Artin-Mazur zeta function
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| - | In [[ | + | In [[maths]], the '''Artin-Mazur [[zeta function|zeta-function]]''' is a tool for studying the [[iterated function]]s that occur in [[dynamical systems]] and [[fractals]]. |
It is defined as the [[formal power series]] | It is defined as the [[formal power series]] | ||
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\\left(\\textrm{Fix} (f^n)\\right)</math> is the [[cardinality]] of this set of fixed points. | \\left(\\textrm{Fix} (f^n)\\right)</math> is the [[cardinality]] of this set of fixed points. | ||
| - | Note that the zeta-function is defined only if set of fixed points is finite. This definition is formal in that it does not always have a positive [[radius of convergence]]. | + | 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 Artin-Mazur zeta-function is not invariant under [[topological conjugation]]. | ||
Current revision as of 16:12, 25 August 2006
In maths, 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.
[edit] See also
[edit] 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)
