nLab Ruelle zeta function

under construction

Idea

For dynamical systems

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

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

For compact hyperbolic manifolds

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

$$R_\rho(s) = \underset{\gamma \neq e}{\prod} \det \left( Id - hol_\rho(\gamma) e^{-s l(\gamma)} \right)$$

and defined from there by analytic continuation.

Here

• the product is over prime geodesics $\gamma$;

• of length $l(\gamma)$;

• and $hol_\rho(\gamma)$ denotes the holonomy of $\rho$ along $\gamma$.

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 $Frob_p$ (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 $R_\rho$ at the origin encodes

• the Reidemeister torsion (Fried 86, theorem 1);

• the volume of the hyperbolic manifold (Mathai 92, Lott 92).

References

Original articles include

• David Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. math. 84, 523-540 (1986) (pdf)

• Ulrich Bunke, Martin Olbrich, Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one (arXiv:dg-ga/9407012)

• David Ruelle, Dynamical zeta functions and transfer operators, Notices

  Amer. Math. Soc. 49 (2002), no. 8, 887895.

Seminar/lecture notes include

• Selberg and Ruelle zeta functions for compact hyperbolic manifolds (pdf)

• V. Baladi, Dynamical zeta functions, arXiv:1602.05873

Relation to the volume of hyperbolic manifolds is discussed in

• Varghese Mathai, section 6 of $L^2$-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386

• John Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 no 2 (1992) (pdf)

The analogy with the Artin L-function is discussed in

• Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

• Masanori Morishita, section 12.1 of Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)