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geometry > geometric figure > fractal > Artin-Mazur zeta function

mathematical physics > dynamical systems theory > dynamical system > Artin-Mazur zeta function

number > number theory > analytic number theory > L-function > Artin-Mazur zeta function

mathematical analysis > complex analysis > meromorphic function > L-function > Artin-Mazur zeta function

Preferred term

Artin-Mazur zeta function

In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals.
It is defined from a given function ${\\displaystyle f}$ as the formal power series

${\\displaystyle \\zeta _{f}(z)=\\exp \\left(\\sum _{n=1}^{\\infty }{\igl |}\\operatorname {Fix} (f^{n}){\igr |}{\ rac {z^{n}}{n}}\ ight),}$

where ${\\displaystyle \\operatorname {Fix} (f^{n})}$ is the set of fixed points of the ${\\displaystyle n}$ th iterate of the function ${\\displaystyle f}$ , and ${\\displaystyle |\\operatorname {Fix} (f^{n})|}$ is the number of fixed points (i.e. the cardinality of that set).
Note that the zeta function is defined only if the set of fixed points is finite for each ${\\displaystyle n}$ . This definition is formal in that the series does not always have a positive radius of convergence.
The Artin–Mazur zeta function is invariant under topological conjugation.
The Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map ${\\displaystyle f}$ is the inverse of the kneading determinant of ${\\displaystyle f}$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Artin%E2%80%93Mazur_zeta_function)

fonction zêta d'Artin-Mazur

French

URI

http://data.loterre.fr/ark:/67375/PSR-Z8FJPBWK-8

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