A Ruelle dynamical zeta function for equivariant flows

TL;DR

This paper introduces an equivariant Ruelle dynamical zeta function for flows on manifolds with group actions, extending classical concepts and exploring its relation to equivariant analytic torsion with concrete examples.

Contribution

It defines a new equivariant Ruelle zeta function for proper group actions, generalizing Guillemin's trace formula and analyzing its properties and applications.

Findings

The equivariant Ruelle zeta function is computed in several examples.

The equivariant Fried conjecture holds when the Laplacian kernel vanishes.

The construction extends classical dynamical zeta functions to equivariant settings.

Abstract

For proper group actions on smooth manifolds, with compact quotients, we define an equivariant version of the Ruelle dynamical $\zeta$-function for equivariant flows satisfying a nondegeneracy condition. The construction is based on an equivariant generalisation of Guillemin's trace formula, obtained in a companion paper. This formula implies several properties of the equivariant Ruelle $\zeta$-function. We ask the question in what situations an equivariant generalisation of Fried's conjecture holds, relating the equivariant Ruelle $\zeta$-function to equivariant analytic torsion. We compute the equivariant Ruelle $\zeta$-function in several examples, including examples where the classical Ruelle $\zeta$-function is not defined. The equivariant Fried conjecture holds in the examples where the condition of the conjecture (vanishing of the kernel of the Laplacian) is satisfied.

Equations

$$P_{\gamma }:T_{\gamma (0)}M/\mathbb{R}{\gamma }^{\prime }(0)\to T_{\gamma (0)}M/\mathbb{R}{\gamma }^{\prime }(0),$$

$$T_{\gamma }^{\#}:=\min \{t>0;\gamma (t)=\gamma (0)\}$$

$$\log R_{\phi ,{\nabla }^F}(\sigma ):=\sum _{\gamma }\frac{e^{-\sigma l}}{l}\mathrm{sgn}(\det (1-P_{\gamma }))\mathrm{tr}(\rho (\gamma ))T_{\gamma }^{\#},$$

$$2\log T({\nabla }^F):=-{\frac{d}{ds}|}_{s=0}\frac{1}{\Gamma (s)}{\int }_0^{\infty }{t}^{s-1}\mathrm{Tr}((-1{)}^N{e}^{-t{\Delta }_F}-P)dt.$$

$$|R_{\phi ,{\nabla }^F}(0)|=T({\nabla }^F).$$

$$\gamma (l)=g\gamma (0)$$

$$T_{\gamma (0)}{\phi }_l\circ g^{-1}:T_{\gamma (0)}M\to T_{\gamma (0)}M$$

$$2\log R_{\varphi,\nabla^{F}}^{g}(\sigma)=-\int_{\mathbb{R}\setminus\{0\}}\operatorname{Tr}_{g}^{\flat}\bigl{(}(-1)^{N}N\varphi_{t}^{*}\bigr{)}\frac{e^{-\sigma|t|}}{|t|}\,dt.$$

$$R_{\phi ,{\nabla }^F}^g(0)=T_g({\nabla }^F).$$

$$\begin{array}{rl}g\cdot (p,Y)& =(gp,\mathrm{Ad}(g)Y);\\ {\phi }_t(p,Y)& =(\exp (tY)p,Y),\end{array}$$

$${\int }_G\chi (xm)dx=1.$$

$$L_x(\phi ):=\{l\in \mathbb{R}\setminus \{0\};\exists m\in M:{\phi }_l(m)=xm\}.$$

$$(s\cdot \gamma )(t)=\gamma (t+s),$$

$${\Gamma }_l^x(\phi ):={\tilde{\Gamma }}_l^x(\phi )/\mathbb{R}.$$

$$P_{\gamma }^x:T_{\gamma (0)}M/\mathbb{R}u(\gamma (0))\to T_{\gamma (0)}M/\mathbb{R}u(\gamma (0))$$

$${\rho }_x(\gamma ):F_{\gamma (0)}\to F_{\gamma (0)}$$

$$T_{\gamma }:={\int }_{I_{\gamma }}\chi (\gamma (s))ds,$$

$$T_{\gamma }={\int }_{\mathbb{R}}\chi (s)ds=1.$$

$$T_{\gamma }^{\#}=\min \{t>0;\gamma (t)=\gamma (0)\}$$

$$\det\bigl{(}(T_{m}(\varphi_{l}\circ g^{-1})-1)|_{T_{m}M/\mathbb{R}u(m)}\bigr{)}\not=0.$$

$$T_m{M}^{g^{-1}\circ {\phi }_l}=\mathbb{R}u(m).$$

$$L_{hg{h}^{-1}}(\phi )=L_g(\phi )$$

$${\Gamma }_l^{hg{h}^{-1}}(\phi )=h\cdot {\Gamma }_l^g(\phi ).$$

$$2\log R_{\varphi,\nabla^{F}}^{g}(\sigma)=\\ \int_{G/Z}\sum_{l\in L_{g}(\varphi)}\frac{e^{-|l|\sigma}}{|l|}\sum_{\gamma\in\Gamma^{hgh^{-1}}_{l}(\varphi)}\operatorname{sgn}\bigl{(}\det(1-P_{\gamma}^{hgh^{-1}})\bigr{)}\operatorname{tr}(\rho_{hgh^{-1}}(\gamma))T_{\gamma}\,d(hZ).$$

$$\begin{array}{rl}{\rho }_x({\gamma }_2)& ={\phi }_s\circ {\rho }_x({\gamma }_1)\circ {\phi }_{-s}\\ P_{{\gamma }_2}^x& ={\phi }_s\circ P_{{\gamma }_1}^x\circ {\phi }_{-s}.\end{array}$$

$$2\log R_{\varphi,\nabla^{F}}^{g}(\sigma)=\\ \int_{G/Z}\sum_{l\in L_{g}(\varphi)}\frac{e^{-|l|\sigma}}{|l|}\sum_{\gamma\in\Gamma^{g}_{l}(\varphi)}\operatorname{sgn}\bigl{(}\det(1-P_{\gamma}^{g})\bigr{)}\operatorname{tr}(\rho_{g}(\gamma))\int_{I_{\gamma}}\chi(h\gamma(s))\,ds\,d(hZ).$$

$$\begin{array}{rl}{T}_{h\gamma (0)}({\phi }_l\circ h{g}^{-1}{h}^{-1})& =T_{\gamma (0)}h\circ T_{\gamma (0)}({\phi }_l\circ g^{-1})\circ T_{h\gamma (0)}{h}^{-1}\\ {\rho }_{hg{h}^{-1}}(h\gamma )& =h\circ {\rho }_g(\gamma )\circ h^{-1}.\end{array}$$

$$\begin{array}{rl}\det (1-P_{h\gamma }^{hg{h}^{-1}})& =\det (1-P_{\gamma }^g)\\ \mathrm{tr}({\rho }_{hg{h}^{-1}}(h\gamma ))& =\mathrm{tr}({\rho }_g(\gamma )),\end{array}$$

$$2\log R_{\varphi,\nabla^{F}}^{g}(\sigma)=\\ \sum_{l\in L_{g}(\varphi)}\frac{e^{-|l|\sigma}}{|l|}\sum_{\gamma\in\Gamma^{g}_{l}(\varphi)}\operatorname{sgn}\bigl{(}\det(1-P_{\gamma}^{g})\bigr{)}\operatorname{tr}(\rho_{g}(\gamma))T^{g}_{\gamma},$$

$$T_{\gamma }^g:={\int }_{G/Z}{\int }_{I_{\gamma }}\chi (h\gamma (s))dsd(hZ)$$

Full text

1. 1 Introduction
2. 2 The equivariant Ruelle dynamical $\zeta$-function
3. 3 The flat $g$-trace
4. 4 A fixed point formula
5. 5 The line and the circle
6. 6 Geodesic flow on $\mathbb{R}^{n}$
7. 7 Geodesic flow on spheres

1. Introduction

The Ruelle dynamical $\zeta$-function is an expression involving the lengths of closed flow curves of a flow on a compact manifold. The Fried conjecture states that, for a reasonable class of flows, the value of this function at zero is equal to the analytic torsion of the manifold. In this paper, we construct an equivariant generalisation of the Ruelle $\zeta$-function for proper, cocompact group actions. We give examples where an equivariant generalisation of the Fried conjecture does and does not hold, and ask the question under what hypotheses this generalisation is true.

Background and motivation

Let $M$ be a compact manifold, and $\varphi$ a flow on $M$. Suppose that $\varphi$ is Anosov, i.e. that the tangent bundle of $M$ decomposes into the image of the generating vector field of $\varphi$, and two sub-bundles on which the derivative of $\varphi$ is exponentially contractive as time goes to plus or minus infinity, respectively. An example of an Anosov flow to keep in mind is the geodesic flow on the unit sphere bundle of the tangent bundle of a compact Riemannian manifold with negative sectional curvature..

If a flow curve $\gamma$ of $\varphi$ satisfies $\gamma(l)=\gamma(0)$, then the linearised Poincaré map

[TABLE]

induced by $T_{\gamma(0)}\varphi_{l}$, does not have $1$ as an eigenvalue. Therefore, $\operatorname{sgn}(\det(1-P_{\gamma}))$ is well-defined; we then say that the flow $\varphi$ is nondegenerate. Let $\nabla^{F}$ be a flat connection on a vector bundle $F\to M$. For $\gamma$ as above, let $\rho(\gamma)$ be the parallel transport map with respect to $\nabla^{F}$ along $\gamma$ from [math] to $l$. Let

[TABLE]

be the primitive period of $\gamma$. Then the Ruelle dynamical $\zeta$-function for $\varphi$ with respect to $\nabla^{F}$ is defined by

[TABLE]

where the sum is over closed flow curves $\gamma$, and $l$ is the period of such a curve $\gamma$. (See [28] for the version introduced by Ruelle, or Definition 2.7 in [33] for a more general definition.) For Anosov flows, the set of such curves is discrete, and the sum on the right in (1.3) converges for $\sigma\in\mathbb{C}$ with large real part. This follows from an exponential bound on the number of flow curves with periods less than a given value; see (8) in [21] or Lemma 2.2 in [7].

Let $\Delta_{F}:=(\nabla^{F})^{*}\nabla^{F}+\nabla^{F}(\nabla^{F})^{*}$ be the Laplacian on differential forms twisted by $F$ associated to $\nabla^{F}$. Let $N$ be the number operator on such forms, equal to $p$ times the identity on $p$-forms. Let $P$ be the orthogonal projection onto the kernel of $\Delta_{F}$. The Ray–Singer analytic torsion of $\nabla^{F}$ is defined by

[TABLE]

The Fried conjecture states [9] that if $\Delta_{F}$ has trivial kernel, then $R_{\varphi,\nabla^{F}}$ has a meromorphic extension to $\mathbb{C}$ that is regular at zero, and

[TABLE]

This was proved by Fried for the geodesic flow on hyperbolic manifolds [8], and has since been proved in several other cases, see for example [3, 6, 23, 24, 29, 30, 32, 34, 35, 41, 43]. This includes versions for non-Anosov flows, such as geodesic flows on unit sphere bundles of compact manifolds whose sectional curvature is non-positive, but not everywhere negative. For such flows a more general version of the definition of the Ruelle $\zeta$-function is used, see for example Definition 2.7 in [33]. The Fried conjecture does not hold in general, see the end of Section 2 in [33], and [31, 40].

In this paper, we define a Ruelle dynamical $\zeta$-function for flows $\varphi$ satisfying an equivariant nondegeneracy condition, on manifolds $M$ with proper actions by groups $G$ such that $M/G$ is compact. The idea is to replace flow curves $\gamma$ satisfying $\gamma(l)=\gamma(0)$ by flow curves $\gamma$ satisfying

[TABLE]

for a fixed element $g\in G$. We study several properties of such a $\zeta$-function, and verify an equivariant version of the Fried conjecture in some examples. Several factors motivate our work on this equivariant extension.

1. (1)

In the compact case, assumptions must be made on $M$ and $\varphi$ to guarantee that the Ruelle $\zeta$-function is well-defined, such as nonpositive sectional curvature for geodesic flows. Already in this case, we show that the equivariant Ruelle $\zeta$-function is often defined under much weaker conditions. For example, we compute the equivariant Ruelle $\zeta$-function for the geodesic flow on the unit sphere bundles of Euclidean space and of the two- and three-spheres in Sections 6 and 7 respectively. In those cases, the classical Ruelle $\zeta$-function is not even defined. Thus we see that even in the compact case, the equivariant Ruelle $\zeta$-function can be used to study flows on a larger class of manifolds than its classical counterpart. 2. (2)

An equivariant Ruelle $\zeta$-function captures information on the symmetries of dynamical systems given by the class of flows we consider. 3. (3)

The construction of equivariant analytic torsion for proper group actions in [13] makes it natural to ask in what cases an equivariant generalisation of Fried’s conjecture holds. See Question 2.31. 4. (4)

In some cases, convergence and meromorphic extension of the equivariant Ruelle $\zeta$-function is automatic, because there are only finitely many flow curves satisfying (1.5). See Lemma 5.2 and Proposition 6.14. 5. (5)

There is a natural decomposition of the classical Ruelle dynamical $\zeta$-function into terms corresponding to conjugacy classes in the fundamental group of the manifold considered, and these terms can be studied separately as equivariant Ruelle $\zeta$-functions. See Proposition 2.28. 6. (6)

Studying an equivariant Ruelle $\zeta$-function for the action by a Lie group $G$ on $G/K$, with $K<G$ maximal compact, allows one to uniformly study Ruelle $\zeta$-functions on compact locally symmetric spaces $\Gamma\backslash G/K$ for different $\Gamma$ at once. See Example 2.39. 7. (7)

The absolute value in (1.4) appears naturally from the definition of the equivariant Ruelle $\zeta$-function, and can be omitted in the equivariant version of the Fried conjecture. See Lemma 2.24.

Construction and results

Let $M$, $\varphi$ and $\nabla^{F}$ be as before, but in this section we don’t assume $M$ is compact and that $\varphi$ is Anosov. Suppose that a unimodular, locally compact group $G$ acts properly on $M$ and that $M/G$ is compact. Suppose $\varphi$ is $G$-equivariant, and that the group action lifts to $F$ and preserves $\nabla^{F}$. Fix an element $g\in G$ such that the quotient of $G$ by the centraliser of $g$ has a nonzero $G$-invariant Borel measure. The Anosov or weaker nondegeneracy condition on $\varphi$ is replaced by the assumption that for all flow curves $\gamma$ such that (1.5) holds, the map

[TABLE]

has one-dimensional eigenspace for eigenvalue $1$ (spanned by $\gamma^{\prime}(0)$). Then we call the flow $g$-nondegenerate.

As mentioned, the idea behind the definition of the equivariant Ruelle $\zeta$-function $R^{g}_{\varphi,\nabla^{F}}$ is to replace closed flow curves in (1.3) by flow curves $\gamma$ satisfying (1.5). We now also allow negative $l$, because in some cases (1.5) may not hold for any positive $l$, see Example 2.7. More importantly, allowing negative $l$ seems to be essential for obtaining a natural equivariant generalisation of Fried’s conjecture; see Remarks 2.34 and 5.3.

There are straightforward generalisations of the maps $P_{\gamma}$ (defined in (1.1)) and $\rho(\gamma)$ to such flow curves. The appropriate generalisation of the primitive period $T^{\#}_{\gamma}$ in (1.2) is less obvious, however, particularly for non-discrete groups. This factor $T^{\#}_{\gamma}$ is replaced by a certain integral involving a cutoff function $\chi\in C^{\infty}_{c}(M)$, see (2.1) and (2.7). The final definition of the equivariant Ruelle $\zeta$-function also involves an integral over the conjugacy class of $g$, making it conjugation-invariant. See Definition 2.13 for details.

The main result in this paper is Theorem 4.9, an expression for the equivariant Ruelle $\zeta$-function in terms of a distributional flat $g$-trace $\operatorname{Tr}_{g}^{\flat}$, see Definition 3.3. This is a common generalisation of the distributional flat trace $\operatorname{Tr}^{\flat}$ used for example in Theorem 19.4.1 in [18] and the orbital integral trace $\operatorname{Tr}_{g}$ used in various places, such as [3, 14, 15, 20, 26]. With slight abuse of notation (treating distributions as functions), Theorem 4.9 is the equality

[TABLE]

The proof is based on Theorem 4.2, an equivariant generalisation of Guillemin’s trace formula [11], which should be of independent interest.

Applications of and motivation for Theorem 4.9 include the following.

1. (1)

It implies that the equivariant Ruelle $\zeta$-function is independent of the cutoff function $\chi$ used; see Proposition 2.18. 2. (2)

It implies a decomposition of the classical Ruelle $\zeta$-function for a flow on a compact manifold into equivariant Ruelle $\zeta$-functions for the action by the fundamental group of the manifold on its universal cover; see Proposition 2.28. 3. (3)

As in the classical case, Theorem 4.9 may be a basis for proofs of further properties of the equivariant Ruelle $\zeta$-function, such as meromorphic continuation (see Section 2.2 in [10], or [7]) and homotopy invariance (see Theorem 2 in [6]). 4. (4)

As the flat $g$-trace is a natural generalisation of the classical flat trace, Theorem 4.9 and its classical counterpart suggest that our equivariant Ruelle $\zeta$-function is a natural generalisation of the classical Ruelle $\zeta$-function. In particular, this suggests that our proposed replacement for the primitive period in (1.3) makes sense. 5. (5)

As in the classical case, the expression (1.6) for the equivariant Ruelle $\zeta$-function suggests similarities with equivariant analytic torsion; see Remark 4.10.

