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.

Formal definition[edit]

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[1]


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

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.[2]

See also[edit]


  1. ^ a b Terras (2010) p. 28
  2. ^ Terras (2010) p. 29