Fractal zeta functions of orbits of parabolic diffeomorphisms

TL;DR

This paper investigates the fractal zeta functions of orbits of parabolic diffeomorphisms, revealing their complex dimensions, poles, and fractal footprints, and relating these to asymptotic expansions and intrinsic properties of the germs.

Contribution

It introduces the meromorphic extension of fractal zeta functions for parabolic orbits and characterizes their complex dimensions and fractal footprints, linking them to dynamical and geometric properties.

Findings

Fractal zeta functions of parabolic orbits can be meromorphically extended to the entire complex plane.

The set of poles (complex dimensions) characterizes the fractal footprint of the orbit.

Parabolic orbits exhibit higher order oscillatory tube functions but lack non-real complex dimensions.

Abstract

In this paper, we prove that fractal zeta functions of orbits of parabolic germs of diffeomorphisms can be meromorphically extended to the whole complex plane. We describe their set of poles (i.e. their complex dimensions) and their principal parts which can be understood as their fractal footprint. We study the fractal footprint of one orbit of a parabolic germ f and extract intrinsic information about the germ f from it, in particular, its formal class. Moreover, we relate complex dimensions to the generalized asymptotic expansion of the tube function of orbits with oscillatory "coefficients" as well as to the asymptotic expansion of their dynamically regularized tube function. Interestingly, parabolic orbits provide a first example of sets that have nontrivial Minkowski (or box) dimension and their tube function possesses higher order oscillatory terms, however, they do not posses non-real complex dimensions and are therefore not called fractal in the sense of Lapidus.