---
title: Ruelle Zeta Function
url: https://www.emergentmind.com/topics/ruelle-zeta-function
type: topic
---

# Ruelle Zeta Function

The Ruelle zeta function is a dynamical zeta function attached to a hyperbolic flow, defined by an Euler product over primitive periodic orbits. For the geodesic flow on a compact hyperbolic surface \(X\), it has the classical form
\[
R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),
\]
with \(\gamma\) ranging over primitive closed geodesics and \(\ell(\gamma)\) their lengths. In geometric and dynamical settings, it encodes the periodic orbit structure of the flow; in many cases its analytic continuation, zeros, poles, and special values are governed by Selberg zeta functions, Pollicott–Ruelle resonances, Laplace-type operators, and torsion invariants [2105.13321][1410.5516][2603.03156].

## 1. Definition and dynamical framework

For an Anosov flow \(e^{tX}\) on a compact manifold \(M\), with primitive periodic orbits \(\gamma^\sharp\) of periods \(T_{\gamma^\sharp}\), a weighted Ruelle zeta function is defined by
\[
\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big),
\qquad \Re \lambda\gg 1,
\]
where \(V_{\gamma^\sharp}\) is the cycle average of a smooth potential \(V\) along the orbit [1410.5516]. The untwisted case \(V\equiv 0\) gives the basic product over primitive periodic orbits. In the geodesic setting, the periodic trajectories are closed geodesics, so the zeta function becomes a generating object for the length spectrum.

On a compact hyperbolic surface \(X\cong \Gamma\backslash\mathbb H^2\), the geodesic flow on the unit tangent bundle \(T^1X\) is an Anosov flow, and each nontrivial conjugacy class in \(\Gamma\) corresponds to a closed geodesic. The classical surface formula
\[
R(s)=\prod_{\gamma}\left(1-e^{-s\ell(\gamma)}\right)
\]
converges absolutely for \(\Re(s)\) sufficiently large [2105.13321]. In this setting the Ruelle zeta function is a dynamical analogue of the Riemann zeta function, with prime numbers replaced by primitive closed geodesics [1606.04560].

A discrete-time analogue appears in symbolic dynamics. For a one-sided topological Markov shift \((\Sigma_A^+,\sigma_A)\) and a super-continuous potential \(f\), the Ruelle zeta function is defined formally by
\[
\zeta_f(z)=\exp\left(\sum_{q=1}^\infty \frac{z^q}{q}\sum_{w\in \mathrm{Per}_q(\sigma_A)} e^{S_qf(w)}\right),
\]
where \(S_qf\) is the \(q\)-step Birkhoff sum [2006.01564]. This discrete formulation and the continuous-time flow formulation share the same organizing principle: periodic orbit data are repackaged as an analytic function whose zeros and poles reflect spectral properties of an associated transfer or evolution operator.

## 2. Hyperbolic geometry and the Selberg relation

In constant negative curvature, the Ruelle zeta function is closely tied to the Selberg zeta function. For a compact hyperbolic surface and a finite-dimensional complex representation \(\chi:\Gamma\to \mathrm{GL}(V_\chi)\), Frahm and Spilioti define the twisted Ruelle zeta function by
\[
R(s;\chi):=\prod_{[\gamma]\neq e}^{\mathrm{prime}}
\det\bigl(\mathrm{Id}-\chi(\gamma)e^{-s\ell(\gamma)}\bigr),
\]
and the twisted Selberg zeta function by
\[
Z(s;\chi):=
\prod_{[\gamma]\neq e}^{\mathrm{prime}}
\prod_{k=0}^{\infty}
\det\bigl(\mathrm{Id}-\chi(\gamma)e^{-(s+k)\ell(\gamma)}\bigr).
\]
A fundamental identity is
\[
R(s;\chi)=\frac{Z(s;\chi)}{Z(s+1;\chi)},
\]
obtained by direct comparison of Euler products [2105.13321]. This relation is the basic mechanism by which analytic properties of \(Z(s;\chi)\) transfer to \(R(s;\chi)\).

