Ruelle Zeta-function
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Ruelle zeta function is a
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * A ...
associated with a dynamical system. It is named after mathematical physicist David Ruelle.


Formal definition

Let ''f'' be a function defined on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''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 isTerras (2010) p. 28 : \zeta(z) = \exp\left( \sum_ \frac \sum_ \operatorname \left( \prod_^ \varphi(f^k(x)) \right) \right)


Examples

In the special case ''d'' = 1, ''φ'' = 1, we have : \zeta(z) = \exp\left( \sum_ \frac m \left, \operatorname(f^m)\ \right) which is the
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 f as th ...
. The Ihara zeta function is an example of a Ruelle zeta function.Terras (2010) p. 29


See also

*
List of zeta functions In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * A ...


References

* * * * {{cite journal , first1=David , last1=Ruelle , author-link=David Ruelle , title=Dynamical Zeta Functions and Transfer Operators , url=https://www.ams.org/notices/200208/fea-ruelle.pdf , journal=Bulletin of AMS , volume=8 , issue=59 , year=2002 , pages=887–895 Zeta and L-functions