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Ruelle zeta function

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In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle.

Contents

Formal definition

Let f be a function defined on a manifold M, such that the set of fixed points Fix(f n) is finite for all n > 1. Further let φ be a function on M with values in d × d complex matrices. The zeta function of the first kind is

ζ ( z ) = exp ( m 1 z m m x Fix ( f m ) Tr ( k = 0 m 1 ϕ ( f k ( x ) ) ) )

Examples

In the special case d = 1, φ = 1, we have

ζ ( z ) = exp ( m 1 z m m | Fix ( f m ) | )

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.

References

Ruelle zeta function Wikipedia