Fractal zeta functions of orbits of parabolic diffeomorphisms

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.

61 pages, corrected typos and a minor mistake in the proof of Proposition 4.2

References in corpus (4)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Zeta Functions and Complex Dimensions of Relative Fractal Drums: Theory, Examples and Applications

Cited by in corpus (1)

• Reading analytic invariants of parabolic diffeomorphisms from their orbits