Reading analytic invariants of parabolic diffeomorphisms from their orbits

TL;DR

This paper introduces a method to determine the analytic invariants of parabolic complex plane diffeomorphisms from a single orbit by using the dynamic theta function, which is shown to be resurgent and related to sectorial Fatou coordinates.

Contribution

It develops the dynamic theta function and fractal theta function as tools to extract analytic invariants from orbits of parabolic germs, advancing the understanding of their complex dynamics.

Findings

The dynamic theta function is $2 extpi i extZ$-resurgent.

Sectorial Fatou coordinates can be obtained via Laplace transform of the theta function.

The fractal theta function encodes the same invariants, generalizing the geometric zeta function.

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.