Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations

Abstract

In this article, we study analyticity properties of (directed) areas of epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using epsilon-neighborhoods of orbits in the simplest formal class. We show that the coefficient in front of epsilon^2 term in the asymptotic expansion in epsilon, which we call the principal part of the area, is a sectorially analytic function of 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 globally analytic solution of this equation. Furthermore, we introduce new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes.