Artin-Mazur zeta function

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Revision as of 13:45, 25 July 2006

In mathematics, the Artin-Mazur zeta-function is a tool for studying the iterated functions that occur in dynamical systems and fractals.

It is defined as the formal power series

Failed to parse (Can't write to or create math temp directory): \\zeta_f(z)=\\exp \\sum_{n=1}^\\infty \\textrm{card} \\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n}{n}

, where Failed to parse (Can't write to or create math temp directory): \\textrm{Fix}(f^n)

is the set of fixed points of the n-th iterate of an iterated function f, and Failed to parse (Can't write to or create math temp directory): \\textrm{card}  \\left(\\textrm{Fix} (f^n)\\right)
is the cardinality of this set of fixed points.

Note that the zeta-function is defined only if the set of fixed points is finite. This definition is formal in that it does not always have a positive radius of convergence.

The Artin-Mazur zeta-function is not invariant under topological conjugation.

The Milnor-Thurston theorem states that the Artin-Mazur zeta-function is the inverse of the kneading determinant of f.

The Artin-Mazur zeta-function is equivalent to the Weil zeta-function when there is a diffeomorphism on a compact manifold.

Under certain cases, the Artin-Mazur zeta-function can be related to the Ihara zeta-function of a graph.

See also

• Lefschetz number
• Lefschtetz zeta-function

References

• M. Artin and B. Mazur, On periodic points, Ann. of Math (2) 81 (1965) 82-99.
• David Ruelle, Dynamical Zeta Functions and Transfer Operators (2002) (PDF)