epsilon-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

In this article, we study the analyticity of (directed) areas of epsilon-neighbourhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using epsilon-neighbourhoods of orbits in the simplest formal class. We show that the coefficient in front of the epsilon(2) term in the asymptotic expansion in epsilon, which we call the principal part of the area, is a sectorially analytic function in the initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms. We give necessary and sufficient conditions on a diffeomorphism f for the existence of a globally analytic solution of this equation. Furthermore, we introduce a new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes.