Reading analytic invariants of parabolic diffeomorphisms from their orbits

Abstract

In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $\mathbb{A}$? We solve this problem for parabolic germs. This is done by associating to the orbit $\mathbb{A}$ a function that we call the dynamic theta function ${\Theta }_{\mathbb{A}}$. We prove that the function ${\Theta }_{\mathbb{A}}$ is $2\pi i\mathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function ${\Theta }_{\mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits. We also define a closely related fractal theta function ${\tilde{\Theta }}_{\mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism.

References in corpus (1)

• Fractal zeta functions of orbits of parabolic diffeomorphisms