under construction
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in physics
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
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
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 $\gamma$.
for the moment see at Selberg zeta 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).
The special value of $R_\rho$ at the origin encodes
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)