Ruelle Zeta Function

• Ruelle zeta function is defined for hyperbolic flows as an Euler product over primitive periodic orbits that systematically encodes dynamical information.
• Its analytic structure is unveiled through methods like meromorphic continuation and spectral analysis of transfer operators in both continuous and symbolic settings.
• The function bridges geometric, topological, and dynamical insights by connecting geodesic flow properties to special value formulas, resonance theory, and torsion invariants.

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 (Frahm et al., 2021, Dyatlov et al., 2014, Spilioti, 3 Mar 2026).

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 $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$0 along the orbit (Dyatlov et al., 2014). The untwisted case $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$1 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 $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$2, the geodesic flow on the unit tangent bundle $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$3 is an Anosov flow, and each nontrivial conjugacy class in $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$4 corresponds to a closed geodesic. The classical surface formula

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$5

converges absolutely for $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$6 sufficiently large (Frahm et al., 2021). In this setting the Ruelle zeta function is a dynamical analogue of the Riemann zeta function, with prime numbers replaced by primitive closed geodesics (Dyatlov et al., 2016).

A discrete-time analogue appears in symbolic dynamics. For a one-sided topological Markov shift $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$7 and a super-continuous potential $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$8, the Ruelle zeta function is defined formally by

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$9

where $\gamma$0 is the $\gamma$1-step Birkhoff sum (Nakagawa, 2020). 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 $\gamma$2, Frahm and Spilioti define the twisted Ruelle zeta function by

$\gamma$3

and the twisted Selberg zeta function by

$\gamma$4

A fundamental identity is

$\gamma$5

obtained by direct comparison of Euler products (Frahm et al., 2021). This relation is the basic mechanism by which analytic properties of $\gamma$6 transfer to $\gamma$7.

In compact odd-dimensional hyperbolic manifolds, the same structure persists in representation-theoretically enriched form. For $\gamma$8, $\gamma$9 odd, a finite-dimensional complex representation $\ell(\gamma)$0, and $\ell(\gamma)$1, the twisted Ruelle zeta function is

$\ell(\gamma)$2

and it can be expressed as an alternating product of Selberg-type factors built from exterior powers of the $\ell(\gamma)$3-action on $\ell(\gamma)$4 (Spilioti, 2015). This formulation makes explicit that the Ruelle zeta function is not isolated from harmonic analysis on $\ell(\gamma)$5; 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: $\ell(\gamma)$6 This directly controls the behavior at $\ell(\gamma)$7 (Frahm et al., 2021).

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

$\ell(\gamma)$8

admits a meromorphic continuation to the whole complex plane (Dyatlov et al., 2014).

The analytic object controlling this continuation is the resolvent of the generator of the flow. In the same framework, the restricted resolvent

$\ell(\gamma)$9

extends meromorphically, and its poles are the Pollicott–Ruelle resonances (Dyatlov et al., 2014). The associated dynamical trace

$e^{tX}$0

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: $e^{tX}$1 Each $e^{tX}$2 is entire, and its zeros encode Pollicott–Ruelle resonances of the Lie derivative on the bundle $e^{tX}$3 of forms annihilated by contraction with the flow vector field. Near $e^{tX}$4, the order of $e^{tX}$5 is therefore an alternating sum of zero-resonance multiplicities,

$e^{tX}$6

reducing the special-value problem to a resonance multiplicity calculation (Dyatlov et al., 2016). 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 $e^{tX}$7 of that operator. Under an additional regularity condition, one obtains a canonical product representation for $e^{tX}$8, making precise the identification of zeros of the zeta function with reciprocals of transfer-operator eigenvalues (Nakagawa, 2020).

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 $e^{tX}$9, the flat bundle

$M$0

carries a twisted Bochner–Laplace operator

$M$1

Its principal symbol is

$M$2

so its spectrum is discrete even though $M$3 is not self-adjoint when $M$4 is non-unitary (Frahm et al., 2021). The zeros of the twisted Selberg zeta function are then organized by the eigenvalues of $M$5.

In compact odd-dimensional hyperbolic manifolds, non-unitary twisting requires a more elaborate analytic package. For a finite-dimensional representation $M$6, one considers the associated flat bundle $M$7, the twisted de Rham complex, the odd-signature operator

$M$8

and the refined analytic torsion $M$9 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 $\gamma^\sharp$0, and identifies $\gamma^\sharp$1 with Cappell–Miller torsion when $\gamma^\sharp$2 is acyclic and close enough to an acyclic unitary representation (Spilioti, 2020).

A further refinement arises on the unit tangent bundle $\gamma^\sharp$3 of an Anosov surface. There, representations of $\gamma^\sharp$4 split into two qualitatively different classes: those that factor through $\gamma^\sharp$5, equivalently $\gamma^\sharp$6, and those that do not, for which irreducibility forces $\gamma^\sharp$7 with $\gamma^\sharp$8. Humbert and Tao show that there exists an open subset $\gamma^\sharp$9 of irreducible representations of $T_{\gamma^\sharp}$0, with complement of complex codimension at least one, on which the zero-resonance spaces can be computed explicitly (Humbert et al., 12 Feb 2026). This factorization/non-factorization dichotomy is decisive for the behavior of the twisted Ruelle zeta function at $T_{\gamma^\sharp}$1.

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

The point $T_{\gamma^\sharp}$2 is the main arithmetic-topological special value of the Ruelle zeta function. For untwisted negatively curved oriented closed surfaces $T_{\gamma^\sharp}$3, Dyatlov and Zworski prove that

$T_{\gamma^\sharp}$4

Equivalently, $T_{\gamma^\sharp}$5 has a zero at $T_{\gamma^\sharp}$6 of order $T_{\gamma^\sharp}$7 (Dyatlov et al., 2016). The order is therefore purely topological, independent of the variable negative curvature metric.

For compact hyperbolic surfaces with an arbitrary finite-dimensional complex representation $T_{\gamma^\sharp}$8, Frahm and Spilioti prove the twisted analogue

$T_{\gamma^\sharp}$9

together with the local expansion

$$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$0

The factor $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$1 shows that, in the factor-through-surface case, the order of vanishing remains Euler-characteristic controlled, now scaled by the rank of the twist (Frahm et al., 2021).

This should be contrasted with acyclic odd-dimensional twists. For compact hyperbolic odd-dimensional manifolds and acyclic $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$2 close to an acyclic unitary representation, Spilioti proves that the twisted Ruelle zeta function is regular at zero and satisfies

$$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$3

where $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$4 is Cappell–Miller torsion and $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$5 is refined analytic torsion (Spilioti, 2020). 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 $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$6 of irreducible representations of $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$7, Humbert and Tao prove that if $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$8 factors through $$\zeta_V(\lambda):=\prod_{\gamma^\sharp}\Big(1-\exp\big(-T_{\gamma^\sharp}(\lambda+V_{\gamma^\sharp})\big)\Big), \qquad \Re \lambda\gg 1,$$9, then

$V_{\gamma^\sharp}$0

whereas if $V_{\gamma^\sharp}$1 does not factor through $V_{\gamma^\sharp}$2, then

$V_{\gamma^\sharp}$3

and

$V_{\gamma^\sharp}$4

This extends Fried’s conjectural dynamical–torsion correspondence to a generic set of acyclic, not necessarily unitary, representations (Humbert et al., 12 Feb 2026). The resulting picture is not uniform across all settings: some regimes force a zero at $V_{\gamma^\sharp}$5, others force a nonzero torsion value.

6. Variants, boundary cases, and alternative formalisms

Several extensions modify the behavior at $V_{\gamma^\sharp}$6 by introducing cusps, orbifold points, boundary, or different dynamical models. For a cofinite hyperbolic Riemann surface of type $V_{\gamma^\sharp}$7, Teo proves

$V_{\gamma^\sharp}$8

and shows that the order at $V_{\gamma^\sharp}$9 is

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$00

where $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$01 is the order of the scattering determinant $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$02 at $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$03. The leading coefficient involves $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$04, $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$05, and the ramification indices $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$06 (Teo, 2019). In the multiplier-system setting on finite-volume hyperbolic Riemann surfaces, a functional equation of the form

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$07

was established, with $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$08 explicitly determined by topological data and $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$09; this yields further special-value formulas and additional cases of Fried’s conjecture (Jorgenson et al., 2024).

For negatively curved oriented surfaces with strictly convex boundary, Hadfield proves that the Ruelle zeta function has a zero at $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$10 of multiplicity

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$11

The proof identifies the relevant zero-resonance space with a relative cohomology group $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$12, so the boundary changes the topological answer from the closed-surface value $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$13 to a relative-cohomological quantity (Hadfield, 2018).

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 $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$14 (Nakagawa, 2020). For the quadratic family $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$15 with $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$16, the relevant Ruelle zeta function coincides with a Fredholm determinant $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$17, and as $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$18 the normalized zero-counting measures converge to the uniform distribution on the circle $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$19 (Eremenko et al., 2017). 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 $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$20 partition function in an unusual contact gauge satisfies

$$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$21

while in the metric gauge it gives analytic torsion. This reframes Fried’s conjecture as a gauge-fixing-invariance statement in the $$R(s)=\prod_{\gamma}\left(1-e^{-s\,\ell(\gamma)}\right),$$22 formalism, with homotopies of Lagrangian submanifolds interpolating between the dynamical and metric descriptions (Hadfield et al., 2020).

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.