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 * ...

associated with a

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

. It is named after mathematical physicist

David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...

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. 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 first ...

is an example of a Ruelle zeta function.Terras (2010) p. 29

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

Formal definition

Examples

See also

References