nLab
Ruelle zeta function

under construction

Context

Theta functions

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in physics

Riemannian geometry

Basic definitions

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Further concepts

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Theorems

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Applications

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Contents

Idea

For dynamical systems

The Ruelle zeta function is defined for a function f:MMf : M \to M on a compact manifold MM. Assume that the set Fix(f m)Fix(f^m) is finite for all m1m \geq 1. Suppose that ϕ:M d×d\phi: M \to \mathbb{C}^{d \times d} is a matrix-valued function. The first type of Ruelle zeta function is defined by

ζ(z)=exp{ m1z mm xFix(f m)Tr( k=0 m1ϕ(f k(x)))}. \zeta(z) = exp \Bigg\{ \sum_{m \geq 1} \frac{z^m}{m} \sum_{x \in Fix(f^m)} Tr \bigg(\prod^{m-1}_{k=0} \phi(f^k(x)) \bigg) \Bigg\}.

For compact hyperbolic manifolds

For X ΓX_\Gamma a hyperbolic manifold of odd dimension, and given a flat GLGL-principal connection ρ\rho on X ΓX_\Gamma, the Ruelle zeta function is the meromorphic function R ρR_\rho given for Re(s)>(dim(X γ)1)/2Re(s) \gt (dim(X_\gamma)-1)/2 by the infinite product (Euler product) over the characteristic polynomials of the monodromy/holonomy of the flat connection along prime geodesics

R ρ(s)=γedet(Idhol ρ(γ)e sl(γ)) R_\rho(s) = \underset{\gamma \neq e}{\prod} \det \left( Id - hol_\rho(\gamma) e^{-s l(\gamma)} \right)

and defined from there by analytic continuation.

Here

Properties

Relation to the Selberg zeta function

for the moment see at Selberg zeta function.

Analogy to the Artin L-function

The Ruelle zeta function is analogous to the standard definition of an Artin L-function if one interprets a) a Frobenius map Frob pFrob_p (as discussed there) as an element of the arithmetic fundamental group of an arithmetic curve and b) a Galois representation as a flat connection.

So under this analogy the Ruelle zeta function for hyperbolic 3-manifolds as well as the Artin L-function for a number field both are like an infinite product over primes (prime geodesics in one case, prime ideals in the other, see also at Spec(Z) – As a 3-dimensional space containing knots) of determinants of monodromies of the given flat connection.

See at Artin L-function – Analogy with Selberg/Ruelle zeta function for more. This analogy has been highlighted in (Brown 09, Morishita 12, remark 12.7).

Relation to volume and analytic torsion

The special value of R ρR_\rho at the origin encodes

References

Original articles include

Seminar/lecture notes include

Relation to the volume of hyperbolic manifolds is discussed in

The analogy with the Artin L-function is discussed in