Ruelle operator and transcendental entire maps

Abstract

We calculate the Ruelle operator of a transcendental entire function f having only a finite set of algebraic singularities. Moreover, under certain topological conditions on the postcritical set we prove (i) if f has a summable critical point, then f is not structurally stable and (ii) if all critical points of f belonging to Julia set are summable, then there do not exist invariant lines fields on the Julia set.