In compact odd-dimensional hyperbolic manifolds, the same structure persists in representation-theoretically enriched form. For \(X=\Gamma\backslash \mathbb H^d\), \(d\) odd, a finite-dimensional complex representation \(\chi:\Gamma\to \mathrm{GL}(V_\chi)\), and \(\sigma\in \widehat M\), the twisted Ruelle zeta function is
\[
R(s;\sigma,\chi)=
\prod_{\substack{[\gamma]\neq e\\ [\gamma]\ \mathrm{primitive}}}
\det\big(\mathrm{Id}-\chi(\gamma)\otimes \sigma(m_\gamma)e^{-s\,l(\gamma)}\big)^{(-1)^{d-1}},
\]
and it can be expressed as an alternating product of Selberg-type factors built from exterior powers of the \(MA\)-action on \(\mathfrak n_\mathbb C\) [1506.04672]. This formulation makes explicit that the Ruelle zeta function is not isolated from harmonic analysis on \(G\); it is a compressed dynamical shadow of a richer Selberg-theoretic structure.

The Selberg relation also underlies functional equations. On compact hyperbolic surfaces, the functional equation for the twisted Selberg zeta yields a corresponding functional equation for the twisted Ruelle zeta:
\[
R(s;\chi)R(-s;\chi)=\bigl(2\sin(\pi s)\bigr)^{2(2g-2)\dim(V_\chi)}.
\]
This directly controls the behavior at \(s=0\) [2105.13321].

## 3. Meromorphic continuation, resonances, and dynamical determinants

The initial Euler products defining Ruelle zeta functions are only convergent in a right half-plane. Their extension beyond that domain is a central analytic problem. For open hyperbolic systems with orientable stable and unstable foliations, including geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, Dyatlov and Guillarmou prove that
\[
\zeta_V(\lambda)=\prod_{\gamma^\sharp}\Big(1-\exp(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp}))\Big)
\]
admits a meromorphic continuation to the whole complex plane [1410.5516].

The analytic object controlling this continuation is the resolvent of the generator of the flow. In the same framework, the restricted resolvent
\[
\mathbf R(\lambda)=1_{\mathcal U}(\mathbf X+\lambda)^{-1}1_{\mathcal U}
\]
extends meromorphically, and its poles are the Pollicott–Ruelle resonances [1410.5516]. The associated dynamical trace
\[
F_{\mathbf X}(\lambda)=
\sum_{\gamma}
\frac{e^{-\lambda T_\gamma}T_\gamma^\sharp \operatorname{tr}\alpha_\gamma}
{|\det(I-\mathcal P_\gamma)|}
\]
has the same poles, and these poles govern the logarithmic derivatives of dynamical zeta functions.

For contact Anosov flows on 3-manifolds, Dyatlov and Zworski factor the Ruelle zeta function through dynamical zeta functions attached to differential forms:
\[
\zeta_R(s)=\frac{\zeta_1(s)}{\zeta_0(s)\,\zeta_2(s)}.
\]
Each \(\zeta_k\) is entire, and its zeros encode Pollicott–Ruelle resonances of the Lie derivative on the bundle \(\Omega_0^k\) of forms annihilated by contraction with the flow vector field. Near \(s=0\), the order of \(\zeta_R\) is therefore an alternating sum of zero-resonance multiplicities,
\[
m_R(0)=m_1(0)-m_0(0)-m_2(0),
\]
reducing the special-value problem to a resonance multiplicity calculation [1606.04560]. This resonance-theoretic description is one of the main structural advances in modern work on Ruelle zeta functions.

A related spectral picture appears in symbolic dynamics. Nakagawa constructs, for every super-continuous potential, a Banach space on which the Ruelle transfer operator is compact, and proves a spectral representation of the zeta function in terms of the nonzero eigenvalues \(\lambda_n(f)\) of that operator. Under an additional regularity condition, one obtains a canonical product representation for \(\zeta_f(z)^{-1}\), making precise the identification of zeros of the zeta function with reciprocals of transfer-operator eigenvalues [2006.01564].

## 4. Twists, flat bundles, and non-unitary spectral theory

Twisting is implemented by a finite-dimensional representation of a fundamental group or related group, and geometrically corresponds to passing from scalar dynamics to dynamics with coefficients in a flat vector bundle. For a compact hyperbolic surface and \(\chi:\Gamma\to \mathrm{GL}(V_\chi)\), the flat bundle
\[
E_\chi=V_\chi\times_\Gamma \mathbb H^2 \to X
\]
carries a twisted Bochner–Laplace operator
\[
\Delta_\chi=-\operatorname{tr}\bigl((\nabla^{E_\chi})^2\bigr).
\]
Its principal symbol is
\[
\sigma(\Delta_\chi)(x,\xi)=\|\xi\|^2\,\mathrm{Id}_{(E_\chi)_x},
\]
so its spectrum is discrete even though \(\Delta_\chi\) is not self-adjoint when \(\chi\) is non-unitary [2105.13321]. The zeros of the twisted Selberg zeta function are then organized by the eigenvalues of \(\Delta_\chi\).

In compact odd-dimensional hyperbolic manifolds, non-unitary twisting requires a more elaborate analytic package. For a finite-dimensional representation \(\chi:\pi_1(X)\to \mathrm{GL}(V_\chi)\), one considers the associated flat bundle \(E_\chi\), the twisted de Rham complex, the odd-signature operator
\[
B_\chi=\Gamma\nabla_\chi+\nabla_\chi\Gamma,
\]
and the refined analytic torsion \(T_\chi\) in the sense of Braverman–Kappeler. In this setting Spilioti proves determinant formulas for the twisted Ruelle zeta function in terms of non-self-adjoint twisted Laplace-type operators \(\Delta_{\chi,k}^\sharp\), and identifies \(R(0;\chi)\) with Cappell–Miller torsion when \(\chi\) is acyclic and close enough to an acyclic unitary representation [2004.13474].

A further refinement arises on the unit tangent bundle \(M=S\Sigma\) of an Anosov surface. There, representations of \(\pi_1(M)\) split into two qualitatively different classes: those that factor through \(\pi_1(\Sigma)\), equivalently \(\rho(c)=\mathrm{Id}\), and those that do not, for which irreducibility forces \(\rho(c)=\zeta\,\mathrm{Id}\) with \(\zeta\neq 1\). Humbert and Tao show that there exists an open subset \(\mathcal U_g\) of irreducible representations of \(\pi_1(M)\), with complement of complex codimension at least one, on which the zero-resonance spaces can be computed explicitly [2602.12166]. This factorization/non-factorization dichotomy is decisive for the behavior of the twisted Ruelle zeta function at \(s=0\).

## 5. Special value at zero, Euler characteristic, and torsion

The point \(s=0\) is the main arithmetic-topological special value of the Ruelle zeta function. For untwisted negatively curved oriented closed surfaces \((\Sigma,g)\), Dyatlov and Zworski prove that
\[
s^{\chi(\Sigma)}\zeta_R(s)
\quad\text{is holomorphic at }s=0,\qquad
s^{\chi(\Sigma)}\zeta_R(s)\big|_{s=0}\neq 0.
\]
Equivalently, \(\zeta_R(s)\) has a zero at \(s=0\) of order \(-\chi(\Sigma)=|\chi(\Sigma)|\) [1606.04560]. The order is therefore purely topological, independent of the variable negative curvature metric.

For compact hyperbolic surfaces with an arbitrary finite-dimensional complex representation \(\chi\), Frahm and Spilioti prove the twisted analogue
\[
\operatorname{ord}_{s=0}R(s;\chi)=\dim(\chi)(2g-2),
\]
together with the local expansion
\[
R(s;\chi)=\pm (2\pi s)^{\dim(V_\chi)(2g-2)}+\text{higher order terms}.
\]
The factor \(2g-2=-\chi(X)\) shows that, in the factor-through-surface case, the order of vanishing remains Euler-characteristic controlled, now scaled by the rank of the twist [2105.13321].

This should be contrasted with acyclic odd-dimensional twists. For compact hyperbolic odd-dimensional manifolds and acyclic \(\chi\) close to an acyclic unitary representation, Spilioti proves that the twisted Ruelle zeta function is regular at zero and satisfies
\[
R(0;\chi)=\tau_\chi
=
T_\chi^2\, e^{2\pi i(\eta(B_\chi^{\mathrm{ev}})-\operatorname{rank}(E_\chi)\eta_{\mathrm{tr}})},
\]
where \(\tau_\chi\) is Cappell–Miller torsion and \(T_\chi\) is refined analytic torsion [2004.13474]. Thus the special value is nonzero and torsion-theoretic rather than vanishing.