Replacing the left hand side of (1.4) by the equivariant Ruelle $\zeta$-function (with no absolute value), and the right hand side by the equivariant analytic torsion $T_{g}(\nabla^{F})$ of $\nabla^{F}$ [13], we obtain an equivariant version of Fried’s conjecture:

[TABLE]

See Question 2.31. If $M$ is compact and $G$ is trivial, then this reduces to the classical Fried conjecture (1.4). We do not expect (1.7) to hold in general, and only ask the question under what conditions it may be expected to hold. Analogously to the non-equivariant case, vanishing of the $L^{2}$-kernel of $\Delta_{F}$ is a condition.

A relevant class of examples where the equivariant Fried conjecture does not hold is the action by the fundamental group of a compact manifold on its universal cover. Already in Fried’s proof of his conjecture for hyperbolic manifolds [8], and also in later proofs of other cases (see the references below (1.4)), the strategy is to decompose both sides of (1.4) into factors corresponding to conjugacy classes in the fundamental group. But only the total expressions are equal, not the factors for individual conjugacy classes. This is easiest to see for the trivial conjugacy class. Then the corresponding factor in the expression for analytic torsion is $L^{2}$-analytic torsion [19, 22] and is generally different from $1$, whereas the corresponding factor for the Ruelle $\zeta$-function is always $1$.

In several other cases, however, we compute the equivariant Ruelle $\zeta$-function explicitly, and find that its value at zero equals equivariant analytic torsion. These cases are:

1. (1)

the real line acting on itself, see Lemma 5.2; 2. (2)

the circle acting on itself, see Lemmas 5.6 and 5.8; 3. (3)

certain discrete subgroups of the Euclidean motion group acting on the unit sphere bundle of $\mathbb{R}^{n}$, with geodesic flow; see Proposition 6.18.

For geodesic flow on the unit sphere bundle of the $n$-sphere, acted on by $\operatorname{SO}(n+1)$, we compute the equivariant Ruelle $\zeta$-function in Proposition 7.10 for $n=2$ and in Proposition 7.14 for $n=3$. In these cases, we do not expect (1.7) to hold, because the kernel of the Laplacian $\Delta_{F}$ is nonzero.

Furthermore, we show that in many cases, the equivariant Ruelle $\zeta$-function has the same behaviour with respect to restriction to subgroups as equivariant analytic torsion; see Lemma 2.21.

These results, together with the suggestive similarities between the equivariant Ruelle $\zeta$-function and equivariant analytic torsion in Remark 4.10, make us wonder if general conditions can be formulated under which the equivariant Fried conjecture is true. We plan to investigate this in the future.

Acknowledgements

We thank Jean-Michel Bismut, Bingxiao Liu, Thomas Schick, Shu Shen and Polyxeni Spilioti for helpful comments and discussions. PH is partially supported by the Australian Research Council, through Discovery Project DP200100729, and by NWO ENW-M grant OCENW.M.21.176. HS was supported by the Australian Research Council, through grant FL170100020.

2. The equivariant Ruelle dynamical $\zeta$-function

We start by introducing the equivariant Ruelle dynamical $\zeta$-function, and state some of its basic properties. These include some relations to the classical Ruelle dynamical $\zeta$-function, Lemmas 2.24 and 2.29, Proposition 2.28. Using the equivariant version of analytic torsion from [13], we state an equivariant version of the Fried conjecture as Question 2.31.

The proofs of two key properties of the equivariant Ruelle dynamical $\zeta$-function, Propositions 2.18 and 2.28, are given at the end of Section 4. These involve an expression of the equivariant Ruelle dynamical $\zeta$-function in terms of the flat $g$-trace introduced in Section 3, based on an equivariant version of Guillemin’s trace formula, Theorem 4.2.

2.1. Definition of the equivariant Ruelle dynamical $\zeta$-function

Let $M$ be a smooth manifold. Let $\varphi$ be the flow of a smooth vector field $u$ on $M$. Suppose that $u$ has no zeroes, and that its flow $\varphi$ is defined for all time. Let $G$ be a unimodular locally compact group, acting properly and cocompactly on $M$. Suppose that for all $t\in\mathbb{R}$, the flow map $\varphi_{t}\colon M\to M$ is $G$-equivariant. This is true in the following special cases.

Lemma 2.1**.**

Suppose that $G$ acts isometrically on a Riemannian manifold $X$, that $M=S(TX)$ is the unit sphere bundle of $TX$, and that $\varphi$ is the geodesic flow. Then $\varphi$ commutes with the action by $G$.

Proof.

As $G$ acts by isometries, its elements map geodesics to geodesics. This implies the claim. ∎

Example 2.2**.**

Let $X$ be a compact manifold. Let $M$ be the universal cover of $X$, acted on by the fundamental group $G$ of $X$. A flow $\varphi_{X}$ on $X$ has a unique $G$-equivariant lift to a flow $\varphi$ on $M$. The generating vector field $u$ of $\varphi$ has no zeroes if and only if the generating vector field of $\varphi_{X}$ has no zeroes.

Example 2.3**.**

Suppose that $G$ is a compact Lie group, acting on a compact manifold $X$. Let $S(\mathfrak{g})$ be the unit sphere in the Lie algebra $\mathfrak{g}$ of $G$, with respect to an $\operatorname{Ad}(G)$-invariant inner product. Let $M=X\times S(\mathfrak{g})$. Define the action by $G$ on $M$, and the flow $\varphi$, by

[TABLE]

for $g\in G$, $p\in X$, $Y\in S(\mathfrak{g})$ and $t\in\mathbb{R}$. Then $\varphi$ is equivariant with respect to the action by $G$. The generating vector field $u$ is given by $u(p,Y)=(Y^{X}_{p},0)$, where $Y^{X}_{p}$ is the tangent vector at $p\in X$ induced by $Y\in S(\mathfrak{g})$. It follows that $u$ has no zeroes if and only if the action is locally free.

Let $\chi\in C^{\infty}_{c}(M)$ be such that for all $m\in M$,

[TABLE]

Such a function exists because the action is proper and $M/G$ is compact. Furthermore, let $F\to M$ be a flat, $G$-equivariant vector bundle, and $\nabla^{F}$ a flat, $G$-invariant connection on $F$.

Definition 2.4**.**

Let $x\in G$.

• (a)

The $x$-delocalised length spectrum of $\varphi$ is

[TABLE]

Let $l\in L_{x}(\varphi)$.

• (b)

A flow curve of $\varphi$ is a curve $\gamma\colon\mathbb{R}\to M$ such that $\gamma(t)=\varphi_{t}(\gamma(0))$ for all $t\in\mathbb{R}$. A flow curve $\gamma$ is said to be $(x,l)$-periodic if $\gamma(l)=x\gamma(0)$. There is an action by $\mathbb{R}$ on the set $\tilde{\Gamma}^{x}_{l}(\varphi)$ of such curves, defined by

[TABLE]

for $\gamma\in\tilde{\Gamma}^{x}_{l}(\varphi)$ and $s,t\in\mathbb{R}$. We write

[TABLE]

We will often tacitly choose representatives of classes in $\Gamma^{x}_{l}(\varphi)$, and implicitly claim that what follows is independent of the choice of this representative; compare Remark 2.14 below.

Let $\gamma\in\Gamma_{l}^{x}(\varphi)$.

• (c)

The linearised $x$-delocalised Poincaré map of $\gamma$ is the map

[TABLE]

induced by $T_{\gamma(0)}(\varphi_{l}\circ x^{-1})$.

• (d)

The map

[TABLE]

is the parallel transport map from $F_{\gamma(0)}$ to $F_{\gamma(-l)}$ along $\gamma$ with respect to $\nabla^{F}$, composed with $x\colon F_{\gamma(-l)}\to F_{\gamma(0)}$.

• (e)

The $\chi$-primitive period of $\gamma$ is

[TABLE]

where $I_{\gamma}$ is an interval such that $\gamma|_{I_{\gamma}}$ is a bijection onto the image of $\gamma$, modulo sets of measure zero.

Remark 2.5**.**

It is possible that the $x$-delocalised length spectrum (2.2) is empty for some $x\in G$, but not for others. It is empty when $g=e$ in the settings of Subsection 2.4 and Section 6, for example. Then we find that using nontrivial group elements still allows us to define a meaningful version of the Ruelle $\zeta$-function.

Remark 2.6**.**

As the domain of a flow curve is taken to be the real line, a single flow curve $\gamma$ can occur in more than one of the sets $\Gamma_{l}^{x}(\varphi)$, if $\gamma(l)=x\gamma(0)$ for more than one combination of $l\in\mathbb{R}\setminus\{0\}$ and $x\in G$. For a flow curve $\gamma\in\Gamma_{l}^{g}(\varphi)$, the maps $P^{x}_{\gamma}$ and $\rho_{x}(\gamma)$ depend on $l$, even though this is not reflected in the notation.

Example 2.7**.**

Let $M=G=\mathbb{R}$, and consider the flow $\varphi_{t}(m)=m+t$. Then the only flow curve of $\varphi$, modulo the action (2.3), is $\gamma(t)=t$. Let $x\in\mathbb{R}\setminus\{0\}$. Then $L_{x}(\varphi)=\{x\}$, and $\Gamma_{x}^{x}(\varphi)=\{x\}$. Here we see that it is relevant to allow negative numbers in $L_{x}(\varphi)$. Now $I_{\gamma}=\mathbb{R}$, so

[TABLE]

Example 2.8**.**

If $G$ is compact (for example, trivial), then we may take $\chi$ to be the constant function $1$. Then $T_{\gamma}$ is the length of $I_{\gamma}$, which is the primitive period

[TABLE]

of $\gamma$.

Example 2.9**.**

For the geodesic flow on the universal cover of a negatively curved manifold, the $\chi$-primitive period of a geodesic equals its primitive period in the usual sense, for well-chosen $\chi$. See the proof of Lemma 2.29.

From now on, we fix $g\in G$. It follows from equivariance of $\varphi$ that $T_{m}(\varphi_{l}\circ g^{-1})(u(m))=u(m)$ for all $m\in M$ such that $\varphi_{l}(m)=gm$. This implies that $T_{m}(\varphi_{l}\circ g^{-1})-1$ descends to a linear endomorphism of $T_{m}M/\mathbb{R}u(m)$, which we denote by $(T_{m}(\varphi_{l}\circ g^{-1})-1)|_{T_{m}M/\mathbb{R}u(m)}$.

Definition 2.10**.**

The flow $\varphi$ is $g$-nondegenerate if for all $l\in\mathbb{R}\setminus\{0\}$ and $m\in M$ such that $\varphi_{l}(m)=gm$,

[TABLE]

In other words, $\varphi$ is $g$-nondegenerate if the map $P_{\gamma}^{g}-1_{T_{\gamma(0)}M/\mathbb{R}u(\gamma(0))}$ is invertible for all $l\in L_{g}(\varphi)$ and $\gamma\in\Gamma_{l}^{g}(\varphi)$.

Remark 2.11**.**

If $g=e$, then an Anosov flow is $e$-nondegenerate, but the latter condition is weaker. (The Anosov condition involves all flow curves, whereas $e$-nondegeneracy is only about closed curves.) In particular, the $g$-nondegeneracy condition does not require the choice of a Riemannian metric.

The following lemma generalises a well-known criterion for fixed points of group actions, or flow curves of a given flow, to be isolated. See page 377 of [2] for a comment on the non-equivariant case.

Lemma 2.12**.**

Suppose that $\varphi$ is $g$-nondegenerate and let $l\in L_{g}(\varphi)$. Then the set $\Gamma^{g}_{l}(\varphi)$ is discrete, in the sense that it is countable, and the images of the curves in $\Gamma^{g}_{l}(\varphi)$ together form the closed subset $M^{g^{-1}\varphi_{l}}\subset M$ of points fixed by $g^{-1}\circ\varphi_{l}$.

Proof.

Since $l\in L_{g}(\varphi)$, the fixed-point set $M^{g^{-1}\circ\varphi_{l}}$ is nonempty. Let $m\in M^{g^{-1}\circ\varphi_{l}}$. A vector $v\in T_{m}M$ is tangent to $M^{g^{-1}\circ\varphi_{l}}$ if and only if $v\in\ker(T_{m}(g^{-1}\circ\varphi_{l})-1)=\mathbb{R}u(m)$. So

[TABLE]

which implies that $M^{g^{-1}\circ\varphi_{l}}$ is one-dimensional.

Let $\gamma(t)=\varphi_{t}(m)$ be the flow curve through $m$. This curve lies in the one-dimensional submanifold $M^{g^{-1}\circ\varphi_{l}}$, and hence its image equals the connected component of $m$. We find that the images of all flow curves in $\Gamma^{g}_{l}(\varphi)$ are connected components of the one-dimensional manifold $M^{g^{-1}\circ\varphi_{l}}$. ∎

It follows from the definitions that for all $h\in G$,

[TABLE]

and that for $l$ in this set,

[TABLE]

So by Lemma 2.12, the set $\Gamma_{l}^{hgh^{-1}}(\varphi)$ is discrete for all $l\in L_{g}(\varphi)$ if $\varphi$ is $g$-nondegenerate.

Let $Z<G$ be the centraliser of $g$, and assume that $G/Z$ has a $G$-invariant measure $d(hZ)$.

Definition 2.13**.**

Suppose that $L_{g}(\varphi)$ is countable, and that $\varphi$ is $g$-nondegenerate. If the following expression (2.11) converges for $\sigma\in\mathbb{C}$ with large real part, then for such $\sigma$, the equivariant Ruelle dynamical $\zeta$-function at $g$ is defined by

[TABLE]

Remark 2.14**.**

The numbers $\det(1-P_{\gamma}^{hgh^{-1}})$ and $\operatorname{tr}(\rho_{hgh^{-1}}(\gamma))$ in (2.11) are independent of the choice of the representative $\gamma$ of a class in $\Gamma_{l}^{hgh^{-1}}(\varphi)$. Indeed, suppose that $x\in G$, $l\in L_{x}(\varphi)$ and that $\gamma_{1},\gamma_{2}\in\tilde{\Gamma}_{l}^{x}(\varphi)$ are in the same $\mathbb{R}$-orbit for the action (2.3). Then

[TABLE]

Furthermore, it follows from Theorem 4.9 that the function $R_{\varphi,\nabla^{F}}^{g}$ does not depend on the choices made in the definition of $T_{\gamma}$, see Proposition 2.18 below.

Remark 2.15**.**

The function $R_{\varphi,\nabla^{F}}^{g}$ only depends on the conjugacy class of $g$.

Remark 2.16**.**

The equivariant Ruelle dynamical $\zeta$-function depends on the $G$-invariant, flat connection $\nabla^{F}$ on $F$. In the classical case, where $G=\{e\}$, such a flat vector bundle $F$ with a flat connection is given by a representation of the fundamental group of $M$ via the Riemann–Hilbert correspondence. In that case, the Ruelle dynamical $\zeta$-function is usually viewed as depending on such a representation.

Remark 2.17**.**

In the general definition of the classical Ruelle dynamical $\zeta$-function, for possibly non-Anosov flows, an orbifold Euler characteristic appears. See e.g. Definitions 2.5 and 2.7 in [33]. In the case of flows that are $g$-nondegenerate that we consider here, this orbifold Euler characteristic equals $1$. This is because the relevant fixed point set quotiented out by the real line (or circle), is a point by Lemma 2.12. Compare the cases of Anosov flows, or geodesic flows on manifolds with negative sectional curvature, in the classical case; see (1.9) in [32]. We consider the $g$-nondegenerate case to obtain an expression for the equivariant Ruelle $\zeta$-function in terms of a distributional trace, see Theorem 4.9.

2.2. Basic properties of the equivariant Ruelle dynamical $\zeta$-function

The individual numbers $T_{\gamma}$ in (2.11) depend on the function $\chi$ in general, but the expression (2.11) does not.

Proposition 2.18**.**

If the expression (2.11) converges, then it does not depend on the choices of representatives of classes in the sets $\Gamma_{l}^{hgh^{-1}}(\varphi)$, the intervals $I_{\gamma}$ in (2.7), or the function $\chi$ satisfying (2.1) for all $m$.

This proposition is a consequence of an expression for the equivariant Ruelle $\zeta$-function in terms of a distributional flat $g$-trace, Theorem 4.9. We prove Proposition 2.18 just after that theorem.

The integrand in (2.11) is clearly invariant under right multiplication of $h$ by an element of $Z$, so that it is well-defined as a function on the quotient $G/Z$. We reformulate (2.11) in a way that makes this right $Z$-invariance property less obvious, but which can be easier to evaluate in practice.

Lemma 2.19**.**

If (2.11) converges, then

[TABLE]

Proof.

Let $h\in G$. Let $l\in L_{g}(\varphi)$, and $\gamma\in\Gamma_{l}^{g}(\varphi)$. Then by (2.10), we have $h\gamma\in\Gamma_{l}^{hgh^{-1}}(\varphi)$. By $G$-equivariance of $\varphi$ and $\nabla^{F}$,

[TABLE]

These equalities imply that

[TABLE]

respectively. Together with (2.9) and (2.10), the claim follows. ∎

Remark 2.20**.**

It is immediate from Lemma 2.19 that if the integral over $G/Z$ in (2.12) may be interchanged with the sums over $L_{g}(\varphi)$ and $\Gamma_{l}^{g}(\varphi)$, then

[TABLE]

where

[TABLE]

plays the role of a $g$-delocalised primitive period of $\gamma$. Here a Borel section of $G\to G/Z$ is used, and the total expression (2.13) does not depend on this section. This is true in particular if (2.11) converges, $L_{g}(\varphi)$ is finite, and $\Gamma^{g}_{l}(\varphi)$ is finite for all $l\in L_{g}(\varphi)$.

The equivariant Ruelle $\zeta$-function behaves naturally with respect to restriction to cocompact subgroups. This is analogous to the corresponding property of equivariant analytic torsion, Proposition 2.4 in [13].

Lemma 2.21**.**

Let $H<G$ be a cocompact, unimodular, closed subgroup containing $g$. Let $Z_{H}:=Z\cap H$ be the centraliser of $g$ in $H$, and suppose that $H/Z_{H}$ has an $H$-invariant measure. The expression (2.11) converges absolutely if and only if the same expression for the restricted action by $H$ on $M$ converges absolutely. In that case, the equivariant Ruelle $\zeta$-function $R^{g,H}_{\varphi,\nabla^{F}}$ for the action by $H$ is related to the equivariant Ruelle $\zeta$-function for the action by $G$ by

[TABLE]

Here the volume $\operatorname{vol}(Z/Z_{H})$ is computed with respect to the measure compatible with the Haar measures on $G$ and $H$ and the given measures on $G/Z$ and $H/Z_{H}$.

*If $G$ is compact, then the same statement is true with absolute convergence replaced by convergence. *

Proof.

Let $\psi\in C^{\infty}_{c}(G)$ be such that for all $x\in G$,

[TABLE]

Define $\chi_{H}\in C^{\infty}_{c}(M)$ by

[TABLE]

for $m\in M$. Then (2.1) holds for all $m\in M$, with $\chi$ replaced by $\chi_{H}$ and $G$ replaced by $H$.

By Lemma 2.19 and right $Z$-invariance of the integrand, the right hand side of (2.11) equals

[TABLE]

This expression converges absolutely if and only if the following expression does, and then the two are equal:

[TABLE]

For all $l\in L_{g}(\varphi)$ and $\gamma\in\Gamma^{g}_{l}(\varphi)$,

[TABLE]

The maps $P_{\gamma}^{g}$ and $\rho_{g}(\gamma)$ are the same for the actions by $G$ and $H$. So by Lemma 2.19,

[TABLE]

By (2.16), the right hand side of (2.17) converges absolutely if and only if (2.15) does, and then $\eqref{eq Ruelle subgroup 2}$ equals $1/\operatorname{vol}(Z/Z_{H})$ times (2.17). This implies the claim for possibly noncompact $G$.

If $G$, and hence $M$ and $H$, is compact, then we may use constant cutoff functions $\chi=1/\operatorname{vol}(G)$ and $\chi_{H}=1/\operatorname{vol}(H)$ (these volumes may be normalised to $1$). Then Lemma 2.19 and Example 2.8 imply that, whenever the following expression converge,

[TABLE]

which implies the claim. ∎

Remark 2.22**.**

If $G$ and $M$ are compact, $g=e$, and $\varphi$ is Anosov, then the last claim in Lemma 2.21 and the exponential bound on the number of flow curves with periods less than a given value [21] imply that (2.11) converges for $\sigma$ with large real part.

Remark 2.23**.**

If $G$ is noncompact, then absolute convergence is important in Lemma 2.21. See Remark 6.15 for an example where convergence is not absolute and the conclusion of Lemma 2.21 does not hold.

2.3. Relations with the classical Ruelle dynamical $\zeta$-function

Lemma 2.24**.**

If $M$ is odd-dimensional and orientable, the flow $\varphi$ is orientation-preserving, all operators (2.6) are unitary for a Hermitian metric on $F$, and $G$ is the trivial group, then $R^{e}_{\varphi,\nabla^{F}}$ is the absolute value of the classical Ruelle dynamical $\zeta$-function.

Remark 2.25**.**

In the proof of Lemma 2.24, we will obtain the the absolute value of the classical Ruelle dynamical $\zeta$-function, because the set $L_{g}(\varphi)$ in (2.11) also contains negative numbers in general. It is natural to include negative numbers in this set in our equivariant setting, because in some cases the equality $\varphi_{l}(m)=xm$ in (2.2) may only hold for negative $l$ (as in Example 2.7). In the Fried conjecture, the absolute value of the Ruelle dynamical $\zeta$-function is used, so for that purpose, $R^{e}_{\varphi,\nabla^{F}}$ is the natural function to consider. See Question 2.31 and Example 2.33.

Suppose for the rest of this subsection that $G=\{e\}$ is trivial and $M$ is orientable. We take $\chi\equiv 1$. For $l\in L_{e}(\varphi)$, we write

[TABLE]

We fix a Hermitian metric on $F$, and say that $\rho_{e}$ is unitary if $\rho_{e}(\gamma)$ is unitary for all $\gamma\in\Gamma^{e}_{l}(\varphi)$.

Lemma 2.26**.**

Let $l\in L_{e}(\varphi)$. Then $-l\in L_{e}(\varphi)$. If $\varphi$ is orientation-preserving and $\rho_{e}$ is unitary, then

[TABLE]

Proof.

The first claim follows from the fact that for all $m\in M$ and $l\in\mathbb{R}$, we have $\varphi_{l}(m)=m$ if and only if $\varphi_{-l}(m)=m$. This also shows that for all $l\in L_{e}(\varphi)$, the sets $\Gamma_{l}^{e}(\varphi)$ and $\Gamma_{-l}^{e}(\varphi)$ are equal.

Recall that the maps $P_{\gamma}^{e}$ and $\rho_{e}(\gamma)$ depend on $l$, if $\gamma\in\Gamma_{l}^{e}(\varphi)$. Fix $l\in L_{e}(\varphi)$ and $\gamma\in\Gamma_{l}^{e}(\varphi)=\Gamma_{-l}^{e}(\varphi)$. We write $(P_{\gamma}^{e})_{\pm l}$ for the map $P_{\gamma}^{e}$, and $\rho_{e}(\gamma)_{\pm l}$ for the map $\rho_{e}(\gamma)$, associated to $\gamma$ when viewed as an element of $\Gamma_{\pm l}^{e}(\varphi)$. Then

[TABLE]

If $\rho_{e}$ is unitary, then the second equality implies that

[TABLE]

Now for any invertible endomorphism $A$ of a finite-dimensional vector space $V$,

[TABLE]

We apply this to $A=(P_{\gamma}^{e})_{l}$, and note that this map has positive determinant because $T\varphi_{l}$ preserves the orientation on the quotient of $TM$ by the span of the image of $u$. This yields

[TABLE]

The claim follows from this equality, (2.18), and the fact that $T_{\gamma}$ does not depend on $l$. ∎

We write $L^{+}(\varphi):=L_{e}(\varphi)\cap(0,\infty)$. The classical Ruelle dynamical $\zeta$-function $R_{\varphi,\nabla^{F}}$ for Anosov flows is defined by

[TABLE]

for $\sigma\in\mathbb{C}$ for which the expression converges. (See also Example 2.8.)

Lemma 2.27**.**

If $\varphi$ is orientation-preserving and $\rho_{e}$ is unitary, then for all large enough $\sigma\in\mathbb{R}$,

[TABLE]

Proof.

By Lemma 2.26,

[TABLE]

Suppose that $\sigma$ is real. If $\dim(M)$ is odd, then (2.20) becomes

[TABLE]

If $\dim(M)$ is even, then (2.20) becomes

[TABLE]

The equalities (2.21) and (2.22) can be rewritten as (2.19). ∎

Lemma 2.24 is the case of Lemma 2.27 where $\dim(M)$ is odd.

There is a more interesting and useful relation between the equivariant Ruelle dynamical $\zeta$-function and its classical counterpart than Lemma 2.24. Suppose that $X$ is a compact manifold, and let $M$ be the universal cover of $X$. Let $F_{X}=M\times_{\rho}\mathbb{C}^{r}\to X$ be the flat vector bundle associated to a representation $\rho\colon\pi_{1}(X)\to\operatorname{GL}(r,\mathbb{C})$. Consider the trivial connection $\nabla^{F}=d$ on $F=M\times\mathbb{C}^{r}$.

Let $\varphi_{X}$ be a flow on $X$, and $\varphi$ its lift to a $\Gamma$-equivariant flow on $M$. Let $R_{\varphi_{X},\rho}$ be the classical Ruelle dynamical $\zeta$-function associated to $\varphi_{X}$ and $\rho$.

Proposition 2.28**.**

Let $\sigma\in\mathbb{C}$. If $R_{\varphi_{X},\rho}(\sigma)$ converges, then so does $\prod_{(g)}R^{g}_{\varphi,\nabla^{F}}(\sigma)$, and

[TABLE]

Here $\prod_{(g)}$ is the product over all conjugacy classes $(g)\subset\pi_{1}(X)$.

We prove this proposition at the end of Subsection 4.4, as a consequence of the expression of the Ruelle $\zeta$-function in terms of the flat $g$-trace, Theorem 4.9.

2.4. The classical decomposition of the Ruelle dynamical $\zeta$-function for geodesic flow

The classical Ruelle dynamical $\zeta$-function for geodesic flows on the sphere bundle of a compact, oriented, negatively curved Riemannian manifold $X$ can be decomposed into contributions from different conjugacy classes of the fundamental group of $X$, analogously to Proposition 2.28. These contributions are equal to the relevant equivariant Ruelle dynamical $\zeta$-functions, as we now discuss.

Let $X$ be a compact, oriented Riemannian manifold with negative sectional curvature. Let $\varphi_{X}$ be the geodesic flow on the unit sphere bundle $S(TX)$. Let $\tilde{X}$ be the universal cover of $X$. Let $\varphi$ be the lift of $\varphi_{X}$ to $M=S(T\tilde{X})$; this is the geodesic flow on the universal cover. Consider the natural action by $G=\pi_{1}(X)$ on $M$. Let $g\in G$ be nontrivial. Let $l$ be the period of $g$ as a closed curve in $X$. Let

[TABLE]

be the primitive period of $g$.

Let $\rho\colon G\to\operatorname{U}(r)$ be a unitary representation. Consider the trivial vector bundle $M\times\mathbb{C}^{r}\to M$, on which $G$ acts by

[TABLE]

for $x\in G$, $m\in M$ and $v\in\mathbb{C}^{r}$. Let $\nabla^{F}=d\otimes 1_{\mathbb{C}^{r}}$ be the natural flat connection on $F$.

Lemma 2.29**.**

For all $\sigma\in\mathbb{C}$,

[TABLE]

Proof.

In this case $L_{g}(\varphi)=\{l\}$. From the assumption that $X$ has negative sectional curvature, it follows that there is a unique geodesic in $X$ in the conjugacy class of $g$. Let $\gamma$ be the corresponding curve in $M$, extended to the real line.

[TABLE]

As $X$ is oriented,

[TABLE]

for all $h\in G$. Because parallel transport with respect to $\nabla^{F}$ is trivial, we have

[TABLE]

for all $h$, so

[TABLE]

The primitive period $T_{\gamma}$ a priori may depend on the choice of the cutoff function $\chi$. As the left hand side of (2.24) does not depend on $\chi$, by Proposition 2.18, we are free to choose this function in a suitable way. We choose $\chi$ to be an approximation of the indicator function of a suitable fundamental domain $Y\subset M$.

The subset $\gamma([0,T^{\#}_{g}))\subset M$ has the property that for all $m\in\gamma([0,T^{\#}_{g}))$ and $x\in G\setminus\{e\}$, the point $xm$ lies outside $\gamma([0,T^{\#}_{g}))$. Therefore, the set $\gamma([0,T^{\#}_{g}))$ is contained in a fundamental domain $Y$ for the action by $G$ on $M$. We choose such a fundamental domain $Y$ such that

1. (1)

$\gamma(\mathbb{R})\cap Y=\gamma([0,T^{\#}_{g}))$; and 2. (2)

the closure of $Y$ equals the closure of the interior of $Y$.

By the second property, the indicator function $1_{Y}$ can be approximated by compactly supported smooth functions arbitrarily close in the $L^{1}$-norm. As $\chi\in C^{\infty}_{c}(M)$ approaches $1_{Y}$ in the $L^{1}$-norm,

[TABLE]

By (2.25), (2.26) and (2.27), the left hand side of (2.24) equals the right hand side of (2.24). ∎

Remark 2.30**.**

Lemma 2.29 means that (possibly up to a factor 2, depending on conventions) $R^{g}_{\varphi,\nabla^{F}}$ is the contribution to the classical Ruelle dynamical $\zeta$-function associated to the conjugacy class of $g$, see for example Definition 5.4 in [32]. This can be used to prove Proposition 2.28 in the case of geodesic flow.

2.5. An equivariant Fried conjecture

Suppose for now that $M$ is a compact, oriented Riemannian manifold, and $G$ is trivial. Let $\rho$ be a unitary representation of $\pi_{1}(M)$. Ray and Singer [27] defined the analytic torsion $T_{\rho}(M)$ of $M$, with respect to $\rho$. In this setting, the Fried conjecture states that, for suitable classes of flows $\varphi$ on $M$, the classical Ruelle dynamical $\zeta$-function $R_{\varphi,\rho}$ extends meromorphically to the complex plane, is regular in a neighbourhood of zero, and satisfies

[TABLE]

This conjecture has been proved in several cases, see for example [3, 6, 8, 23, 24, 29, 30, 32, 34, 35, 41, 43], this includes versions for non-Anosov flows. The equality (2.28) is known to not hold in general, see the end of Section 2 in [33] and [31, 40].

In [13], we defined an equivariant version of analytic torsion for proper group actions with compact quotients. It is therefore natural to ask under what conditions an equivariant version of (2.28) holds, see Question 2.31 below.

To state this question more explicitly, we return to the equivariant setting considered before. We now also assume that $M$ has an orientation and a $G$-invariant Riemannian metric, and that $F$ has a $G$-invariant Hermitian metric preserved by $\nabla^{F}$. Consider the Laplacian

[TABLE]

where we view $\nabla^{F}$ as an operator on the space $\Omega^{*}(M;F)$ of differential forms on $M$, twisted by $F$, and $(\nabla^{F})^{*}$ is its formal adjoint, defined in the usual way in terms of an extension of the Hodge $*$-operator extended to differential forms twisted by $F$. Let $P_{p}$ be the orthogonal projection from the Hilbert space of square integrable $p$-forms twisted by $F$ onto the kernel of $\Delta_{F}^{p}$ inside this space.

For a $G$-equivariant bounded operator $T$ on the space of $L^{2}$-sections of a Hermitian $G$-vector bundle over $M$, we define its $g$-trace as

[TABLE]

if $T\chi$ is trace-class and the integral converges. This trace has been used in various places besides [13], see for example [14, 15, 20, 38].

Assuming that the following expressions converge, we write

[TABLE]

for $t>0$, and

[TABLE]

In the first term, we use meromorphic continuation from $s$ with large real part to $s=0$, again assuming the resulting expression is well-defined. Then $T_{g}(\nabla^{F})$ is the equivariant analytic torsion of $\nabla^{F}$ at $g$. For results on convergence and meromorphic extension of the integrals involved, see Section 3 of [13]. For independence of the Riemannian metric and other properties, see Sections 4 and 5 of [13]. For metric independence, a condition is that $\Delta_{F}^{p}$ has trivial kernel for all $p$.

Question 2.31** (Equivariant Fried conjecture).**

Suppose that $M$ is odd-dimensional, and $\Delta_{F}^{p}$ has trivial $L^{2}$-kernel for all $p$. Under what further conditions does

1. (1)

the expression (2.11) for $R^{g}_{\varphi,\nabla^{F}}(\sigma)$ converge when $\sigma$ has large real part; 2. (2)

this have a meromorphic extension to the complex plane that is regular in a neighbourhood of zero; and 3. (3)

the equality

[TABLE]

hold?

Remark 2.32**.**

The definition of the equivariant Ruelle $\zeta$-function does not involve a Riemannian metric, but the definition of equivariant analytic torsion does. We expect equivariant analytic torsion to be independent of the choice of this metric, see Section 4 in [13] for results in this direction. Then the equality (2.32) for one $G$-invariant Riemannian metric implies this equality for all such metrics.

Example 2.33**.**

In the classical setting, where $G$ is trivial and $M$ is compact, the left hand side of (2.32) is the absolute value of the classical Ruelle dynamical $\zeta$-function at zero by Lemma 2.24, while the right hand side is classical Ray–Singer analytic torsion. So in this case, the equality (2.32) is the usual Fried conjecture (2.28). Then the answer to Question 2.31 is yes in cases where the classical Fried conjecture was proved.

Slightly more generally, if $G$, and hence $M$, is compact, and $g=e$, then (2.32) is equivalent to the classical Fried conjecture.

Remark 2.34**.**

In the non-equivariant case, analytic torsion is a positive number. This is consistent with the absolute value on the left-hand side of (2.32). In the equivariant case, analytic torsion need not be positive; see the examples in Sections 5 and 6. Allowing negative $l$ in the definition of the equivariant Ruelle $\zeta$-function lets us formulate the equivariant Fried conjecture (2.32) without absolute values. For a version of the equivariant Ruelle $\zeta$-functions that only includes positive $l$, the analogue of (2.32) does not even hold with absolute values on both sides in basic cases; see Remark 5.3 for an example.

Remark 2.35**.**

In Question 2.31, we assume that $M$ is odd-dimensional, because the question then reduces to the classical Fried conjecture if $G$ is trivial, as in Example 2.33.

If $M$ is even-dimensional, then by Lemma 2.27, the equivariant Ruelle $\zeta$-function for trivial groups becomes the phase of the classical Ruelle $\zeta$-function, rather than its absolute value. It then follows that (2.32) does not reduce to the classical Fried conjecture. For $M$ even-dimensional, as in the classical case, the equivariant analytic torsion of $M$ is trivial by Proposition 5.6 in [13], so the case where $M$ is odd-dimensional is the most interesting. In the important example of the geodesic flow on the sphere bundle $M$ on a Riemannian manifold, the manifold $M$ is odd-dimensional.

Example 2.36**.**

In several proofs of the Fried conjecture in different settings, both sides of (2.28) are decomposed as sums of terms corresponding to conjugacy classes in the fundamental group. The term in the decomposition of the Ruelle dynamical $\zeta$-function corresponding to a conjugacy class is an equivariant Ruelle dynamical $\zeta$-function by Lemma 2.29, in the case of the geodesic flow on a negatively curved manifold. The terms on the two sides of the Fried conjecture corresponding to the same conjugacy class are not equal in general, even if the sums over all conjugacy classes are equal.

This is perhaps clearest for the trivial conjugacy class $\{e\}$. In the setting of Proposition 2.28, the number $T_{e}(\nabla^{F_{X}})$ is the $L^{2}$-analytic torsion of $X$ with respect to $\rho$ [19, 22]. This is generally different from $1$. However, the lift of a suitable flow on $X$ to $M$ may not have any closed loops, so that $R^{e}_{\varphi,\nabla^{F}}$ is the constant $1$. In fact, in Fried’s proof of his conjecture in the case of hyperbolic manifolds [8], this difference exactly compensates for a functional relation satisfied by certain $\zeta$-functions related to $R_{\varphi_{X},\rho}$, see (FE) and (FE)’ on page 531 of [8]. In such cases, the answer to Question 2.31 is no for the action by $G=\pi_{1}(X)$ on $M$, for $g=e$ and at least one other conjugacy class.

For examples where the answer to Question 2.31 is yes, see Sections 5 and 6. For geodesic flow on the two-sphere, discussed in Section 7, the equality (2.32) does not hold, but then the $L^{2}$-kernels of the operators $\Delta_{F}^{p}$ are nonzero. So this is not a counterexample to Question 2.31.

In the setting of Lemma 2.21, the equivariant Ruelle $\zeta$-function has the same behaviour with respect to restriction to cocompact subgroups as equivariant analytic torsion, see Proposition 2.4 in [13]. This has the following immediate consequence.

Proposition 2.37**.**

In the setting of Lemma 2.21, the answer to Question 2.31 is yes for the action by $G$ on $M$ if and only if the answer is yes for the action by the cocompact subgroup $H$.

Example 2.38**.**

If $G$ is compact, then Question 2.31 is equivalent to the same question for the action by the closure of the subgroup generated by $g$.

Example 2.39**.**

Suppose that $G$ is a Lie group, $K<G$ is maximal compact, and $\Gamma<G$ is discrete and cocompact, and contains $g$. Suppose that $M=G/K$. If the equivariant Ruelle $\zeta$-function converges absolutely for large enough $\sigma$, then Question 2.31 for the action by $G$ is equivalent to the same question for the action by $\Gamma$. In this way, the equivariant Fried conjecture for the action by $G$ on $G/K$ contains information about all compact locally symmetric spaces $\Gamma\backslash G/K$, for $\Gamma<G$ as in this example.

2.6. Extensions and further work

2.6.1. Flows that are not $g$-nondegenerate

In this paper, we consider $g$-nondegenerate flows; see Definition 2.10. The geodesic flow on the tangent sphere bundle of a compact, oriented manifold has this property if the manifold has negative sectional curvature. In this $g$-nondegenerate case, we view several results (Propositions 2.18 and 2.28) and examples (Lemmas 2.24, 2.29, 5.2, 5.6 and 5.8 and Propositions 7.10 and 6.14) as indications that Definition 2.13 of the equivariant Ruelle $\zeta$-function is sensible. Several of these follow from the flat $g$-trace formula, Theorem 4.9, which depends on the $g$-nondegeneracy condition in an essential way. The latter theorem in itself can be seen as another indication that our definition is a natural generalisation to the equivariant case. In fact, proving this result was our initial approach to finding a suitable definition of an equivariant Ruelle dynamical $\zeta$-function.

To go beyond the $g$-nondegenerate case, the sum over $\gamma\in\Gamma_{l}^{hgh^{-1}}(\varphi)$ in (2.11) would have to be replaced by a sum over the connected components $Y$ of the quotient of the fixed-point set $M^{hg^{-1}h^{-1}\varphi_{l}}$ by the action by $\mathbb{R}$ defined by the flow. For $g$-nondegenerate flows, such connected components are single points by Lemma 2.12. One can show that the numbers $\operatorname{sgn}(\det(1-P_{\gamma}^{hgh^{-1}}))$ and $\operatorname{tr}(\rho_{hgh^{-1}}(\gamma))$ in (2.11) are the same for all flow lines $\gamma$ in the same connected component $Y$.

One generalisation of the $\chi$-primitive period is

[TABLE]

where $I_{y}\subset\mathbb{R}$ is an interval on which the curve $s\mapsto\varphi_{s}(y)$ is bijective onto its image, up to sets of measure zero. Then if $G$ is trivial, the quotient $T_{Y}/|l|$ is the multiplicity used in the definition of the classical Ruelle dynamical $\zeta$-function; see e.g. (2.16) in [33].

A further ingredient in an equivariant Ruelle dynamical $\zeta$-function for flows that are not $g$-nondegenerate is a generalisation of the orbifold Euler characteristic used in the classical case; see e.g. (2.18) in [33]. One generalisation is the number $\chi_{\operatorname{orb}}(Y)$, for a connected component $Y$ of $M^{g^{-1}\varphi_{l}}/\mathbb{R}$ as before.

For the trivial group, the resulting expression for the equivariant Ruelle dynamical $\zeta$-function reduces to the absolute value of the classical Ruelle dynamical $\zeta$-function, analogously to Lemma 2.24. However, in more general situations, such as the geodesic flow on the tangent sphere bundle of a compact, oriented manifold with non-positive (but not everywhere negative) sectional curvature, it is not clear to us if this construction yields the right result. For this reason, we do not propose a definition of an equivariant Ruelle dynamical $\zeta$-function for flows that are not $g$-nondegenerate in this paper.

2.6.2. Convergence and meromorphic extension

Convergence and particularly meromorphic extension and regularity at zero of the classical Ruelle $\zeta$-function are highly nontrivial questions in general. In proofs of various cases of the Fried conjecture, proving meromorphic extension and regularity at zero is part of the problem. Separate proofs of meromorphic extension have also been given, such as the ones in [4, 7, 10, 36]. In the case $G=\mathbb{Z}$, a result related to convergence and meromorphic extension of the equivariant Ruelle $\zeta$-function is proved for geodesic flows on sphere bundles of surfaces in [5].

We make no general comments on convergence or meromorphic extension of the equivariant Ruelle dynamical $\zeta$-function in this paper. In Sections 5 and 6, we show by hand that this function converges and has a meromorphic extension that is regular at zero in some examples. It is possible that the flat $g$-trace formula for the equivariant Ruelle $\zeta$-function, Theorem 4.9, can be used as a tool in proofs of meromorphic continuation, as in the non-equivariant case.

An important comment here is that in the equivariant case, convergence and meromorphic extension of the Ruelle $\zeta$-function are trivial in some relevant cases, because the delocalised length spectrum and sets of flow curves may be finite. This happens in Lemmas 2.29 and 5.2, and in Proposition 6.14.

3. The flat $g$-trace

We express the equivariant Ruelle dynamical $\zeta$-function in terms of the flat $g$-trace that we define in this section; see Theorem 4.9. This will allow us to prove Propositions 2.18 and 2.28. This flat $g$-trace is a common generalisation of the distributional trace used for example in Theorem 19.4.1 in [18], and a $g$-trace defined in terms of orbital integrals, and used for example in [14, 15, 16]. A version of the latter trace for discrete groups was used before to construct higher invariants, see for example [20, 38, 42].

In Section 4, we obtain a fixed-point formula for the flat $g$-trace of the relevant type of flow, generalising Guillemin’s trace formula [11]. This is used to prove Theorem 4.9.

As we did in Subsection 2.1, we assume that $M$ is a manifold, on which a unimodular, locally compact group $G$ acts properly and cocompactly. Fix a $G$-equivariant vector bundle $E\to M$. (In the application to the equivariant Ruelle $\zeta$-function, we will take $E=\textstyle{\bigwedge}T^{*}M\otimes F$.) Let $g\in G$, $Z<G$ and $d(hZ)$ be as in Definition 2.13.

For notational simplicity, we fix a $G$-invariant, nonvanishing density $dm$ on $M$, and use this to identify the space $\Gamma^{-\infty}(E)$ of generalised sections of $E$ with the continuous dual of $\Gamma_{c}^{\infty}(E^{*})$. What follows does not depend on the choice of this density, see for example Remark 3.5. We use analogous identifications for other vector bundles as well. For example, we identify

[TABLE]

where the prime on the right hand side denotes the continuous dual space.

3.1. Distributional kernels on the diagonal

We define a flat $g$-trace of operators on $\Gamma^{\infty}(E)$ in Definition 3.3. We later apply this to flows in (3.11).

The identity operator $\operatorname{Id}_{E}$ on $E$ defines an element of

[TABLE]

where the isomorphism is swapping the components in $E$ and $E^{*}$.

Definition 3.1**.**

The fibre-wise trace of $\kappa\in\Gamma^{-\infty}((E\boxtimes E^{*})|_{\Delta(M)})$ is the distribution $\operatorname{tr}(\kappa)\in\mathcal{D}^{\prime}(\Delta(M))$ defined by

[TABLE]

for $f\in C^{\infty}_{c}(\Delta(M))$.

Example 3.2**.**

In the case where $\kappa\in\Gamma^{\infty}((E\boxtimes E^{*})|_{\Delta(M)})$ is smooth, let $m\in M$, let $\{e_{j}\}_{j=1}^{\operatorname{rank}(E)}$ be a basis of $E_{m}$, and $\{e^{j}\}_{j=1}^{\operatorname{rank}(E)}$ be the dual basis of $E^{*}_{m}$. Then under the isomorphism (3.1),

[TABLE]

Therefore, viewing $\kappa(m,m)$ as an endomorphism of $E_{m}$, we have

[TABLE]

where $\operatorname{tr}$ is the fibre-wise operator trace. It follows that $\operatorname{tr}(\kappa)$ is indeed the fibre-wise trace of $\kappa$, and lies in $C^{\infty}(\Delta(M))$.

Let $T\colon\Gamma^{\infty}_{c}(E)\to\Gamma^{\infty}(E)$ be a $G$-equivariant, continuous linear operator. Let $K$ be its Schwartz kernel. More generally, for $x\in G$, we write $x\cdot K$ for the Schwartz kernel of $x\circ T$, where $x$ acts on sections in the usual way. Let $\mathcal{N}^{*}\to\Delta(M)$ be the conormal bundle.

Let $\chi$ be as in (2.1). Let $p_{2}\colon M\times M\to M$ be projection onto the second coordinate. Then $(p_{2}^{*}\chi)|_{\Delta(M)}$ has compact support. Note that the function $(p_{2}^{*}\chi)|_{\Delta(M)}$ is just the function on $\Delta(M)$ corresponding to $\chi$ via the diagonal diffeomorphism $\Delta\colon M\to\Delta(M)$. The wave front set of a distribution is denoted by $\operatorname{WF}$; see Definition 8.1.2 in [17].

Definition 3.3**.**

Suppose that for all $h\in G$,

[TABLE]

Suppose also that for all $f\in C^{\infty}_{c}(M)$, the integral

[TABLE]

converges absolutely, and that this defines a distribution on $M$. Then we say that $T$ is flat $g$-trace class, and define its flat $g$-trace by

[TABLE]

Remark 3.4**.**

The condition (3.2) implies that the restriction $hgh^{-1}\cdot K|_{\Delta(M)}$ in (3.3) and (3.4) is well-defined. See e.g. Theorem 8.2.4 in [17].

Remark 3.5**.**

If we view $K$ as a generalised section in $\Gamma^{-\infty}(E\boxtimes E^{*})$, this implicitly involves the density $dm\times dm$. The identification of $\Gamma^{-\infty}(E\boxtimes E^{*})$ with $\Gamma_{c}^{\infty}(E^{*}\boxtimes E)^{\prime}$ via the same density, used in (3.4), cancels out the choice of density in a sense, so that $\operatorname{Tr}_{g}^{\flat}(T)$ does not depend on $dm$. Compare Lemma 2.9 in [13] in the case where $K$ is smooth.

Example 3.6**.**

If $G$ is the trivial group, so $g=e$, then $\chi$ may be taken to be the constant $1$. So the flat $e$-trace is the classical flat trace, used for example in Theorem 19.4.1 in [18].

Example 3.7**.**

If $K$ is smooth, then by Example 3.2,

[TABLE]

If $T\chi$ is a trace-class operator on $L^{2}(E)$, then this equals the $g$-trace $\operatorname{Tr}_{g}(T)$ in (2.29), when this converges. In this sense, the flat $g$-trace $\operatorname{Tr}_{g}^{\flat}$ is a generalisation of the $g$-trace $\operatorname{Tr}_{g}$, in the same way that the classical flat trace, see Example 3.6, generalises the operator trace in some settings.

Example 3.8**.**

If $g=e$, then $\operatorname{Tr}_{e}^{\flat}=\langle\operatorname{tr}(K|_{\Delta(M)}),(p_{2}^{*}\chi)|_{\Delta(M)}\rangle$, a distributional version of the von Neumann trace [1, 39].

We will show in Proposition 3.11 that the flat $g$-trace is independent of the function $\chi$. The proof is based on Lemmas 3.9 and 3.10.

Lemma 3.9**.**

For all $f\in C^{\infty}_{c}(M)$ and all $x,h\in G$,

[TABLE]

Proof.

If $K$ is smooth, then by Example 3.2, a substitution $m\mapsto xm$, equivariance of $T$ and the trace property of the fibre-wise trace, the left hand side of (3.5) equals

[TABLE]

The expression on the right is the right hand side of (3.5), so the equality holds for smooth kernels. The general case follows by continuity with respect to $K$ of both sides of (3.5). ∎

Lemma 3.10**.**

Let $\eta$ be a $G$-invariant distribution on $M$. Let $f\in C^{\infty}_{c}(M)$ be such that for all $m\in M$,

[TABLE]

Let $\psi\in C^{\infty}_{c}(G)$. Then

[TABLE]

Proof.

We prove the claim for $\eta\in C^{\infty}(M)^{G}$, the general case follows because the left hand side of (3.6) depends continuously on $\eta$. Suppose furthermore that $M=G\times_{K}N$ for a compact subgroup $K<G$ and a compact, $K$-invariant submanifold $N\subset M$. The general case follows from an application of Palais’ slice theorem [25] and the use of a partition of unity. Since $M/G$ is compact, it can be covered by finitely many open sets of the form $G\times_{K}N$. A sum over these sets commutes with the integrals that occur in the argument below.

Choose the density $dm$ so that for all $a\in C_{c}(M)$,

[TABLE]

for a $K$-invariant density $dn$ on $N$. Then the left hand side of (3.6) equals

[TABLE]

Since $\psi$ and $f$ have compact supports, the action is proper, and $N$ is compact, the integrand has compact support. So we may freely interchange the order of integration. Interchanging the inner integrals over $G$ and $N$, and using left invariance of $dy$, we find that (3.7) equals

[TABLE]

∎

Proposition 3.11**.**

The integral (3.4) is independent of the function $\chi\in C^{\infty}_{c}(M)$ satisfying (2.1).

Proof.

Let $f\in C^{\infty}_{c}(M)$, and suppose that for all $m\in M$,

[TABLE]

We will show that the integral (3.3) equals zero; this implies the claim.

For $x\in G$, let $xKx^{-1}$ be the Schwartz kernel of $x\circ T\circ x^{-1}$. By $G$-equivariance of $T$, we have $xKx^{-1}=K$. Let $\psi\in C^{\infty}_{c}(G)$ be a function with integral equal to one. Then (3.3) equals

[TABLE]

In the second equality, we used Lemma 3.9. Since the integral (3.3) converges absolutely and $\psi$ has compact support, the above integrals converge absolutely. Hence the right hand side of (3.8) equals

[TABLE]

The distribution

[TABLE]

on $M$ is $G$-invariant by Lemma 3.9 and $G$-invariance of $d(hZ)$. Lemma 3.10 then implies that (3.9) equals zero and the proof is complete. ∎

3.2. The case of equivariant flows

As we did in Subsection 2.1, we consider a $G$-equivariant flow $\varphi$ on $M$, generated by a nonvanishing vector field $u$. We assume that $\Phi$ is a fibre-wise linear flow on $E$ that lifts $\varphi$. For a function $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$, consider the operator $\Phi^{*}_{\psi}$ on $\Gamma^{\infty}(E)$ defined by

[TABLE]

for $s\in\Gamma^{\infty}(E)$ and $m\in M$.

Let $A\in\operatorname{End}(E)$ be $G$-equivariant, and suppose that $A$ commutes with $\Phi_{t}$ for all $t\in\mathbb{R}$. In the application to the Ruelle dynamical $\zeta$-function, we will take $A=(-1)^{N}N$, with $N$ the number operator on differential forms. We will obtain a criterion for the operator $A\Phi^{*}_{\psi}$ to be flat $g$-trace class for $g$-nondegenerate flows and $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$, see Corollary 3.17 and Theorem 4.2. In that case we write

[TABLE]

We will see in Theorem 4.2 that this defines a distribution on $\mathbb{R}\setminus\{0\}$ for suitable $A$.

Fix $x\in G$, and let x\cdot AK\in\Gamma^{\infty}_{c}\bigl{(}\mathbb{R}\setminus\{0\}\times M\times M,\mathbb{R}\setminus\{0\}\times E^{*}\boxtimes E\bigr{)}^{\prime} be defined by

[TABLE]

for $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$, $\xi\in\Gamma^{\infty}_{c}(E^{*})$ and $s\in\Gamma^{\infty}_{c}(E)$.

We will show in Proposition 3.16 that the condition (3.2) holds for the operator $A\Phi_{\psi}^{*}$ if $\varphi$ is $g$-nondegenerate. Consider the map

[TABLE]

given by

[TABLE]

Lemma 3.12**.**

The derivative of $\Psi_{x}$ at $(t,m,m^{\prime})$ is given by

[TABLE]

for all $s\in T_{t}\mathbb{R}=\mathbb{R}$, $v\in T_{m}M$ and $v^{\prime}\in T_{m^{\prime}}M$.

Proof.

This is a direct computation. ∎

Consider the set

[TABLE]

Lemma 3.13**.**

The set $S_{x}$ is a smooth submanifold, and its tangent bundle is

[TABLE]

Proof.

If $t\in\mathbb{R}\setminus\{0\}$ and $m\in M$, then Lemma 3.12 implies that the image of $T_{(t,m,x\varphi_{t}(m))}\Psi_{x}$ contains $\{0\}\times T_{x\varphi_{t}(m)}M$. So the sum of this image and

[TABLE]

is the whole tangent space $T_{\Psi_{x}(t,m,x\varphi_{t}(m))}(M\times M)$. This implies that $\Psi_{x}$ is transversal to $\Delta(M)$, and that $S_{x}$ is a smooth submanifold of $\mathbb{R}\setminus\{0\}\times M\times M$.

The tangent bundle to $TS_{x}$ is

[TABLE]

By Lemma 3.12, this equals the right hand side of (3.14). ∎

The following lemma is used on page 24 of [7] for the identity element in $G$, where $V(x)$ should be replaced by $V(\varphi_{t}(x))$.

Lemma 3.14**.**

The conormal bundle of $S_{x}$ is

[TABLE]

Proof.

Let $a\in\mathbb{R}=T^{*}_{t}\mathbb{R}$, $\zeta\in T^{*}_{m}M$ and $\xi\in T^{*}_{x\varphi_{t}(m)}M$. By Lemma 3.13, we have $(a,\zeta,\xi)\in\mathcal{N}^{*}(S_{x})$ if and only if for all $s\in\mathbb{R}$ and $v\in T_{m}M$,

[TABLE]

So

[TABLE]

The second equality implies that $\xi=-(T_{m}\varphi_{-t}x^{-1})^{*}\zeta$, and then the first becomes

[TABLE]

In the last equality, we used the fact that

[TABLE]

∎

Lemma 3.15**.**

Suppose that the flow $\varphi$ is $x$-nondegenerate. Then

[TABLE]

Proof.

By Lemma 3.14, an element of $\mathcal{N}^{*}(S_{x^{-1}})$ is of the form

[TABLE]

for $t\in\mathbb{R}\setminus\{0\}$ and $\zeta\in T_{m}M$. Suppose that this element also lies in $\mathcal{N}^{*}(\mathbb{R}\setminus\{0\}\times\Delta(M))$. Then $(m,x\varphi_{t}(m))\in\Delta(M)$, so $x\circ\varphi_{t}(m)=m$. Furthermore,

[TABLE]

and it follows that

[TABLE]

We thus see that the $x$-nondegeneracy property of $\varphi$ implies that $\zeta=0$. ∎

Proposition 3.16**.**

Suppose that the flow $\varphi$ is $g$-nondegenerate. Then for all $h\in G$,

[TABLE]

Proof.

Since $hgh^{-1}\cdot AK$ is a $\delta$-distribution on the set $S_{hg^{-1}h^{-1}}$ in (3.13),

[TABLE]

(See Example 8.2.5 in [17].) By equivariance of $\varphi_{t}$, the condition (2.8) implies that \det\bigl{(}(T_{m}(hg^{-1}h^{-1}\circ\varphi_{t})-1)_{T_{m}M/\mathbb{R}u(m)}\bigr{)}\not=0 for all $h\in G$. So $\varphi$ is $hgh^{-1}$-nondegenerate for all $h\in G$. Hence the proposition follows from Lemma 3.15. ∎

If $\varphi$ is $g$-nondegenerate, then Proposition 3.16 implies that $hgh^{-1}\cdot AK$ has a well-defined restriction to $\mathbb{R}\setminus\{0\}\times\Delta(M)$ for all $h\in G$. Analogously to Definition 3.1, we define

[TABLE]

by

[TABLE]

for $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$ and $f\in C^{\infty}_{c}(\Delta(M))$.

Proposition 3.16 has the following consequence.

Corollary 3.17**.**

Suppose that the flow $\varphi$ is $g$-nondegenerate, and that the integral

[TABLE]

converges absolutely for $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$ and $f\in C^{\infty}_{c}(M)$. Then $A\Phi^{*}_{\psi}$ is flat $g$-trace class for all such $\psi$, and (3.11) equals

[TABLE]

In the special case where $E$ is the trivial line bundle, with the trivial action by $G$ on fibres, and with $A$ the identity, we will use an equivalent expression for $\operatorname{Tr}^{\flat}_{g}(\Phi^{*})$. If $x\in G$, then we write

[TABLE]

Lemma 3.18**.**

Suppose that $E=M\times\mathbb{C}$, with the trivial $G$-action on fibres. If $\varphi$ is $g$-nondegenerate,then $K$ has a well-defined restriction to $\mathbb{R}\setminus\{0\}\times\Delta_{hg^{-1}h^{-1}}$ for all $h\in G$. Furthermore,

[TABLE]

for all $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$.

Proof.

Analogously to the proof of Proposition 3.16, one can show that

[TABLE]

and that for all $x\in G$

[TABLE]

This implies that $\operatorname{WF}(K)\cap\mathcal{N}^{*}(\mathbb{R}\setminus\{0\}\times\Delta_{x}(M))=\emptyset$, as in the proof of Proposition 3.16.

For $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$ and $f\in C^{\infty}_{c}(M)$, and a smooth kernel $\kappa$, the equality

[TABLE]

follows from the definitions. This extends to distributional kernels by continuity. ∎

3.3. Relation with the classical flat trace

In Example 3.6, we saw that for the trivial group, the flat $g$-trace of Definition 3.3 reduces to the classical flat trace. However, there is also a more refined relation between the flat $g$-trace and the classical flat trace, in terms of universal covers.

We consider the setting of Example 2.2. Suppose that $M$ is the universal cover of a compact manifold $X$, and $G=\Gamma=\pi_{1}(X)$. Let $\varphi_{X}$ be a flow on $X$, and $\Phi_{X}$ a fibre-wise linear lift of $\varphi_{X}$ to $E_{X}$. Let $\varphi$ be the lift of $\varphi_{X}$ to $M$, and let $\Phi$ be the $\Gamma$-equivariant lift of $\Phi_{X}$ to $E$.

Let $u_{X}$ be the generating vector field of $\varphi_{X}$, and let $u$ be the generating vector field of $\varphi$.

Let $\Phi_{X}$ be the flow on $E_{X}=E/\Gamma\to X$ induced by a $\Gamma$-equivariant lift $\Phi$ of $\varphi$ to $E$, and $K_{X}$ the corresponding Schwartz kernel. Let $q_{M}\colon M\to X$ be the covering map. We use similar notation for the covering maps between other spaces, such as $q_{\mathbb{R}\setminus\{0\}\times M\times M}\colon\mathbb{R}\setminus\{0\}\times M\times M\to\mathbb{R}\setminus\{0\}\times X\times X$.

Lemma 3.19**.**

Let $t\in\mathbb{R}\setminus\{0\}$. Then the following two conditions are equivalent.

1. (1)

For all $x\in X$ such that $(\varphi_{X})_{t}(x)=x$, we have on $T_{x}X/\mathbb{R}u_{X}(x)$ that

[TABLE] 2. (2)

For all $\gamma\in\Gamma$, and all $m\in M$ such that $\varphi_{t}(m)=\gamma m$, we have on $T_{m}M/\mathbb{R}u(m)$ that

[TABLE]

Proof.

Let $t\in\mathbb{R}\setminus\{0\}$, $m\in M$ and $\gamma\in\Gamma$ be such that $\varphi_{t}(m)=\gamma m$. Write $x:=\Gamma m$. The the equality $u_{X}(x)=T_{m}q(u(m))$ and commutativity of the diagram below imply that the determinants (3.17) and $\eqref{eq det M}$ are equal:

[TABLE]

∎

Proposition 3.20**.**

Under the equivalent conditions of Lemma 3.19,

[TABLE]

where the sum is over the conjugacy classes in $\Gamma$. In particular, the sum on the right converges in $\mathcal{D}^{\prime}(\mathbb{R}\setminus\{0\})$.

The proof of this proposition will be given after we establish two key lemmas.

Since $q_{M}$ is a submersion, any distributional section can be pulled back along $q_{M}$.

Lemma 3.21**.**

For $v\in\Gamma^{\infty}(E_{X}^{*})^{\prime}$, and $s\in\Gamma^{\infty}_{c}(E^{*})$,

[TABLE]

Proof.

If $v\in\Gamma^{\infty}(E_{X})$, then

[TABLE]

Here $\chi$ is as in (2.1), but in the current setting it may also be replaced by the indicator function of a fundamental domain. In the pairings in the integrands, we identified $E_{m}\cong(E_{X})_{\Gamma m}\cong(E/\Gamma)_{\Gamma m}$ via $w\mapsto\Gamma w$, for $w\in E$.

Substituting $m^{\prime}=\gamma m$, we find that (3.20) equals

[TABLE]

which equals the right hand side of (3.19).

The case for a general distributional section $v$ follows by continuity of both sides of (3.19) in $v$. ∎

Let $K_{X}\in\Gamma_{c}^{\infty}(\mathbb{R}\setminus\{0\}\times X\times X,\mathbb{R}\setminus\{0\}\times E_{X}^{*}\boxtimes E_{X})^{\prime}$ be defined as in (3.12) (with $x=e$ and $A$ the identity), for the flow $\Phi_{X}$. Similarly, let the distributional sections $K$ and $\gamma\cdot K$, for $\gamma\in\Gamma$, in $\Gamma_{c}^{\infty}(\mathbb{R}\setminus\{0\}\times M\times M,\mathbb{R}\setminus\{0\}\times E^{*}\boxtimes E)^{\prime}$ be as in (3.12). The main computation in the proof of Proposition 3.20 is the following.

Lemma 3.22**.**

We have

[TABLE]

In particular, the right hand side converges in \Gamma_{c}^{\infty}\bigl{(}\mathbb{R}\setminus\{0\}\times M\times M,(\mathbb{R}\setminus\{0\})\times E^{*}\boxtimes E\bigr{)}^{\prime}.

Proof.

Let $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$, $\xi\in\Gamma^{\infty}_{c}(E^{*})$ and $s\in\Gamma^{\infty}_{c}(E)$. Then by Lemma 3.21 (with $M$ replaced by $\mathbb{R}\setminus\{0\}\times M\times M$) and unpacking definitions, we find that

[TABLE]

By choosing a cutoff function $\chi$ close enough to the indicator function of a fundamental domain, we can make the latter expression arbitrarily close to

[TABLE]

Substituting $m^{\prime}=\gamma^{-1}m$, $\gamma^{\prime\prime}=\gamma^{-1}\gamma^{\prime}$, and using $\Gamma$-invariance of the metric on $E$ and $\Gamma$-equivariance of $\Phi_{t}$ and $\varphi_{t}$, we find that (3.21) equals

[TABLE]

∎

Proof of Proposition 3.20..

Commutativity of the diagram

[TABLE]

and Lemma 3.22 imply that

[TABLE]

Let $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$. We view the $\Gamma$-invariant function $\sum_{\gamma\in\Gamma}\gamma\cdot(p_{2}^{*}\chi)|_{\Delta(M)}$ on $\Delta(M)$ (which is constant $1$) as a function on $\Delta(X)$. Using this identification, and then Lemma 3.21 and (3.22), we find that

[TABLE]

∎

4. A fixed point formula

Guillemin’s trace formula [11] is an expression for the (non-equivariant) flat trace of the pullback map of functions along a flow, in terms of the closed orbits of this flow. Our goal in this section is to prove an equivariant generalisation of this formula, Theorem 4.2. This involves the flat $g$-trace from Section 3.

This equivariant trace formula may be of independent interest. Our main application of it will be in expressing the equivariant Ruelle dynamical $\zeta$-function in terms of the flat $g$-trace, in Theorem 4.9. This will then be used to prove Propositions 2.18 and 2.28.

The special case of Theorem 4.9 where $G$ is trivial is a fundamental property of the classical Ruelle dynamical $\zeta$-function. See for example (6.17) in [33]. This is the basis for proofs of several analytic properties of the Ruelle dynamical $\zeta$-function, such as meromorphic continuation and homotopy invariance, see [6, 7]. We expect that Theorem 4.9 can serve as a similar basis for proofs of analytic properties of the equivariant Ruelle $\zeta$-function.

Let $A\in\operatorname{End}(E)$ be $G$-equivariant, and suppose that it commutes with the flow $\Phi$. For $x\in G$, let $L_{x}(\varphi)$ and $\Gamma^{x}_{l}(\varphi)$, for $l\in L_{x}(\varphi)$, be as in Definition 2.4.

Lemma 4.1**.**

Let $x\in G$, $l\in L_{x}(\varphi)$ and $\gamma\in\Gamma^{x}_{l}(\varphi)$. Then the number

[TABLE]

is independent of $t$.

Proof.

For all $t$, we have a commuting diagram

[TABLE]

Here we use the assumption that $A$ commutes with $x$ and $\Phi_{t}$. We find that $\operatorname{tr}(Ax(\Phi_{-l})_{\gamma(t)})=\operatorname{tr}(Ax(\Phi_{-l})_{\gamma(0)})$. ∎

We denote the number (4.1) by $\operatorname{tr}(Ax\Phi_{-l}|_{\gamma})$.

Recall that we have chosen $g\in G$ such that $G/Z$ has a $G$-invariant measure $d(hZ)$. Recall Definition 2.10 of the $g$-nondegenerate property.

Theorem 4.2** (Equivariant Guillemin trace formula).**

Suppose that the flow $\varphi$ is $g$-nondegenerate. Then the operator $\Phi^{*}_{\psi}$ is flat $g$-trace class for all $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$. Furthermore, (3.11) defines a distribution on $\mathbb{R}\setminus\{0\}$, equal to

[TABLE]

Here $\delta_{l}$ is the Dirac $\delta$ distribution at $l$.

4.1. A local expression near individual fixed points

Until stated otherwise in Subsection 4.3, we assume that $E=M\times\mathbb{C}$ with the trivial action on fibres. We take $A$ to be the identity. We will use the expression for $\operatorname{Tr}_{g}^{\flat}(\Phi^{*})$ in Lemma 3.18. Let the Schwartz kernel $K$ be as in (3.12), for $x=e$. From now on, we will often use notation as if $K$ were a function.

We fix a $G$-invariant Riemannian metric on $M$, and use it to identify $T_{m}M/\mathbb{R}u(m)\cong u(m)^{\perp}$ for $m\in M$. When $x\in G$ and $l\in\mathbb{R}\setminus\{0\}$ are given, we also assume that this Riemannian metric is preserved by $\varphi$ along flow curves in $\Gamma_{l}^{x}(\varphi)$. This is possible by Lemma 2.12 and the use of a suitable partition of unity. Then for all such flow curves $\gamma$, the map $T_{\gamma(0)}(\varphi_{l}\circ x^{-1})$ preserves $u(\gamma(0))^{\perp}\subset T_{\gamma(0)}M$, and $P^{x}_{\gamma}$ can be identified with

[TABLE]

We start with a generalisation of the local computation carried out in Lemma B.1 in [7].

Lemma 4.3**.**

Let $m_{0}\in M$, $x\in G$ and $l\in\mathbb{R}\setminus\{0\}$ be such that $\varphi_{l}(m_{0})=xm_{0}$. Suppose that $\varphi$ is $x$-nondegenerate. Then there exists a density $dm$ on $M$, an $\varepsilon>0$, and a neighbourhood $U$ of $m_{0}$ such that $\varphi_{s}(m_{0})\in U$ for all $s\in(-\varepsilon,\varepsilon)$, such that for all $f\in C_{c}^{\infty}((l-\varepsilon,l+\varepsilon)\times U)$,

[TABLE]

Proof.

The proof below is similar to the proof of Lemma B.1 in [7].

Choose a neighbourhood $V$ of $m_{0}$, an $r>0$, and a chart $\kappa\colon V\to B^{n}(0,r)$ such that

[TABLE]

Here $B^{n}(0,r)$ is the open ball around [math] in $\mathbb{R}^{n}$ of radius $r$. Fix $\varepsilon\in(0,r)$ such that the neighbourhood $U:=\kappa^{-1}(B^{n}(0,\varepsilon))$ has the property that for all $t\in(l-2\varepsilon,l+2\varepsilon)$,

[TABLE]

Define the map

[TABLE]

by commutativity of the diagram

[TABLE]

Let $F_{1}\colon B^{n}(0,\varepsilon)\to(-r,r)$ be the first component of $F$, and $F^{\prime}\colon B^{n}(0,\varepsilon)\to B^{n-1}(0,r)$ the last $n-1$ components.

Let $t\in(l-\varepsilon,l+\varepsilon)$. For all $x_{1}\in\mathbb{R}$ and $x^{\prime}\in\mathbb{R}^{n-1}$ such that $(x_{1},x^{\prime})\in B^{n}(0,\varepsilon)$, the facts that the flow $\varphi$ commutes with $x$ and that $|x_{1}|$ and $|l-t+x_{1}|$ are smaller than $2\varepsilon$ imply that

[TABLE]

Choose a density $dm$ on $M$ such that $\kappa_{*}(dm)$ is the Lebesgue measure on $V$. Then the above computation implies that the Schwartz kernel $K^{x}_{t}$ of the pullback along $\varphi_{t}\circ x^{-1}$ is locally given by

[TABLE]

Note that

[TABLE]

So for $f\in C_{c}((l-\varepsilon,l+\varepsilon)\times U)$,

[TABLE]

We have a commuting diagram

[TABLE]

So

[TABLE]

and in particular the right hand side is nonzero. This implies that $\det(1-T_{x^{\prime}}F^{\prime})\not=0$ for all $x^{\prime}\in B^{n-1}(0,\varepsilon)$, if we choose $\varepsilon$ small enough. Then, again for small enough $\varepsilon$, the only point $x^{\prime}\in B^{n-1}(0,\varepsilon)$ such that $(1-F^{\prime})(x^{\prime})=0$ is the origin. For such an $\varepsilon$, we find that the right hand side of (4.4) equals

[TABLE]

which equals the right hand side of (4.3). ∎

4.2. Globalising Lemma 4.3

In this section we will establish a global version of Lemma 4.3. Recall that we write $I_{\gamma}$ for an interval on which a curve $\gamma\colon\mathbb{R}\to M$ is bijective onto its image, up to sets of measure zero.

