + D(z) for a suitable kneading operator D(z). It is a natural idea to try to replace monotone maps of intervals of R by holomorphic diieomorphisms of domains of C and to transpose the results of 1] to this new situation. This program of transposition to the complex has been carried through in part, and the present preprint shows our results. In a suitable function-theoretical setting one can analyze the spectral properties of the generalized transfer operator M and relate formally the sharp determinant Det # (1 ? zM) to the determinant of 1 + D(z) where D(z) is a kneading operator. Unfortunately the trace of D(z) might diverge, and only regularized determinants like Det 3 (1 + D(z)) make sense a priori. The present preprint leaves unnn-ished the task of either bounding Tr D(z) or making suitable subtractions in the deenition of Det # (1 ? zM). Nevertheless it has appeared useful to record the existing non trivial results obtained, in view of their later utilisation. The present preprint is a result of extensive discussions between the authors. It consists of three parts. The rst part contains an identity between formal power series. The second part is about spectral properties, and the third part (traces and determinants) concerns bounds on Tr D(z) 2 and higher traces.