Poincare series and instability of exponential maps

Abstract

We relate the properties of the postsingular set for the exponential family regarding stability questions. We calculate the action of the Ruelle operator for the exponential family, and we prove that if the asymptotic (or singular) value is a summable point and its orbit satisfies certain topological conditions, the map is unstable. Hence there are no Beltrami differentials in the Julia set. We also show that if the Julia set is the whole sphere and the postsingular set is a compact set, then the singular value is summable and the map is unstable.