# 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 .

## 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.