Artin-Mazur's zeta function

In mathematics , the Artin-Mazur zeta function named after Michael Artin and Barry Mazur is an aid for studying iterated functions in dynamic systems . It is sometimes referred to as the topological zeta function .

Artin and Mazur introduced this zeta function in 1965. This function was then further explored and made popular by Stephen Smale .

The Artin-Mazur zeta function is defined as a formal power series :

${\displaystyle \zeta _{f}(z)=\exp \sum _{n=1}^{\infty }{\textrm {card}}\left({\textrm {Fix}}(f^{n})\right){\frac {z^{n}}{n}},}$

Here referred to the set of fixed points of th iteration of the function , and the cardinality of the set of checkpoints. Only finite cardinalities are allowed here. ${\displaystyle {\textrm {Fix}}(f^{n})}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle {\textrm {card}}({\textrm {Fix}}(f^{n})}$

The Artin-Mazur zeta function is a topological invariant, that is, it is invariant under topological conjugations . It connects local properties of the function with global properties of the manifold generated by the discrete trajectories (orbits) . ${\displaystyle f}$

Extensive convergence studies have been carried out by William Parry and Mark Pollicott .

A further development of the Artin-Mazurian zeta function in the theory of dynamic systems was carried out by David Ruelle , Viviane Baladi and others to the Ruelle zeta function and dynamic zeta function .

Web links

• John Baez : This Week's Finds in Mathematical Physics (Week 216). 2005 (for connection with other Zeta functions).
• David Ruelle : Dynamical Zeta Functions and Transfer Operators. 2002 (PDF; 233 kB).
• Predrag Cvitanović et al .: Chaos: Classical and Quantum. 2009 (including Chapter 15 on application in theoretical physics ).

Individual evidence

1. ↑ Michael Artin, Barry Mazur: On periodic points. In: Annals of Mathematics. 81, 1965, pp. 82-99.
2. ↑ Stephen Smale: Differential dynamical systems. In: Bulletin of the American Mathematical Society. 73, 1967, pp. 747-817.
3. ^ William Parry, Mark Pollicott: Zeta functions and the periodic orbit structure of hyperbolic dynamics. In: Astérisque. vol. 187-188, 1990, Société Mathématique de France, Paris.