Ruelle's Zeta Functions in Hyperbolic Dynamics

• Ruelle zeta functions are defined as Euler products over primitive periodic orbits that capture the length spectrum in hyperbolic and dynamical systems.
• They connect closely with Selberg zeta functions and operator-theoretic models, using transfer operators and trace formulas to reveal spectral and geometric properties.
• Their analytic features include meromorphic continuation, finite-order growth, and functional equations that reflect topological invariants and scattering data.

Ruelle zeta functions are dynamical zeta functions built from primitive periodic orbits. In the geodesic-flow setting they are Euler products over primitive closed geodesics, such as

$$\zeta_R(s)=\prod_{\gamma\in\mathcal G}\bigl(1-e^{-s\ell_\gamma}\bigr)$$

for a compact negatively curved surface, while in twisted settings each factor is replaced by a determinant involving a representation of the relevant fundamental group (Dyatlov et al., 2016, Spilioti, 2015, Frahm et al., 2021). Across compact and noncompact hyperbolic surfaces, odd-dimensional hyperbolic manifolds, locally symmetric spaces, Axiom A flows, Ruelle-expanding maps, and non-Archimedean rational dynamics, these functions encode length-spectrum or periodic-point data while interacting with Selberg zeta functions, transfer operators, Pollicott–Ruelle resonances, analytic torsion, and Reidemeister–Turaev torsion (Avdispahic et al., 2014, Wright, 2010, Jiang et al., 26 Aug 2025).

1. Definitions and geometric frameworks

In the classical hyperbolic surface setting, the Ruelle zeta function is attached to the geodesic flow and is defined by an Euler product over primitive closed geodesics. For a compact connected oriented negatively curved surface $(\Sigma,g)$, the product is indexed by primitive closed geodesics $\gamma$ with lengths $\ell_\gamma$; negative curvature implies that the geodesic flow on $M=S^*\Sigma$ is Anosov (Dyatlov et al., 2016). For a cofinite hyperbolic Riemann surface

$X=\Gamma\backslash \mathbb H$

of type $(g;n;m_1,\dots,m_v)$, meaning genus $g$, $n$ cusps, and $v$ ramification points of orders $(\Sigma,g)$0, the Ruelle zeta function is similarly defined by

$(\Sigma,g)$1

with $(\Sigma,g)$2 running over primitive closed geodesics (Teo, 2019).

The same construction persists under twisting. On a compact hyperbolic surface $(\Sigma,g)$3 of genus $(\Sigma,g)$4, a finite-dimensional complex representation

$(\Sigma,g)$5

defines a twisted Ruelle zeta function

$(\Sigma,g)$6

where $(\Sigma,g)$7 ranges over primitive conjugacy classes, equivalently prime closed geodesics (Frahm et al., 2021). In compact odd-dimensional hyperbolic manifolds

$(\Sigma,g)$8

with $(\Sigma,g)$9 or $\gamma$0, $\gamma$1 odd, and $\gamma$2, the twisted theory also allows an auxiliary $\gamma$3-type $\gamma$4, producing

$\gamma$5

Here $\gamma$6 is the standard $\gamma$7-decomposition associated with the closed geodesic $\gamma$8 (Spilioti, 2015).

These formulations already display the two persistent themes of the subject. First, Ruelle zeta functions are orbit-counting objects: they package the primitive length spectrum or periodic-point data. Second, they are not restricted to a single curvature model or coefficient system. The data block shows compact and noncompact surfaces, manifolds of arbitrary odd dimension, orbisurfaces, open systems with boundary, and non-unitary coefficient systems, all within a common zeta-function framework (Dyatlov et al., 2016, Teo, 2019, Spilioti, 2015).

2. Selberg factorization and operator-theoretic models

A central structural feature is the relation to Selberg zeta functions. For hyperbolic surfaces, including compact surfaces, cofinite surfaces with cusps and elliptic points, and compact hyperbolic orbisurfaces, one has the standard quotient formula

$\gamma$9

This identity converts questions about a dynamical Euler product into questions about the corresponding Selberg zeta function and its functional equation (Teo, 2019, Frahm et al., 2021, Bénard et al., 2021).

In compact odd-dimensional hyperbolic manifolds and, more generally, compact odd-dimensional rank-one locally symmetric spaces, the relation is an alternating product over shifted Selberg zeta functions. Writing

$\ell_\gamma$0

with the factors $\ell_\gamma$1 built from the $\ell_\gamma$2-decomposition of $\ell_\gamma$3, one obtains the Ruelle zeta function from finitely many Selberg zeta functions (Spilioti, 2015). In Bunke–Olbrich’s formalism this becomes

$\ell_\gamma$4

which is the main bridge from the dynamical side to the spectral side (Avdispahic et al., 2014).

