# Fractal zeta functions of orbits of parabolic diffeomorphisms

@article{Mardevsic2020FractalZF, title={Fractal zeta functions of orbits of parabolic diffeomorphisms}, author={P. Mardevsi'c and Goran Radunovi'c and M. Resman}, journal={arXiv: Dynamical Systems}, year={2020} }

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… Expand

#### References

SHOWING 1-10 OF 53 REFERENCES

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

- Mathematics, Physics
- 2017

Abstract We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function ζ A ( s ) : = ∫ A δ d ( x , A ) s −… Expand

Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms

- Mathematics
- 2012

In this article we study the dynamics generated by germs of parabolic diffeomorphisms f : (C; 0)->(C; 0) tangent to the identity. We show how formal classification of a given parabolic diffeomorphism… Expand

Zeta Functions and Complex Dimensions of Relative Fractal Drums: Theory, Examples and Applications

- Mathematics, Physics
- 2016

In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions', associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It… Expand

Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations

- Mathematics
- 2013

In this article, we study analyticity properties of (directed) areas of epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using… Expand

Distance and tube zeta functions of fractals and arbitrary compact sets

- Mathematics, Physics
- 2017

Abstract Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N-dimensional Euclidean space R N , for any… Expand

Iterated function systems and the global construction of fractals

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1985

Iterated function systems (i. f. ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i. f. ss and occur as the supports of probability… Expand

Orthogonal Polynomials and Special Functions

- Mathematics
- 1975

Asymptotics of Jacobi matrices for a family of fractal measures GÖKALP APLAN BILKENT UNIVERSITY, TYRKEY There are many results concerning asymptotics of orthogonal polynomials and Jacobi matrices… Expand

Anosov flows and dynamical zeta functions

- Mathematics
- 2012

We study the Ruelle and Selberg zeta functions for C r Anosov ows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being (a) for C 1 ows the zeta function is… Expand

Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality

- Mathematics, Physics
- 2016

We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is… Expand

The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums

- Mathematics
- 1993

Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's… Expand