Poincare Series and instability of exponential maps

Abstract

We relate the properties of the postsingular set for the exponential family to the questions of stability. We calculate the action of the Ruelle operator for the exponential family. We prove that if the asymptotic 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. Also we show that if the postsingular set is a compact set, then the singular value is summable.