Artin–Mazur zeta function

In mathematics, named after Michael Artin and Barry Mazur Artin Mazursche zeta function is an aid in the study of iterated functions in dynamic systems. It is sometimes called topological zeta function.

Artin and Mazur have introduced this zeta function in 1965. This function was then examined by Stephen Smale further and made ​​widely known.

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

It referred Fix ( ƒn ) the amount of the fixed points of the n - th iteration of the function ƒ, and card ( Fix ( ƒn )) is the cardinality of this set of fixed points. Only finite cardinalities are allowed here.

The Artin - Mazursche zeta function is a topological invariant, ie, it is invariant under topological conjugation. Thus it combines the local properties of the function f with the global properties of the discrete trajectories ( orbits) manifold generated.

Extensive convergence tests were carried out by William Parry and Mark Pollicott.

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