Ruelle zeta function

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In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system.

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]

 \zeta(z) = \exp\left({
                               \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)} 
                               \mathrm{Tr}
                                    \left({ \prod_{k=0}^{m-1} \phi(f^k(x)) 
                                          }\right)
                             }\right)

Examples[edit]

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

 \zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right)

which is the Artin–Mazur zeta function.

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

See also[edit]

References[edit]

  1. ^ a b Terras (2010) p. 28
  2. ^ Terras (2010) p. 29
  • Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006). Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Springer Monographs in Mathematics. New York, NY: Springer-Verlag. ISBN 0-387-33285-5. Zbl 1119.28005. 
  • Kotani, Motoko; Sunada, Toshikazu (2000). "Zeta functions of finite graphs". J. Math. Sci. Univ. Tokyo 7: 7–25.