Lemma 4.4**.**

Let $x\in G$. Suppose that $\varphi$ is $x$-nondegenerate. Suppose that $L_{x}(\varphi)$ is discrete. Then for all $f\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\}\times M)$,

[TABLE]

Proof.

Let $\Delta_{x^{-1}}(m)=(x^{-1}m,m)$, for $m\in M$. The support of $K|_{\mathbb{R}\setminus\{0\}\times\Delta_{x^{-1}}(M)}$ is

[TABLE]

The unions are disjoint. The set $L_{x}(\varphi)$ is discrete by assumption, and for each $l\in L_{x}(\varphi)$, the set $\Gamma_{l}^{x}(\varphi)$ is discrete by Lemma 2.12. Via a partition of unity, the left hand side of (4.5) can be written as a sum of contributions from $l\in L_{x}(\varphi)$ and $\gamma\in\Gamma_{l}^{x}(\varphi)$. We will use Lemma 4.3 to show that such a contribution equals the corresponding term on the right hand side of (4.5).

Fix $l\in L_{x}(\varphi)$ and $\gamma\in\Gamma_{l}^{x}(\varphi)$. The set $\gamma([0,l])$ is compact, so there are finitely many $t_{1},\ldots,t_{k}\in(0,l)$, $\varepsilon_{1},\ldots,\varepsilon_{k}>0$ and neighbourhoods $U_{1},\ldots,U_{k}$ of $\gamma(t_{1}),\ldots,\gamma(t_{k})$, respectively, as in Lemma 4.3 applied with $m_{0}=\gamma(t_{1}),\ldots,\gamma(t_{k})$, such that

[TABLE]

In fact, to simplify notation, we choose all the numbers $\varepsilon_{j}$ to be equal to the same number $\varepsilon$. We also choose the number $\varepsilon$ small enough so that $(l-\varepsilon,l+\varepsilon)\cap L_{x}(\varphi)=\{l\}$.

Since $\Gamma_{l}^{x}(\varphi)$ is discrete, we can choose the sets $U_{j}$ so that

[TABLE]

i.e. $\bigcup_{j=1}^{k}U_{j}$ does not intersect any curves in $\Gamma_{l}^{x}(\varphi)$ apart from $\gamma$. For $n\in\mathbb{Z}$ and $j=1,\ldots,k$, set

[TABLE]

The fact that $\gamma([nl,(n+1)l])=x^{-n}\gamma([0,l])$ then implies that

[TABLE]

And the number $\varepsilon>0$ and the neighbourhoods $U_{n,j}$ of $\gamma(t_{n,j})$ are as in Lemma 4.3.

We choose a sub-collection of the sets $U_{n,j}$ that cover $\operatorname{im}(\gamma)$ in a locally finite way. The interval $I_{\gamma}$ is either all of $\mathbb{R}$ (if $\gamma$ has no loops) or bounded (if $\gamma$ has loops). If $I_{\gamma}=\mathbb{R}$, then we use all sets $U_{n,j}$, and we set

[TABLE]

If $I_{\gamma}$ is bounded, we can choose it to be of the form $[0,T]$ for some $T>0$. Then let $n_{1},n_{2}\in\mathbb{Z}$ be nonnegative such that $n_{2}\leq k$, $t_{n_{1},n_{2}}\geq T$ and $t_{n_{1},n_{2}-1}<T$ (if $n_{2}\geq 1$) or $t_{n_{1}-1,k}<T$ (if $n_{2}=1$). Then

[TABLE]

We now set

[TABLE]

The middle component is omitted if $n_{1}=0$, which happens if $T<l$. Then the set $\{t_{\alpha};\alpha\in J\}$ consists of consecutive elements of the set of all $t_{n,j}$, and

[TABLE]

Let $\zeta_{j}\in C^{\infty}_{c}(U_{j})$ and set $\zeta_{n,j}:=(x^{-n})^{*}\zeta_{j}$. By construction of the set $J$ in both the cases $I_{\gamma}=\mathbb{R}$ and $I_{\gamma}$ bounded, we can choose these functions so that

[TABLE]

equals $1$ on the image of $\gamma$, and zero on the images of all other curves in $\Gamma_{l}^{x}(\varphi)$. Let $\psi\in C_{c}^{\infty}(l-\varepsilon,l+\varepsilon)$ be such that $\psi(l)=1$. Then Lemma 4.3 implies that for all $f\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\}\times M)$,

[TABLE]

Now for all $n$ and $j$

[TABLE]

so

[TABLE]

For any interval $I\subset\mathbb{R}$, we write $1_{I}$ for the function on $\mathbb{R}$ that is $1$ on $I$ and [math] outside $I$. The expression

[TABLE]

approaches the number $\int_{I_{\gamma}}f(l,\gamma(s))\,ds$ arbitrarily closely if the smooth functions $\zeta_{j}$ are chosen so that the function

[TABLE]

approaches $1_{I_{\gamma}}$ closely enough in $L^{1}$-norm. The left hand side of (4.7) is independent of the functions $\zeta$ and $\psi$ with the properties mentioned, and therefore equals

[TABLE]

which completes the proof. ∎

4.3. Proof of Theorem 4.2

We will start by proving Theorem 4.2 in the case where $E$ is the trivial line bundle.

Proposition 4.5**.**

Suppose that $\varphi$ is $g$-nondegenerate. Then the operator $\Phi^{*}_{\psi}$ is flat $g$-trace class for all $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$. Furthermore,

[TABLE]

Proof.

By Lemmas 3.18 and 4.4, we have for all $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$,

[TABLE]

Here we have also used (2.9). ∎

Remark 4.6**.**

Suppose that $G=\{e\}$, so $\chi\equiv 1$. Then by Example 2.8, the equality (4.8) in Proposition 4.5 becomes

[TABLE]

This is Guillemin’s trace formula (II.17) in [11]; see also (1.5) in [7].

Proposition 4.5 is the special case of Theorem 4.2 when $E$ is the trivial line bundle. We now show how this special case implies Theorem 4.2 for general vector bundles $E$ (compare also the end of Appendix B in [7]). Fix $x\in G$. Let $A\in\operatorname{End}(E)$, and suppose $A$ commutes with both the $G$-action and with $\Phi$. Consider the function $\operatorname{tr}(x\cdot A\Phi)\in C^{\infty}(\mathbb{R}\setminus\{0\}\times\Delta(M))$, given in terms of a local frame $\{e_{j}\}_{j=1}^{\operatorname{rank}(E)}$ for $E$, with dual frame $\{e^{j}\}_{j=1}^{\operatorname{rank}(E)}$, by

[TABLE]

for $t\in\mathbb{R}\setminus\{0\}$ and $m\in M$. Let $x\cdot K^{0}$ be the Schwartz kernel of the operator given by pulling back scalar functions along $x^{-1}\circ\varphi_{t}$, as in (3.12) with $E=M\times\mathbb{C}$.

Lemma 4.7**.**

Suppose that $\varphi$ is $g$-nondegenerate. We have the following equality in $\mathcal{D}^{\prime}(\mathbb{R}\setminus\{0\}\times\Delta(M))$:

[TABLE]

Proof.

Consider a local frame $\{e_{j}\}_{j=1}^{\operatorname{rank}(E)}$ for $E$ defined in an open set $U\subset M$, with dual frame $\{e^{j}\}_{j=1}^{\operatorname{rank}(E)}$ for $E^{*}$. Extending (4.9), we define $\tilde{\operatorname{tr}}(x\cdot A\Phi)\in C^{\infty}(\mathbb{R}\setminus\{0\}\times U\times U)$ by

[TABLE]

for $t\in\mathbb{R}\setminus\{0\}$ and $m,m^{\prime}\in U$. Observe that this function is constant in the second factor $U$.

Define $\tilde{\operatorname{tr}}(x\cdot AK)\in\mathcal{D}^{\prime}(\mathbb{R}\setminus\{0\}\times U\times U)$ by

[TABLE]

for $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$, and $f_{1},f_{2}\in C^{\infty}_{c}(U)$. Then for all such $\psi$, $f_{1}$ and $f_{2}$, (3.10) and (3.12) imply that

[TABLE]

so

[TABLE]

The restrictions of both sides to $\mathbb{R}\setminus\{0\}\times\Delta(U)$ are well-defined by Proposition 3.16. Taking these restrictions and using

[TABLE]

and partitions of unity with respect to sets like $U$, we obtain (4.10). ∎

Theorem 4.2 follows from Proposition 4.5, Lemma 4.7 with $A=(-1)^{N}N$, and the fact that for all $m\in M$ such that $\varphi_{t}(m)=xm$,

[TABLE]

the trace of the linear endomorphism $A\circ x\circ(\Phi_{-t})_{m}$ of $E_{m}$.

4.4. An expression for the equivariant Ruellle dynamical $\zeta$-function

We will use some well-known linear algebra.

Lemma 4.8**.**

Let $V$ be a finite-dimensional vector space, and $A\in\operatorname{End}(V)$. If $\dim(\ker(1-A))=1$, then

[TABLE]

Proof.

For all $t\in\mathbb{R}$,

[TABLE]

The derivative at $t=1$ of the left hand side is the left hand side of (4.11). If $\lambda_{1},\ldots,\lambda_{\dim(V)}$ are the eigenvalues of $A$, repeated by multiplicities, then the derivative at $t=1$ of the right hand side of (4.12) is

[TABLE]

If exactly one of the eigenvalues $\lambda_{j}$ equals $1$, then this is the right hand side of (4.11). ∎

For all $m\in M$ and $t\in\mathbb{R}$, let $\tau^{\nabla^{F}}_{t}\colon F_{m}\to F_{\varphi_{t}(m)}$ be parallel transport with respect to $\nabla^{F}$, along the flow curve from $m$ to $\varphi_{t}(m)$. Define the flow $\Phi$ on $\textstyle{\bigwedge}T^{*}M\otimes F$ lifting $\varphi$ by

[TABLE]

for all $t\in\mathbb{R}$, $m\in M$, $\omega\in\textstyle{\bigwedge}T^{*}_{m}M$ and $v\in F_{m}$. Let $N$ be the number operator on $\textstyle{\bigwedge}T^{*}M$, acting as the scalar $p$ on $\textstyle{\bigwedge}^{*}T^{p}M$. As before, we assume that $\varphi$ is $g$-nondegenerate.

Consider the smooth function $\psi_{\sigma}(t)=e^{-\sigma|t|}/|t|$ on $\mathbb{R}\setminus\{0\}$, for $\sigma\in\mathbb{C}$. It is not compactly supported, so a priori the pairing $\langle\operatorname{Tr}_{g}^{\flat}((-1)^{N}N\Phi^{*}),\psi_{\sigma}\rangle$ is not defined. We define this number, if it exists, by the property that for all $\varepsilon>0$, there is an $R>0$ such that for all $\psi\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$ with values in $[0,1]$, for which $\psi(t)=1$ if $\frac{1}{R}<|t|<R$, we have

[TABLE]

The following fact is an equivariant generalisation of a well-known expression for the Ruelle dynamical $\zeta$-function in terms of the flat trace. See e.g. (5.4) in [6], (2.5) in [7] or (6.17) in [33].

Theorem 4.9**.**

Suppose that $L_{g}(\varphi)$ is countable. Then the pairing

[TABLE]

is well-defined if and only if (2.11) converges, and then

[TABLE]

Proof.

It follows from Lemma 4.8 that for all $l\in L_{g}(\varphi)$ and $\gamma\in\Gamma_{l}^{g}(\varphi)$,

[TABLE]

Let $r>0$ be such that $\pm r,\pm 1/r\not\in L_{g}(\varphi)$. Let

[TABLE]

Let $\psi_{r}\in C^{\infty}_{c}(\mathbb{R}\setminus\{0\})$ be such that $\psi_{r}(l)=1$ for all $l\in L_{g}^{r}(\varphi)$ and $\psi_{r}(l)=0$ for all $l\in L_{g}(\varphi)\setminus L_{g}^{r}(\varphi)$.

From (4.14), Theorem 4.2 implies that

[TABLE]

The theorem follows in the limit $r\to\infty$. ∎

Proof of Proposition 2.18..

The right hand side of (4.13) does not involve choices of representatives of classes in sets of the form (2.4), or intervals $I_{\gamma}$ as in (2.7). And Proposition 3.11 implies that the right hand side of (4.13) does not depend on $\chi$. ∎

Proof of Proposition 2.28..

Let $N_{X}$ be the number operator on $\textstyle{\bigwedge}^{*}T^{*}X$. Proposition 3.20 implies that

[TABLE]

Let $\nabla^{F_{X}}$ be the connection on $F_{X}$ induced by $\nabla^{F}=d$. Then by Theorem 4.9 and (4.15),

[TABLE]

whenever the left hand side converges. Hence the claim follows from Lemma 2.24. ∎

Remark 4.10**.**

The equality (4.13) may be written suggestively as

[TABLE]

If $\Delta_{F}$ has trivial kernel, then (2.31) means that $2\log T_{g}(\nabla^{F})$ is a regularisation of the divergent integral

[TABLE]

The superficial similarity between (4.16) and (4.17) is an equivariant generalisation of an analogous similarity in the classical case, compare for example (1.8) and (6.18) in [33]. This may be viewed as evidence that an equivariant Fried conjecture (2.32) may hold in certain cases.

5. The line and the circle

In the rest of this paper, we compute the equivariant Ruelle dynamical $\zeta$-function in several classes of examples. We will see that this function is defined for some flows for which the classical Ruelle $\zeta$-function is not defined. We will also see that the equivariant Fried equality (2.32) holds in the examples where both sides are defined.

We start with the most basic, one-dimensional examples: the line and the circle. We will then show that the equivariant Fried conjecture is true, by explicit computations.

5.1. The real line

Consider the action of $G=\mathbb{R}$ on $M=\mathbb{R}$ by addition, and the flow

[TABLE]

on $\mathbb{R}$. Consider the trivial line bundle $F=\mathbb{R}\times\mathbb{C}\to\mathbb{R}$. Let $\alpha\in i\mathbb{R}$, and consider the connection $\nabla^{F}:=d+\alpha dx$ on $F$. We let $\mathbb{R}$ act trivially on the fibres of $F$.

Remark 5.1**.**

One obtains the same result for the trivial connection $d$ and the nontrivial action by $\mathbb{R}$ on $F$ given by

[TABLE]

for $g,x\in\mathbb{R}$ and $z\in\mathbb{C}$. See Remark 6.9 in [13].

Lemma 5.2**.**

If $g\in\mathbb{R}\setminus\{0\}$, then for all $\sigma\in\mathbb{C}$,

[TABLE]

and

[TABLE]

Proof.

Let $\gamma\colon\mathbb{R}\to\mathbb{R}$ be the unique flow curve of $\varphi$ modulo the action (2.3): $\gamma(t)=t.$ Suppose first that $g\in\mathbb{R}\setminus\{0\}$. Then $L_{g}(\varphi)=\{g\}$, and $\Gamma^{g}_{g}(\varphi)=\{\gamma\}$. Now $u(x)^{\perp}=\{0\}$ for all $x\in\mathbb{R}$, so the flow is nondegerenate, and

[TABLE]

For all $a<b$, parallel transport with respect to $\nabla^{F}$ along $\gamma|_{[a,b]}$ is given by

[TABLE]

for $z\in\mathbb{C}=F_{a}$. So

[TABLE]

Now $T_{\gamma}=1$ by Example 2.7, so

[TABLE]

and (5.1) follows.

For $g=0$, we have $L_{0}(\varphi)=\emptyset$, which implies (5.2). ∎

Proposition 6.2 in [13] states that the equivariant analytic torsion of $\nabla^{F}$ equals

[TABLE]

for $g\in\mathbb{R}\setminus\{0\}$, while

[TABLE]

It follows by Lemma 5.2 that the answer to Question 2.31 is yes in this example and the equivariant Fried conjecture holds.

Remark 5.3**.**

The two sides of (2.32) are generally not real in this example. The same is true in other cases; see Lemma 5.6 and Proposition 6.14. This shows that it is desirable not to have absolute values in (2.32).

A version of the equivariant Ruelle $\zeta$-function involving only positive $l$ would be equal to $1$ for negative $g$ in this example, see Example 2.7. Because equivariant analytic torsion (5.3) is different from $1$ for such $g$, the equivariant Fried conjecture would not hold for such a definition of the equivariant Ruelle $\zeta$-function. The two sides of (2.32) do not even have the same absolute value in that case.

5.2. The circle

The example in this subsection generalises Example 2.10 in [33].

We will use the following fact from complex analysis.

Lemma 5.4**.**

Let $r\in\mathbb{R}\setminus\mathbb{Z}$, and $\alpha\in i\mathbb{R}$. For $z\in\mathbb{C}$ with positive real part, define

[TABLE]

*Then for all $z\in\mathbb{C}\setminus(\pm\alpha+2\pi i\mathbb{Z})$, there is a neighbourhood of $z$ to which $F$ continues analytically. In particular, if $\alpha\not\in 2\pi i\mathbb{Z}$, then $F$ has an analytic continuation to a neighbourhood of zero. *

Proof.