A second framework uses transfer operators. For Axiom A flows, the Ruelle transfer operator

$\ell_\gamma$5

controls periodic-orbit sums

$\ell_\gamma$6

and Ruelle’s Lemma gives a quantitative comparison between $\ell_\gamma$7 and transfer-operator iterates $\ell_\gamma$8 (Wright, 2010). For Ruelle-expanding maps on compact metric spaces, a finite symbolic model leads to an alternating trace formula

$\ell_\gamma$9

from which the zeta function is rational, explicitly a quotient of finite determinants (Magalhães, 2010).

The non-Archimedean theory retains the transfer-operator viewpoint. For hyperbolic and subhyperbolic rational maps on $M=S^*\Sigma$0 or more generally on non-Archimedean local fields of characteristic $M=S^*\Sigma$1, transfer operators acting on $M=S^*\Sigma$2-valued analytic function spaces are shown to be $M=S^*\Sigma$3-nuclear, and their determinants produce meromorphic Ruelle-type zeta functions on $M=S^*\Sigma$4 (Jiang et al., 26 Aug 2025).

3. Meromorphic continuation, finite-order growth, and functional equations

Meromorphic continuation is one of the foundational analytic properties of Ruelle zeta functions. For compact hyperbolic odd-dimensional manifolds and arbitrary finite-dimensional, not necessarily unitary, twists $M=S^*\Sigma$5, both Selberg and Ruelle zeta functions extend meromorphically to all of $M=S^*\Sigma$6 (Spilioti, 2015). On compact hyperbolic surfaces, the same holds for arbitrary finite-dimensional complex representations $M=S^*\Sigma$7 of $M=S^*\Sigma$8 (Frahm et al., 2021). For compact odd-dimensional locally symmetric spaces of rank one and negative curvature, the meromorphic continuations of the associated Selberg and Ruelle zeta functions are even more rigid: they can be written as quotients of entire functions of finite order at most the dimension $M=S^*\Sigma$9 of the manifold (Avdispahic et al., 2014).

This finite-order phenomenon sharpens mere continuation into a global growth statement. If

$X=\Gamma\backslash \mathbb H$0

with $X=\Gamma\backslash \mathbb H$1 and $X=\Gamma\backslash \mathbb H$2 entire of Hadamard order $X=\Gamma\backslash \mathbb H$3, then the zero and pole sets have exponent of convergence compatible with the ambient dimension. The cited work derives this from spectral discreteness of the relevant elliptic or Dirac-type operators, Weyl-law growth of eigenvalues, and Hadamard factorization (Avdispahic et al., 2014). In even-dimensional compact locally symmetric spaces, the singularity distribution can be counted more explicitly: the number $X=\Gamma\backslash \mathbb H$4 of singularities of the Ruelle zeta function in a vertical strip satisfies

$X=\Gamma\backslash \mathbb H$5

while the Selberg singularity count has leading term of order $X=\Gamma\backslash \mathbb H$6 (Avdispahic et al., 2014).

Functional equations make the continuation highly structured. For compact hyperbolic surfaces and arbitrary finite-dimensional twists,

$X=\Gamma\backslash \mathbb H$7

and this directly determines the vanishing order at the origin (Frahm et al., 2021). For cofinite hyperbolic Riemann surfaces with cusps and ramification points,

$X=\Gamma\backslash \mathbb H$8

so the scattering determinant $X=\Gamma\backslash \mathbb H$9 and the elliptic orders $(g;n;m_1,\dots,m_v)$0 enter explicitly (Teo, 2019). In compact odd-dimensional hyperbolic manifolds with non-unitary twists, the functional equations involve exponential terms and, in the non-Weyl-invariant setting, eta-invariant phases; for example,

$(g;n;m_1,\dots,m_v)$1

in the Weyl-invariant case (Spilioti, 2015).

These results place Ruelle zeta functions at the intersection of dynamical orbit products, spectral determinants, and classical entire-function theory. The cited works also situate this control within prime geodesic analysis, where zero and pole distribution affects error terms (Avdispahic et al., 2014, Avdispahic et al., 2014).

