The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic

  title={The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic},
  author={Andrew Bridy},
  journal={arXiv: Number Theory},
  • Andrew Bridy
  • Published 21 June 2013
  • Mathematics
  • arXiv: Number Theory
A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine self-map of $\mathbb{P}^1(k)$ for $k$ an algebraically closed field of positive characteristic. 

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain


This is a special case of an Artin-Mazur zeta function, which is defined for certain dynamical systems (and in general counts the number of isolated fixed points). Note that we are, as usual, not

Periodic points and tail lengths of split polynomial maps modulo primes

This work includes a detailed analysis of the structure of the directed graph for Chebyshev polynomials of non-prime degree in dimension 1 and the powering map in any dimension.

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$

A general criterion for the P\'{o}lya-Carlson dichotomy and application

. We prove a general criterion for an irrational power series f ( z ) = ∞ X n =0 a n z n with coefficients in a number field K to admit the unit circle as a natural boundary. As an application, let F be

Endomorphisms of positive characteristic tori: entropy and zeta function

. Let F be a finite field of order q and characteristic p . Let Z F = F [ t ], Q F = F ( t ), R F = F ((1 /t )) equipped with the discrete valuation for which 1 /t is a uniformizer, and let T F = R F /

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These



Zeta Functions of Rational Functions Are Rational

We prove that if f is a rational function of degree at least 2 in the Riemann sphere then its zeta function exp f P 1 n=1 N n t n =ng is a rational function. Here N n denotes the number of distinct

Degree-growth of monomial maps

Abstract For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion

Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

  • J. Silverman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2012
Abstract Let φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a

Algebraic Entropy

An entropy is defined, which is a global index of complexity for the evolution map, and its basic properties and its relations to the singularities and the irreversibility of the map are analyzed.

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The

Introduction to finite fields and their applications: List of Symbols

An introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications, especially information theory, algebraic coding theory and cryptology and a chapter on applications within mathematics, such as finite geometries.

The Arithmetic of Dynamical Systems

* Provides an entry for graduate students into an active field of research * Each chapter includes exercises, examples, and figures * Will become a standard reference for researchers in the field

On Lattès Maps

An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 1. The Lattès paper. 2. Finite Quotients of Affine Maps 3. A Cyclic