The recent generic surface-unit-tangent-bundle theory makes this dichotomy explicit. On an open subset \(\mathcal U_g\) of irreducible representations of \(\pi_1(S\Sigma)\), Humbert and Tao prove that if \(\rho\) factors through \(\pi_1(\Sigma)\), then
\[
\operatorname{ord}_{s=0}\zeta_{g,\rho}(s)=\dim(\rho)(2G-2),
\]
whereas if \(\rho\) does not factor through \(\pi_1(\Sigma)\), then
\[
\operatorname{ord}_{s=0}\zeta_{g,\rho}(s)=0
\]
and
\[
\zeta_{g,\rho}(0)^{-1}
=
\pm \tau_{\mathfrak e_{\mathrm{geod}},\mathfrak o}(\rho)
=
\pm \det(\mathrm{Id}-\rho(c))^{2G-2}.
\]
This extends Fried’s conjectural dynamical–torsion correspondence to a generic set of acyclic, not necessarily unitary, representations [2602.12166]. The resulting picture is not uniform across all settings: some regimes force a zero at \(0\), others force a nonzero torsion value.

## 6. Variants, boundary cases, and alternative formalisms

Several extensions modify the behavior at \(s=0\) by introducing cusps, orbifold points, boundary, or different dynamical models. For a cofinite hyperbolic Riemann surface of type \((g;n;m_1,\dots,m_v)\), Teo proves
\[
R(s)=\frac{Z(s)}{Z(s+1)}
\]
and shows that the order at \(s=0\) is
\[
\operatorname{ord}_{s=0}R(s)=2g-2+n-n_0,
\]
where \(n_0\) is the order of the scattering determinant \(\varphi(s)\) at \(s=0\). The leading coefficient involves \((2\pi)^{2g-2+n}\), \(\tilde\varphi(0)^{-1}\), and the ramification indices \(\prod_j m_j\) [1901.07898]. In the multiplier-system setting on finite-volume hyperbolic Riemann surfaces, a functional equation of the form
\[
R(s,\chi)\varphi(s,\chi)=R(-s,\chi)\varphi(s,\chi)H(s,\chi)
\]
was established, with \(H(s,\chi)\) explicitly determined by topological data and \(\sin(s)\); this yields further special-value formulas and additional cases of Fried’s conjecture [2402.02959].

For negatively curved oriented surfaces with strictly convex boundary, Hadfield proves that the Ruelle zeta function has a zero at \(0\) of multiplicity
\[
1-\chi(\Sigma).
\]
The proof identifies the relevant zero-resonance space with a relative cohomology group \(H^1(M,\partial M)\), so the boundary changes the topological answer from the closed-surface value \(-\chi(\Sigma)\) to a relative-cohomological quantity [1803.10982].

Outside geodesic flows, the same analytic patterns reappear. In one-sided topological Markov shifts with super-continuous potentials, Nakagawa constructs a Banach space on which the transfer operator is compact and obtains a spectral product expansion for \(\zeta_f(z)\) [2006.01564]. For the quadratic family \(f_c(z)=z^2+c\) with \(c<-2\), the relevant Ruelle zeta function coincides with a Fredholm determinant \(D_c(\lambda)\), and as \(c\to -2\) the normalized zero-counting measures converge to the uniform distribution on the circle \(\{|\lambda|=4\}\) [1707.03441]. These cases show that the term “Ruelle zeta function” encompasses both flow zeta functions and transfer-operator determinants for expanding maps.

A distinct reinterpretation comes from topological field theory. On a compact contact manifold with Reeb-Anosov flow, the abelian \(BF\) partition function in an unusual contact gauge satisfies
\[
Z(\mathbb S_{BF},\mathbb L_X)=|\zeta_\rho(0)|^{(-1)^{n+1}},
\]
while in the metric gauge it gives analytic torsion. This reframes Fried’s conjecture as a gauge-fixing-invariance statement in the \(BV\) formalism, with homotopies of Lagrangian submanifolds interpolating between the dynamical and metric descriptions [2002.03952].

The Ruelle zeta function therefore occupies a junction of hyperbolic dynamics, trace formulas, resonance theory, spectral analysis of non-self-adjoint elliptic operators, and torsion invariants. Across compact, finite-volume, orbifold, boundary, symbolic, and field-theoretic settings, its defining feature remains the same: periodic orbit data are reorganized into an analytic function whose singularities and special values retain precise geometric and topological content.

Source: https://www.emergentmind.com/topics/ruelle-zeta-function