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 nonreal complex dimensions and are therefore not called fractal in the sense of Lapidus.
 Publication:

Analysis and Mathematical Physics
 Pub Date:
 October 2022
 DOI:
 10.1007/s13324022007243
 arXiv:
 arXiv:2010.05955
 Bibcode:
 2022AnMP...12..114M
 Keywords:

 Fractal zeta functions;
 Complex dimensions;
 Parabolic germs;
 Formal classification;
 Fractal analysis of orbits;
 Tube functions;
 Minkowski dimension and content;
 Mathematics  Dynamical Systems;
 Mathematics  Complex Variables;
 11M41;
 28A75;
 28A80;
 37C05;
 37C15;
 44A15;
 37C25
 EPrint:
 61 pages, corrected typos and a minor mistake in the proof of Proposition 4.2