4. The point $(g;n;m_1,\dots,m_v)$2: surfaces, resonances, and cohomology

For compact oriented negatively curved surfaces, the behavior at the origin is topological. If $(g;n;m_1,\dots,m_v)$3 is compact, connected, oriented, and negatively curved, then $(g;n;m_1,\dots,m_v)$4 has a zero at $(g;n;m_1,\dots,m_v)$5 of order

$(g;n;m_1,\dots,m_v)$6

Equivalently, $(g;n;m_1,\dots,m_v)$7 is holomorphic and nonzero at $(g;n;m_1,\dots,m_v)$8 (Dyatlov et al., 2016). The proof factors the zeta function as

$(g;n;m_1,\dots,m_v)$9

where the zeros of $g$0 are governed by Pollicott–Ruelle resonances of $g$1 on $g$2-forms annihilated by the flow direction. At the origin,

$g$3

so the vanishing order becomes $g$4 (Dyatlov et al., 2016).

For compact oriented negatively curved surfaces with strictly convex boundary, the corresponding order is modified to

$g$5

The mechanism is again resonance-theoretic, but now the relevant degree-one resonant space is identified with relative cohomology: $g$6 while the degree $g$7 and $g$8 contributions vanish: $g$9 The boundary therefore changes the zero order from the closed formula $n$0 to the relative-cohomological formula $n$1 (Hadfield, 2018).

Twisting scales the order by the coefficient dimension in the compact hyperbolic case. For a compact hyperbolic surface of genus $n$2 and arbitrary finite-dimensional complex $n$3,

$n$4

and

$n$5

The sign depends on the multiplicity of the zero eigenvalue of the twisted Laplacian (Frahm et al., 2021).

A more recent resonance-theoretic refinement treats irreducible representations of the unit tangent bundle group for closed Anosov surfaces. There exists an open subset $n$6 of irreducible representations, whose complement has complex codimension at least $n$7, such that for $n$8 the twisted Ruelle zeta function has order $n$9 at $v$0 if $v$1 factors through $v$2, and order $v$3 otherwise (Humbert et al., 12 Feb 2026). The governing identity is

$v$4

which expresses the vanishing order in terms of generalized twisted Pollicott–Ruelle resonant states at zero (Humbert et al., 12 Feb 2026).

Taken together, these results show that there is no single universal “surface formula” independent of geometric setting. Closed surfaces, surfaces with boundary, twisted compact hyperbolic surfaces, and generic twisted Anosov-surface settings all exhibit topological control at $v$5, but the controlling cohomology and the resulting order depend on whether one is in the absolute, relative, or twisted regime (Dyatlov et al., 2016, Hadfield, 2018, Frahm et al., 2021, Humbert et al., 12 Feb 2026).

5. Cusps, elliptic points, orbifolds, and torsion

For cofinite hyperbolic Riemann surfaces with cusps and ramification points, the origin is no longer governed solely by the Euler characteristic. Writing the scattering determinant as

$v$6

and defining

$v$7

the exact asymptotic at $v$8 is

$v$9

Hence $(\Sigma,g)$00 has order $(\Sigma,g)$01 at the origin, and each elliptic point contributes multiplicatively through the factor $(\Sigma,g)$02 (Teo, 2019). The same work determines the order at other integers: $(\Sigma,g)$03 has a simple zero at $(\Sigma,g)$04, a zero of order $(\Sigma,g)$05 at $(\Sigma,g)$06, is regular and nonzero at positive integers $(\Sigma,g)$07, and at negative integers $(\Sigma,g)$08 with $(\Sigma,g)$09 has order

$(\Sigma,g)$10

The paper remarks that $(\Sigma,g)$11 can be negative, so poles may occur at some negative integers (Teo, 2019).

Compact hyperbolic orbisurfaces lead to a different torsion picture. Let

$(\Sigma,g)$12

be a compact hyperbolic orbisurface with cone points of orders $(\Sigma,g)$13, and let $(\Sigma,g)$14 be its unit tangent bundle. For a representation

$(\Sigma,g)$15

the twisted Ruelle zeta function is defined over prime closed geodesics on $(\Sigma,g)$16 (Bénard et al., 2021). If $(\Sigma,g)$17, where $(\Sigma,g)$18 is the generator of the central fiber subgroup in

$(\Sigma,g)$19

then the representation is acyclic,

$(\Sigma,g)$20

the zeta function is regular at $(\Sigma,g)$21, and

$(\Sigma,g)$22

where $(\Sigma,g)$23 is the sign-refined Reidemeister–Turaev torsion computed with the Euler structure induced by the geodesic flow and the natural homology orientation of $(\Sigma,g)$24 (Bénard et al., 2021).