If $\operatorname{Re}(z)>0$, then

[TABLE]

The last term is analytic, so it is enough to show that the two series on the right hand side have the desired properties. These series can be written in terms of the hypergeometric function ${{}_{2}F_{1}}$. Using the Pochhammer symbol $(w)_{n}=w\cdot(w+1)\cdots(w+n-1)$, and $(w)_{0}=1$, we have for all integers $n\geq 0$,

[TABLE]

The equality (5.5) implies that the first series on the right hand side of (5.4) equals

[TABLE]

Similarly, (5.6) implies that the second series on the right hand side of (5.4) equals

[TABLE]

The function ${}_{2}F_{1}(a,b;c;\relbar)$ has an analytic continuation to a neighbourhood of any $z\in\mathbb{C}\setminus\{1\}$; this is a standard fact, see e.g. Section 5.2 of [37]. This implies the claim. We point out that the condition $\operatorname{Re}(c-a-b)>0$ for the function ${}_{2}F_{1}(a,b;c;\relbar)$ to extend to $z=1$ does not apply in our situation, as $c-a-b=0$ for both combinations of $a$, $b$ and $c$ that occur. ∎

We consider the action by the circle $\mathbb{R}/\mathbb{Z}$ on itself by left multiplication, and the flow $\varphi$ given by

[TABLE]

for $x,t\in\mathbb{R}$. We consider the connection $\nabla^{F}=d+\alpha dx$ on the trivial line bundle $F\to\mathbb{R}/\mathbb{Z}$, with the trivial action by $\mathbb{R}/\mathbb{Z}$, for $\alpha\in i\mathbb{R}$.

Remark 5.5**.**

Analogously to Remark 5.1, one can equivalently consider the connection $d$ on the line bundle associated to the unitary representation of $\mathbb{Z}=\pi_{1}(\mathbb{R}/\mathbb{Z})$ defined by $\alpha$.

Lemma 5.6**.**

If $g=r+\mathbb{Z}\in\mathbb{R}/\mathbb{Z}$ for $r\not\in\mathbb{Z}$, then for all $\sigma\in\mathbb{C}$ with positive real part,

[TABLE]

*If $\alpha\not\in 2\pi i\mathbb{Z}$, then this has an analytic continuation to a neighbourhood of $\sigma=0$. *

Proof.

Let $\gamma$ be the unique flow curve of $\varphi$, modulo the action (2.3): $\gamma(t)=t+\mathbb{Z}$. Let $r\not\in\mathbb{Z}$ and let $g=r+\mathbb{Z}$. Then $L_{g}(\varphi)=\{n+r;n\in\mathbb{Z}\}$. For all $n\in\mathbb{Z}$, we have $\Gamma_{n+r}^{g}(\varphi)=\{\gamma\}$. Now $u(x+\mathbb{Z})^{\perp}=\{0\}$ for all $x\in\mathbb{R}$, so the flow is nondegerenate, and

[TABLE]

For all $a<b$, parallel transport with respect to $\nabla^{F}$ along $\gamma|_{[a,b]}$ is given by

[TABLE]

for $z\in\mathbb{Z}=F_{a+\mathbb{Z}}$. For $|a-b|<1$, this follows from the solution of the equation $s^{\prime}+\alpha s=0$ for parallel sections; in general this follows by composing parallel transport over intervals of lengths smaller than $1$. So if $\gamma$ is viewed as an element of $\Gamma_{n+r}^{g}(\varphi)$, then

[TABLE]

Now $T_{\gamma}=1$ by Example 2.8. Therefore, we have

[TABLE]

whenever the above converges. Hence the first claim follows. The second claim follows by Lemma 5.4. ∎

Example 5.7**.**

Using the Taylor series of the function $x\mapsto\frac{\tanh^{-1}(x)}{\sqrt{x}}$, one finds that for $z\in\mathbb{C}$ with negative real part,

[TABLE]

Together with the equality $e^{2w}=\frac{1+\tanh(w)}{1-\tanh(w)}$, this implies that

[TABLE]

So as a special case of Lemma 5.6, we find that for $\sigma\in\mathbb{C}$ with positive real part,

[TABLE]

If $\sigma$ is a positive real number, then this equals $\left|\frac{1+e^{\alpha-\sigma}}{1-e^{\alpha-\sigma}}\right|=|\tanh\left(\frac{\alpha-\sigma}{2}\right)|^{-1}$.

Suppose that $\alpha\not\in 2\pi i\mathbb{Z}$; this is equivalent to the condition that $\ker(\Delta_{F}^{p})=\{0\}$ in Question 2.31. Then by Proposition 6.6 in [13], we have for $g=r+\mathbb{Z}$, with $r\not\in\mathbb{Z}$,

[TABLE]

By Lemma 5.6 and (5.7), and the substitution $n\mapsto-n$,

[TABLE]

So the answer to Question 2.31 is yes in this case.

The following computation is analogous to Example 2.10 in [33].

Lemma 5.8**.**

Suppose that $\alpha\not\in 2\pi i\mathbb{Z}$. Then for $e=0+\mathbb{Z}\in G$ and all $\sigma\in\mathbb{C}$ with positive real part,

[TABLE]

Proof.

Now $L_{e}(\varphi)=\mathbb{Z}\setminus\{0\}$. For all $n\in\mathbb{Z}\setminus\{0\}$, we have $\Gamma_{n}^{e}(\varphi)=\{\gamma\}$, where $\gamma$ is the unique flow curve as in the proof of Lemma 5.2. Analogously to the proof of that lemma, we find that whenever the following converges,

[TABLE]

Using the Taylor series of the function $x\mapsto\log(1+x)$, we find that if $\operatorname{Re}(\sigma)>0$,

[TABLE]

which implies the claim. ∎

If $\alpha\not\in 2\pi i\mathbb{Z}$, then by Proposition 6.6 in [13], we have for $e=0+\mathbb{Z}$,

[TABLE]

By Lemma 5.8 and (5.9),

[TABLE]

So also in this case, the answer to Question 2.31 is yes.

Now consider the situation of Lemma 5.2, with the group $\mathbb{R}$ replaced by $\mathbb{Z}$. Then by arguments entirely analogous to the proof of that lemma, we find that for all $g\in\mathbb{Z}\setminus\{0\}$ and $\sigma\in\mathbb{C}$,

[TABLE]

and that

[TABLE]

Together with Lemma 5.8, this allows us to illustrate Proposition 2.28.

By (5.10) and (5.11), we have for all $\sigma\in\mathbb{C}$ for which the following converges,

[TABLE]

By (5.8), the right hand side equals $2\log R^{0+\mathbb{Z}}_{\varphi_{\mathbb{R}/\mathbb{Z}},\nabla^{F_{\mathbb{R}/\mathbb{Z}}}}(\sigma)$, where $\varphi_{\mathbb{R}/\mathbb{Z}}$ and $\nabla^{F_{\mathbb{R}/\mathbb{Z}}}$ are induced by $\varphi$ and $\nabla^{F}$, respectively. By Lemma 2.24, we find that

[TABLE]

where $\rho$ is the unitary representation of $\pi_{1}(\mathbb{R}/\mathbb{Z})=\mathbb{Z}$ defined by $e^{\alpha}$. So we directly see that (2.23) holds in this example.

6. Geodesic flow on $\mathbb{R}^{n}$

The classical Ruelle $\zeta$-function is trivial for the geodesic flow on $\mathbb{R}^{n}\times S^{n-1}=S(T\mathbb{R}^{n})$, because there are no closed geodesics. We show that this flow is $g$-nondegenerate for the action by the Euclidean motion group on $S(T\mathbb{R}^{n})$, for group elements $g$ satisfying a regularity property. The expression for the equivariant Ruelle $\zeta$-function diverges, however. We show that for certain discrete subgroups of the Euclidean motion group, the equivariant Ruelle $\zeta$-function does converge, and we compute it explicitly. In that case, there are no issues with convergence and meromorphic continuation, because the sums that occur are finite. We conclude in Proposition 6.18 that the equivariant Fried conjecture is true in this case.

6.1. Geodesic flow on $S(T\mathbb{R}^{n})$

Consider the natural action by the oriented Euclidean motion group $G=\mathbb{R}^{n}\rtimes\operatorname{SO}(n)$ on the Euclidean space $\mathbb{R}^{n}$, given by

[TABLE]

for $x,y\in\mathbb{R}^{n}$ and $r\in\operatorname{SO}(n)$. Let $M=\mathbb{R}^{n}\times S^{n-1}$, viewed as the unit sphere bundle of $T\mathbb{R}^{n}$. For the rest of this section, the letters $x$ and $y$ denote elements of $\mathbb{R}^{n}$, $v$ denotes an element of $S^{n-1}$, $r$ an element of $\operatorname{SO}(n)$ and $t$ and $l$ real numbers.

The lift of the action to $M$ is given by

[TABLE]

The exponential map of $\mathbb{R}^{n}$ is

[TABLE]

So the geodesic flow $\varphi$ on $M$ is given by

[TABLE]

The generating vector field $u$ of this flow is

[TABLE]

Consistent with Lemma 2.1, the flow $\varphi$ is $G$-invariant:

[TABLE]

We denote the $n\times n$ identity matrix by $I$.

Lemma 6.1**.**

Let $r\in\operatorname{SO}(n)$, $x,y\in\mathbb{R}^{n}$, $v\in S^{n-1}$ and $l\in\mathbb{R}$. Then $\varphi_{l}(x,v)=(y,r)(x,v)$ if and only if $rv=v$ and

[TABLE]

where $w$ is the component of $x$ orthogonal to $v$.

Proof.

Write $x=\lambda v+w$, with $\lambda\in\mathbb{R}$ and $w\perp v$. Then

[TABLE]

The right hand sides are equal if and only if $rv=v$ and

[TABLE]

The latter condition is equivalent to (6.1). ∎

Example 6.2**.**

If $l=0$, then Lemma 6.1 states that the fixed points of $g=(y,r)\in G$ are the points $(\lambda v+w,v)\in M$, for $\lambda\in\mathbb{R}$, $v\in S^{n-1}$ such that $rv=v$, and $w\perp v$ such that $(I-r)w=y$.

Lemma 6.3**.**

Let $g=(y,r)\in G$, for $y\in\mathbb{R}^{n}$ and $r\in\operatorname{SO}(n)$ such that $\ker(r-I)$ is one-dimensional. Then the flow $\varphi$ is $g$-nondegenerate.

Proof.

Let $v_{0}\in\ker(r-I)$ be a unit vector. Let $(x,v)\in M$, and suppose that $\varphi_{l}(x,v)=g(x,v)$. Let $a\in T_{x}\mathbb{R}^{n}=\mathbb{R}^{n}$ and $b\in T_{v}S^{n-1}\subset\mathbb{R}^{n}$. Then a direct computation involving the equalities $rv=v$ and $rx+y=x+lv$ from Lemma 6.1 yields

[TABLE]

We find that, as a map from $v_{0}^{\perp}\oplus v_{0}^{\perp}$ to itself,

[TABLE]

which implies that

[TABLE]

Since $I-r$ is invertible on $v_{0}^{\perp}$, the number $\det((I-r)|_{v_{0}^{\perp}})^{2}$ is positive. ∎

Remark 6.4**.**

In this example, the group action has fixed points, while the vector field generating the flow has no zeroes. By Example 6.2, the fixed points of $g=(y,r)$, with $\ker(I-r)$ spanned by $v_{0}$ and $y\perp v_{0}$, are of the form $(\lambda v_{0}+w,\pm v_{0})$, for $w\perp v_{0}$ such that $(I-r)w=y$. By (6.2) for $l=0$, we have

[TABLE]

So if $(x,v)=(\lambda v_{0}+w,\pm v_{0})\in M^{g}$, then

[TABLE]

as expected.

Remark 6.5**.**

By a computation analogous to Lemma 6.3, the equivariant Ruelle $\zeta$-function is not well-defined for the action by the group $\mathbb{R}^{n}<G$ on $M$ if $n\geq 2$, because the flow is not $g$-nondegenerate for $g\in\mathbb{R}^{n}<G$.

Remark 6.6**.**

An element $r\in\operatorname{SO}(n)$ for which $\ker(r-I)$ is one-dimensional exists if and only if $n$ is odd. If $n=1$, this element $r$ is the identity element. So for nontrivial examples, $n=2k+1$ is an odd integer greater than or equal to $3$. In such cases, every element of $\operatorname{SO}(n)$ is conjugate to a block-diagonal element of $\operatorname{SO}(2)^{k}\times\{1\}$. The condition that $\ker(r-I)$ is one-dimensional means that all $k$ components in $\operatorname{SO}(2)$ are nontrivial, which is true for elements of an open dense subset of $\operatorname{SO}(n)$. This implies that the condition on $r$ holds generically.

6.2. Ingredients of the equivariant Ruelle $\zeta$-function

Lemma 6.7**.**

Let $r\in\operatorname{SO}(n)$ be such that $\ker(r-I)$ is one-dimensional. Let $v_{0}$ be one of the two unit vectors in $\ker(r-I)$. Let $y\in\mathbb{R}^{n}$, and write $g:=(y,r)\in G$. Then

[TABLE]

if $y=lv_{0}+w^{\prime}$, for $w^{\prime}\perp v_{0}$.

Proof.

Let $l\in\mathbb{R}\setminus\{0\}$. Then $l\in L_{g}(\varphi)$ if and only if there are $x\in\mathbb{R}^{n}$ and $v\in S^{n-1}$ such that $\varphi_{l}(x,v)=g(x,v)$. By Lemma 6.1, this is the case if and only if $rv=v$ and $(r-I)w=lv-y$, where $w$ is the component of $x$ orthogonal to $v$.

Since $\ker(r-I)=\{\pm v_{0}\}$, the condition $rv=v$ means $v=\pm v_{0}$. And then $(r-I)w=lv-y$ means

[TABLE]

Write $y=\lambda v_{0}+w^{\prime}$, with $w^{\prime}\perp v_{0}$. Then (6.3) becomes

[TABLE]

As $\ker(r-I)$ is one-dimensional, $I-r$ is invertible on the orthogonal complement of $v_{0}$. So there is a unique $w$ such that $w^{\prime}=(I-r)w$. ∎

Example 6.8**.**

In the setting of Lemma 6.7, we have $L_{g}(\varphi)=\emptyset$ for $g$ with a fixed point. Indeed, if $g$ fixes a point, then we saw in Example 6.2 that there is a $w\perp v_{0}$ such that $(I-r)w=y$. This implies that $y\perp v_{0}$, so $l=0$ in Lemma 6.7.

Fix a group element $g=(l_{0}v_{0}+w^{\prime},r)$, for $r\in\operatorname{SO}(n)$ with $\ker(r-I)$ one-dimensional, spanned by a unit vector $v_{0}$, $l_{0}\in\mathbb{R}\setminus\{0\}$ and $w^{\prime}\perp v_{0}$. By Lemma 6.7, we have $L_{g}(\varphi)=\{\pm l_{0}\}$.

Lemma 6.9**.**

For $l=\pm l_{0}\in L_{g}(\varphi)$, the set $\Gamma^{g}_{l}(\varphi)$ of $(g,l)$-periodic flow curves consists of a single flow curve of $\varphi$: the one through the point $(w,\pm v_{0})$, for the unique $w\perp v_{0}$ such that $w^{\prime}=(I-r)w$.

Proof.

A point $(x,v)$ lies on a flow curve in $\Gamma^{g}_{l}(\varphi)$ precisely if $\varphi_{l}(x,v)=g(x,v)$. By Lemma 6.1, this means that $rv=v$ and $x=\lambda v+w$, with $w\perp v$ such that (6.1) holds. The condition $rv=v$ means that $v=\pm v_{0}$. We distinguish the four cases $l=\pm l_{0}$ and $v=\pm v_{0}$.

• •

If $l=l_{0}$ and $v=v_{0}$, then (6.1) becomes

[TABLE]

This has a unique solution $w\perp v_{0}$, because $r-I$ is invertible on $v_{0}^{\perp}$.

• •

If $l=l_{0}$ and $v=-v_{0}$, then (6.1) becomes

[TABLE]

This has no solutions, because $l\not=0$ and $(r-I)w$ and $w^{\prime}$ are orthogonal to $v_{0}$.

• •

If $l=-l_{0}$ and $v=v_{0}$, then there are no solutions, similarly to the second point.

• •

If $l=-l_{0}$ and $v=-v_{0}$, then analogously to the first point, we find a unique $w\perp v_{0}$ satisfying (6.1).

So if $l=l_{0}$, then $(x,v)$ lies on a flow curve in $\Gamma^{g}_{l}(\varphi)$ if and only if $v=v_{0}$ and $x=\lambda v_{0}+w$, for the unique $w\perp v_{0}$ such that $(r-I)w=-w^{\prime}$. Since

[TABLE]

this means that $(x,v)$ lies on the unique flow curve through $(w,v_{0})$.

The case where $l=-l_{0}$ is analogous. We then find that the point $(x,v)$ lies on a flow curve in $\Gamma^{h}_{l}(\varphi)$ if and only if $v=-v_{0}$, and $x=\lambda v_{0}+w$, for the unique $w\perp v_{0}$ such that $(r-I)w=-w^{\prime}$. This means that $(x,v)$ lies on the flow curve through $(w,-v_{0})$. ∎

Let $F$ be the trivial line bundle on $M$. Consider the flat connection

[TABLE]

on $F$, for $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{C}^{n}$ and a closed one-form $\beta$ on $S^{n-1}$. We will later assume that $\nabla^{F}$ is invariant under actions by certain subgroups of $G$.

Lemma 6.10**.**

Let $\gamma$ be a flow curve of $\varphi$, and write $\gamma(0)=(x,v)$. Then parallel transport along $\gamma$, from $t=0$ to $t=l$, with respect to $\nabla^{F}$, is given by multiplication by $e^{-l(\alpha,v)}$.

Proof.

Let $s\in C^{\infty}(M)$, viewed as a section of $E$. Then for all $t\in\mathbb{R}$,

[TABLE]

This equals zero if $s$ is given by

[TABLE]

for a constant $C$, and $x^{\prime}\in\mathbb{R}^{n}$ and $v^{\prime}\in S^{n-1}$.

The parallel transport map to be computed maps $s(x,v)$ to

[TABLE]

∎

If we let $G$ act trivially on the fibres of $F$, then Lemma 6.10 implies that that for the curves $\gamma_{\pm}$ in Lemma 6.9,

[TABLE]

independent of the sign.

