Artin-Mazur zeta function
From Thikipedia
| Line 2: | Line 2: | ||
It is defined as the [[formal power series]] | It is defined as the [[formal power series]] | ||
| - | :<math>\\zeta_f(z)=\\exp \\sum_{n= | + | :<math>\\zeta_f(z)=\\exp \\sum_{n=1}^\\infty \\textrm{card} |
| - | \\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n | + | \\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n}{n}</math>, |
where <math>\\textrm{Fix}(f^n)</math> is the set of [[fixed point]]s of the ''n''-th iterate of an [[iterated function]] ''f'', and <math>\\textrm{card} | where <math>\\textrm{Fix}(f^n)</math> is the set of [[fixed point]]s of the ''n''-th iterate of an [[iterated function]] ''f'', and <math>\\textrm{card} | ||
\\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. | ||
Revision as of 10:48, 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 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
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)
