In
mathematics, the Ruelle zeta function is a
zeta function associated with a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
. 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 ...
''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
[Terras (2010) p. 28]
:
Examples
In the special case ''d'' = 1, ''φ'' = 1, we have
[
:
which is the Artin–Mazur zeta function.
The ]Ihara zeta function In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was firs ...
is an example of a Ruelle zeta function.[Terras (2010) p. 29]
See also
* List of zeta functions
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