If instead $(\Sigma,g)$25, so that $(\Sigma,g)$26 factors through $(\Sigma,g)$27, then the zeta function has a zero whose order and leading coefficient reflect the orbifold holonomy. With $(\Sigma,g)$28, $(\Sigma,g)$29, and $(\Sigma,g)$30, the vanishing order is

$(\Sigma,g)$31

and the leading term involves $(\Sigma,g)$32, $(\Sigma,g)$33, and the determinants $(\Sigma,g)$34 (Bénard et al., 2021).

In compact odd-dimensional hyperbolic manifolds, a parallel torsion formula holds for representations close to acyclic unitary ones. For $(\Sigma,g)$35 in a $(\Sigma,g)$36-open neighborhood of the acyclic unitary locus, the twisted Ruelle zeta function is regular at $(\Sigma,g)$37 and

$(\Sigma,g)$38

the Cappell–Miller complex torsion. Using Braverman–Kappeler refined analytic torsion,

$(\Sigma,g)$39

For unitary acyclic $(\Sigma,g)$40, this reduces to

$(\Sigma,g)$41

in the sense described in the paper (Spilioti, 2020).

These results show that once cusps, elliptic points, orbifold singularities, or non-unitary twists are introduced, the origin of the zeta function is controlled not just by topology, but by scattering data, local isotropy data, fixed subspaces of elliptic holonomy, eta invariants, and torsion refinements (Teo, 2019, Bénard et al., 2021, Spilioti, 2020).

6. Extensions beyond classical geodesic flows

Ruelle zeta functions are not confined to constant-curvature geodesic flows. For Axiom A flows on compact smooth manifolds, the zeta function

$(\Sigma,g)$42

is related to periodic-orbit sums

$(\Sigma,g)$43

and Ruelle’s Lemma provides an inequality comparing $(\Sigma,g)$44 with transfer-operator iterates in a way suitable for analytic continuation arguments (Wright, 2010). The cited paper emphasizes that the higher-dimensional case requires Hölder rather than Lipschitz control and replaces interval-based arguments by Markov leaves and cylinder estimates (Wright, 2010).

For Ruelle-expanding maps $(\Sigma,g)$45 on compact metric spaces, the zeta function

$(\Sigma,g)$46

is rational. The proof constructs a finite symbolic cover, derives the alternating trace identity

$(\Sigma,g)$47

and concludes that $(\Sigma,g)$48 is a quotient of finite determinants $(\Sigma,g)$49 (Magalhães, 2010). This places Ruelle zeta functions for expanding systems squarely inside finite-dimensional algebraic combinatorics.

The non-Archimedean theory extends the subject to rational maps on $(\Sigma,g)$50, with $(\Sigma,g)$51 a non-Archimedean local field of characteristic $(\Sigma,g)$52. For hyperbolic and subhyperbolic maps, finite Markov decompositions of the Julia set lead to transfer operators on Banach spaces $(\Sigma,g)$53 of $(\Sigma,g)$54-valued analytic functions on discs (Jiang et al., 26 Aug 2025). If $(\Sigma,g)$55 is $(\Sigma,g)$56-nuclear, then $(\Sigma,g)$57 is analytic on $(\Sigma,g)$58, and the resulting periodic-orbit expansion defines a zeta function that is a quotient of entire functions, hence meromorphic on $(\Sigma,g)$59 (Jiang et al., 26 Aug 2025). The same framework yields a Levin–Sodin–Yuditski type identity and information on Julia-set geometry, including a Hausdorff dimension formula

$(\Sigma,g)$60

for subhyperbolic rational functions on $(\Sigma,g)$61, where $(\Sigma,g)$62 is algebraic and $(\Sigma,g)$63 (Jiang et al., 26 Aug 2025).

A plausible implication is that “Ruelle zeta function” designates not a single rigid object but a family of orbit-product constructions whose analytic realization depends on the ambient dynamical category. In geodesic and locally symmetric settings, Selberg theory, scattering, and torsion dominate; in Axiom A and expanding settings, transfer operators and symbolic codings dominate; in the non-Archimedean setting, $(\Sigma,g)$64-nuclear operators and $(\Sigma,g)$65-analytic determinants replace classical complex-analytic Fredholm theory (Spilioti, 2015, Wright, 2010, Magalhães, 2010, Jiang et al., 26 Aug 2025).