Antiholomorphic perturbations of Weierstrass Zeta functions and Green’s function on tori (original) (raw)

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

Journal of Physics A: Mathematical and Theoretical, 2008

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated to conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζfunctions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

Zeta Functions and Asymptotic Formulae for Preperiodic Orbits of Hyperbolic Rational Maps

Mathematische Nachrichten, 2009

For a hyperbolic rational map R of the Riemann sphere of degree d 2 2, restricted to its . l d b set J(R), we define a %eta function C R ( d ) , which counts the prepenodic orbib of R, according to Lhe weight function IR'I : J(R) -+ C . An analysis of the analytic domain of (~( d ) , using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R. We describe m application to diophantine number theory. 1991 Mathemafica Subject Classification. Primary 30D05, Secondary 58320. Keywords and phrases. btional maps of the Riemann sphere, dynamical zeta functions.

Zeros of dynamical zeta functions for hyperbolic quadradic maps

arXiv (Cornell University), 2022

We prove that the dynamical zeta function Z(s) associated to z 2 + c with c < −3.75 has essential zero-free strips of size 1/2+, that is, for every > 0, there exist only finitely many zeros in the strip Re(s) > 1/2 +. We also present some numerical plots of zeros of Z(s) based on the method proposed in [JP02].

On the Lefschetz zeta function for quasi-unipotent maps on the n -dimensional torus

Journal of Difference Equations and Applications, 2014

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the (n + 1)-th cyclotomic polynomial, when n + 1 is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on T n are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for C 1 quasi-unipotent maps on T n , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse-Smale diffeomorphisms of T n .

On atypical values and local monodromies of meromorphic functions

Arxiv preprint math/ …, 1998

A meromorphic function on a compact complex analytic manifold defines a C ∞ locally trivial fibration over the complement of a finite set in the projective line CP 1. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on C n are described. §1.-Introduction We want to consider fibrations defined by meromorphic functions. In order to have more general statements we prefer to use the notion of a meromorphic function slightly different from the standard one. Let M be an n-dimensional compact complex analytic manifold.

Dynamics of a Family of Non-Critically Finite Even Transcendental Meromorphic Functions

2004

The dynamics of one parameter family of non-critically finite even transcendental meromorphic function ξ λ (z) = = λ sinh 2 z z 4 , λ > 0 is investigated in the present paper. It is shown that bifurcations in the dynamics of the function ξ λ (x) for x ∈ R \ {0} occur at two critical parameter values λ = x 5 1 sinh 2 x 1 (≈ 1.26333) and λ =x 5 sinh 2x (≈ 2.7.715), where x1 andx are the unique positive real roots of the equations tanh x = 2x 3 and tanh x = 2x 5 respectively. For certain ranges of parameter values of λ, it is proved that the Julia set of the function ξ λ (z) contains both real and imaginary axes. The images of the Julia sets of ξ λ (z) are computer generated by using the characterization of the Julia set of ξ λ (z) as the closure of the set of points whose orbits escape to infinity under iterations. Finally, our results are compared with the recent results on dynamics of (i) critically finite transcendental meromorphic functions λ tan z having polynomial Schwarzian Derivative [10, 15, 19] and (ii) non-critically finite transcendental entire functions λ e z − 1 z [14].

Fractal zeta functions of orbits of parabolic diffeomorphisms

Analysis and Mathematical Physics

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.

Meromorphic functions of one complex variable. A survey

2008

This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970. The literature on meromorphic functions 1 is very large. There is a comprehensive survey [62] that contains everything that was reviewed on the topic in the Soviet "Referativnyi Zhurnal" in 1953-1970, and a later large survey [67]. More recent surveys [30], [80] and [48] are shorter and have narrower scope. Some books on specific topics in the theory of meromorphic functions published after 1970 are [18], [20], [79], [100], [137], [134], [138], [162]. A survey of the fast developing subject of iteration of meromorphic functions is [7].