6.3. The equivariant Ruelle $\zeta$-function for a discrete subgroup

It turns out that for the action by the $G$ on $M$, the equivariant Ruelle $\zeta$-function diverges. We compute this function for a discrete subgroup and an element $g$ for which it does converge.

Lemma 6.11**.**

Suppose that $\nabla^{F}$ is $G$-invariant, so $\beta$ is $\operatorname{SO}(n)$-invariant and $\alpha=0$. Let $g=(l_{0}v_{0}+w^{\prime},r)\in G$, for $r\in\operatorname{SO}(n)$ with $\ker(r-I)$ one-dimensional, spanned by a unit vector $v_{0}$, $l_{0}\in\mathbb{R}\setminus\{0\}$ and $w^{\prime}\perp v_{0}$. Then the right hand side of (2.12) diverges, for the action by $G$ on $M$.

Proof.

Let $\tilde{\chi}\in C^{\infty}_{c}(\mathbb{R}^{n})$ be $\operatorname{SO}(n)$-invariant and such that

[TABLE]

The pullback $\chi$ of $\tilde{\chi}$ to $M$ is a cutoff function for the action by $G$. So by Lemmas 2.19, 6.3, 6.7, 6.9 and equality (6.5), the right hand side of (2.12) equals

[TABLE]

where we have used that $\alpha=0$. By right $Z$-invariance of the integrand in the definition of the Ruelle $\zeta$-function and compactness of $Z_{\operatorname{SO}(n)}(r)$, this equals

[TABLE]

Let $w\perp v_{0}$ be such that $(I-r)w=w^{\prime}$. Then

[TABLE]

For a given $r^{\prime}\in\operatorname{SO}(3)$, define $\Phi\colon v_{0}^{\perp}\times\mathbb{R}\to\mathbb{R}^{n}$ by $\Phi(x^{\prime},t)=tr^{\prime}v_{0}+x^{\prime}$. This is a linear isomorphism if $r^{\prime}v_{0}\not\in v_{0}^{\perp}$. Its matrix representation with respect to the decomposition $\mathbb{R}^{n}=v_{0}^{\perp}\oplus\mathbb{R}v_{0}$ is

[TABLE]

The derivative of this map is the map itself, and its determinant is $(v_{0},r^{\prime}v_{0})$. So (6.6) equals

[TABLE]

The integral on the right diverges. ∎

From now on, let $r\in\operatorname{SO}(n)$ be an element of finite order. We assume, as before, that $\ker(r-I)$ is one-dimensional, spanned by a unit vector $v_{0}$. Let $a>0$, and let $\Gamma^{\prime}<v_{0}^{\perp}$ be an $r$-invariant, discrete, cocompact subgroup. Consider the closed, cocompact subgroups

[TABLE]

Note that $\Gamma$ is invariant under $r$, so $H$ is well-defined. Consider the action by $H$ on $M$.

Let $g=(al_{0}v_{0}+w^{\prime},r^{m})\in H$, for $l_{0}\in\mathbb{Z}\setminus\{0\}$, $w^{\prime}\in\Gamma^{\prime}$ and $m\in\mathbb{Z}$. We suppose that $\ker(r^{m}-I)$ is one-dimensional, and hence equal to $\ker(r-I)=\mathbb{R}v_{0}$. Let $Z<H$ be the centraliser of $g$.

Lemma 6.12**.**

The map from $\Gamma^{\prime}$ to $H/Z$ mapping $\gamma^{\prime}\in\Gamma^{\prime}$ to $(\gamma^{\prime},I)Z$, is a bijection.

Proof.

To determine $Z<H$, suppose that $\lambda\in\mathbb{Z}$, $\gamma^{\prime}\in\Gamma^{\prime}$, $k\in\mathbb{Z}$, and consider $(a\lambda v_{0}+\gamma^{\prime},r^{k})\in H$. By a direct computation, we find that this element lies in $Z$ if and only if $(r^{m}-I)\gamma^{\prime}=(r^{k}-I)w^{\prime}$. So

[TABLE]

This implies the lemma. ∎

Let $Y\subset M$ be an $r$-invariant fundamental domain for the action by $\Gamma$. Define the function $\chi$ on $M$ by

[TABLE]

This function is clearly not smooth, but it has the property (2.1).

Lemma 6.13**.**

If $\gamma_{\pm}$ are as in Lemma 6.9, then

[TABLE]

Proof.

By Lemma 6.12, the left hand side of (6.10) equals

[TABLE]

The point \bigl{(}\bigl{(}(I-r)|_{v_{0}^{-1}}\bigr{)}^{-1}w^{\prime}+\gamma^{\prime}\pm sv_{0},\pm v_{0}\bigr{)} lies in $Y$ for a unique $\gamma^{\prime}\in\Gamma^{\prime}$, and for $s$ in a subset of $\mathbb{R}$ of length $a$. Hence (6.10) follows. ∎

Consider the connection (6.4) on the trivial line bundle on $M$. Now suppose that $\nabla^{F}$ is $H$-invariant; this means that $\beta$ is $r$-invariant and $\alpha\in\mathbb{C}v_{0}$.

Proposition 6.14**.**

For all $\sigma\in\mathbb{C}$,

[TABLE]

Proof.

Using an approximation of the function $\chi$ by smooth, compactly supported functions, and applying Lemmas 2.19, 6.3, 6.7 and 6.9 and equality (6.5), we find that the right hand side of (2.12) equals

[TABLE]

Hence the claim follows from Lemma 6.13. ∎

Remark 6.15**.**

In this example, Lemma 2.21 does not apply, since (2.11) does not converge absolutely.

6.4. Equivariant analytic torsion

In this subsection, we compute equivariant analytic torsion in the setting of Subsection 6.3, and conclude that the answer to Question 2.31 is yes.

Let $r\in\operatorname{SO}(n)$ be as in Subsection 6.3, i.e. of finite order and with $\ker(r-I)=\mathbb{R}v_{0}$. Consider the closed, cocompact subgroup

[TABLE]

It decomposes as a Cartesian product:

[TABLE]

Furthermore, the action by $G^{\prime}$ on $M$ is the Cartesian product of the action by $\mathbb{R}v_{0}$ on iself and the action by $v_{0}^{\perp}\rtimes\langle r\rangle$ on $v_{0}^{\perp}\times S^{n-1}$. This means that we can use the product formula for equivariant analytic torsion, Proposition 5.8 in [13] to compute the equivariant analytic torsion for the action by $G^{\prime}$ on $M$.

This product formula involves an equivariant Euler characteristic, see Definition 5.1 in [13]. Suppose that a group $B$ acts properly, cocompactly and isometrically on a manifold $N$. Let $b\in B$, and fix a measure $dz$ on its centraliser $Z$. Let $N^{b}$ be the fixed-point set of $b$. Let $\chi_{Z}\in C^{\infty}(N^{b})$ be such that for all $n\in N^{b}$,

[TABLE]

Then the equivariant Euler characteristic of $N$ with respect to $b$ is

[TABLE]

Here $R^{N^{b}}$ is the curvature of the Levi-Civita connection on $N^{g}$, and $\operatorname{Pf}$ denotes the Pfaffian. (Note that the letter $\chi$ is part of the notation for both the cutoff function and the Euler characteristic.)

Let $g=(l_{0}v_{0}+w^{\prime},r^{m})\in G^{\prime}$, for $l_{0}\in\mathbb{R}\setminus\{0\}$, $w^{\prime}\in v_{0}^{\perp}$ and $m\in\mathbb{Z}$.

Lemma 6.16**.**

We have

[TABLE]

Proof.

We determine the fixed point set $(v_{0}^{\perp}\times S^{n-1})^{(w^{\prime},r^{m})}$ and the centraliser $Z<G^{\prime}$. If $x\in v_{0}^{\perp}$ and $v\in S^{n-1}$, then the point $(x,v)$ is fixed by $(w^{\prime},r^{m})$ if and only if

[TABLE]

So

[TABLE]

consists of two points.

To determine $Z$, let $u\in v_{0}^{\perp}$ and $k\in\mathbb{Z}$. Then $(u,r^{\prime}k)\in Z$ if and only if $(I-r^{m})u=(I-r^{k})w^{\prime}$. So

[TABLE]

is cyclic of order $\#\langle r\rangle$. So the constant function $\chi_{Z}$ equal to $\frac{1}{\#\langle r\rangle}$ has the property (6.12). Using this function and (6.14), we find that (6.13) holds. ∎

Let $\nabla^{F}$ be the connection 6.4 on the trivial line bundle on $M$, now with $\alpha\in i\mathbb{R}^{n}$. We assume that $\nabla^{F}$ is $G^{\prime}$-invariant, which means that $\beta$ is $r$-invariant and $\alpha\in i\mathbb{R}v_{0}$.

Proposition 6.17**.**

The equivariant analytic torsion $T_{g}(\nabla^{F})$ for the action by $G^{\prime}$ on $M$ converges, and is given by

[TABLE]

Proof.

For $t>0$, let $\mathcal{T}^{M}_{g}(t)$ be as in (2.30), for the action by $G^{\prime}$ on $M$. Let $\mathcal{T}^{\mathbb{R}v_{0}}_{l_{0}v_{0}}(t)$ and $\mathcal{T}_{(w^{\prime},r^{m})}^{v_{0}^{\perp}\times S^{n-1}}(t)$ be the analogous numbers for the action by $\mathbb{R}v_{0}$ on itself and the action by $v_{0}^{\perp}\rtimes\langle r\rangle$ on $v_{0}^{\perp}\times S^{n-1}$, respectively. Because of the product structure (6.11) of $G^{\prime}$ and its action on $M$, the proof of Proposition 5.8 in [13] implies that for all $t>0$

[TABLE]

Now $\chi_{l_{0}v_{0}}(\mathbb{R}v_{0})=0$, because $(\mathbb{R}v_{0})^{l_{0}v_{0}}=\emptyset$. So by Lemma 6.16, the right hand side of (6.16) equals

[TABLE]

The number on the right is computed with respect to the connection $d+(\alpha,v_{0})dx$ on $\mathbb{R}v_{0}\cong\mathbb{R}$. For that connection, with $(\alpha,v_{0})$ imaginary, it was shown in Proposition 6.2 in [13] that equivariant analytic torsion converges, and is given by

[TABLE]

We conclude that $T_{g}(\nabla^{F})$ converges, and

[TABLE]

which is the right hand side of (6.15). ∎

We can now explicitly verify the equivariant Fried conjecture in this setting.

Proposition 6.18**.**

The kernel of the Laplacian on $M$ associated to the connection $\nabla^{F}$ is trivial. Let $g$ be as above Lemma 6.12. Suppose that $\alpha$ in (6.4) is imaginary. Let $H<G^{\prime}$ be as in (6.8). For the action by $H$ on $M$,

[TABLE]

Proof.

The $L^{2}$-kernel of the Laplacian on $M$ is the tensor product of the $L^{2}$-kernels of the Laplacians on $\mathbb{R}^{n}$ and on $S^{n-1}$. The former of these is zero, hence so is the $L^{2}$-kernel of the Laplacian on $M$.

Let $Z_{G^{\prime}}$ be the centraliser of $g$ in $G^{\prime}$, and $Z_{H}$ the centraliser of $g$ in $H$. Let $T_{g}^{G^{\prime}}(\nabla^{F})$ be the analytic torsion of $\nabla^{F}$ with respect to the action by $G^{\prime}$. Proposition 2.4 in [13] states that the analytic torsion of $\nabla^{F}$ with respect to the action by $H$ equals

[TABLE]

The centraliser $Z_{H}$ is given by (6.9). By a similar argument,

[TABLE]

So $\operatorname{vol}(Z_{G^{\prime}}/Z_{H})=\operatorname{vol}(\mathbb{R}/a\mathbb{Z})=a$. Hence by (6.18) and Proposition 6.17,

[TABLE]

So (6.17) follows from Proposition 6.14. ∎

7. Geodesic flow on spheres

For geodesic flow on the sphere bundle on a sphere, the classical Ruelle $\zeta$-function is not defined because there are uncountably many closed geodesics. This flow is not Anosov (which would imply that the Ruelle $\zeta$-function is well-defined), in contrast to the geodesic flow in negative curvature. In this section, we onsider the action by $\operatorname{SO}(n+1)$ on the sphere bundle $S(TS^{n})$. For $n=2$ and $n=3$, we show that this flow is $g$-nondegenerate for suitable group elements, and compute the equivariant Ruelle $\zeta$-function for the trivial connection on the trivial line bundle. We expect these computations to generalise to arbitrary $n$, with some additional combinatorics involved. In these cases, the kernel of $\Delta_{F}$ is nonzero, so the hypotheses of the equivariant Fried problem, Question 2.31, are not satisfied.

7.1. Homogeneous spaces of compact Lie groups

We assume in this subsection that $G$ is a Lie group, and $H<G$ is a closed subgroup. Consider the action by $G\times H$ on $G$ given by

[TABLE]

for $x,y\in G$ and $h\in H$. Let $X_{0}\in\mathfrak{g}$ be such that $\operatorname{ad}(X_{0})|_{\mathfrak{h}}=0$. For $t\in\mathbb{R}$, we define $\tilde{\varphi}_{t}\colon G\to G$ by

[TABLE]

for $x\in G$. Because $\tilde{\varphi}_{t}$ is $H$-equivariant, it descends to a map $\varphi_{t}\colon G/H\to G/H$.

Example 7.1**.**

Geodesic flow on the sphere bundle of a sphere occurs in this way, see Lemma 7.3.

Fix $g\in G$. If $y\in G$, and $V\subset\mathfrak{g}$ is a linear subspace, then we denote the map from $\mathfrak{g}/V$ to $\mathfrak{g}/\operatorname{Ad}(y)V$ induced by $\operatorname{Ad}(y)$ by $\operatorname{Ad}(y)$ as well.

Lemma 7.2**.**

Let $l\in\mathbb{R}$ and $x\in G$ be such that $\varphi_{l}(xH)=gxH$. Then the following diagram commutes:

[TABLE]

Proof.

Let $l$ and $x$ be as in the statement of the lemma. Since the map $\varphi_{l}$ is $G$-equivariant,

[TABLE]

The equality $\varphi_{l}(xH)=gxH$ implies that $\exp(lX_{0})H=x^{-1}gxH$. So for all $y\in G$,

[TABLE]

It follows that the right hand side of (7.1) is $\operatorname{Ad}(x^{-1}g^{-1}x)$, and the lemma follows. ∎

7.2. Geodesic flow on $S(TS^{n})$

Let $M=S(TS^{n})$, acted on by $G=\operatorname{SO}(n+1)$ in the natural way. Let $\varphi$ be the geodesic flow on $M$. By Lemma 2.1 it is equivariant.

Consider the subspace

[TABLE]

Let $S(\mathfrak{m})\subset\mathfrak{m}$ be the unit sphere with respect to the Euclidean metric on $\mathfrak{m}\cong\mathbb{R}^{n}$. For a given $p\in S^{n}$, we have the natural $\operatorname{SO}(n+1)$-equivariant diffeomorphism

[TABLE]

mapping $[x,X]$ to

[TABLE]

for $x\in\operatorname{SO}(n+1)$ and $X\in S(\mathfrak{m})$.

The adjoint action by $\operatorname{SO}(n)$ on $\mathfrak{m}\cong\mathbb{R}^{n}$ is the standard action by $\operatorname{SO}(n)$. So for a given $X_{0}\in S(\mathfrak{m})$, the action by $\operatorname{SO}(n)$ on $X_{0}$ defines an $\operatorname{SO}(n)$-equivariant diffeomorphism $\operatorname{SO}(n)/\operatorname{SO}(n-1)\to S(\mathfrak{m})$. Combining this with (7.2) yields an $\operatorname{SO}(n+1)$-equivariant diffeomorphism

[TABLE]

Lemma 7.3**.**

Under the diffeomorphism (7.3), the geodesic flow on $M$ is given by

[TABLE]

for $t\in\mathbb{R}$ and $x\in\operatorname{SO}(n+1)$.

Proof.

We have $[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{so}(n)$, so $S^{n}=\operatorname{SO}(n+1)/\operatorname{SO}(n)$ is a Riemannian symmetric space. The Riemannian exponential map is therefore given by

[TABLE]

for $X\in\mathfrak{m}\cong T_{e\operatorname{SO}(n)}(\operatorname{SO}(n+1)/\operatorname{SO}(n))$. See e.g. page 832 of [12]. Here the map $\exp_{e\operatorname{SO}(n)}$ on the left hand side is the Riemannian exponential map, and the map $\exp$ on the right hand side is the Lie-theoretic one. Then (7.5) and equivariance of the Riemannian exponential map imply that

[TABLE]

for $t\in\mathbb{R}$, $p\in S^{n}$ and $X\in S(\mathfrak{m})$. An application of the diffeomorphism (7.2) then implies that

[TABLE]

for $x\in\operatorname{SO}(n+1)$. Finally, (7.4) follows by an application of (7.3). ∎

From now on, we make the specific choice

[TABLE]

We write $n=2k$ if $n$ is even, and $n=2k-1$ if $n$ is odd. Then $T:=\operatorname{SO}(2)^{k}$ embeds into $\operatorname{SO}(n+1)$ as a maximal torus. If $n$ is odd, then we embed $T$ into $\operatorname{SO}(n+1)$ as diagonal $2\times 2$ blocks. If $n$ is even, we embed $T$ into $\operatorname{SO}(n+1)$ as diagonal $2\times 2$ blocks with a $1$ in the bottom-right entry. For $\theta\in\mathbb{R}$, we write

[TABLE]

Let $\theta_{1},\ldots,\theta_{k}\in\mathbb{R}$, and suppose that $g=(r(\theta_{1}),\ldots,r(\theta_{k}))\in T$.

Lemma 7.4**.**

We have

[TABLE]

Proof.

Suppose that $l\in L_{g}(\varphi)$. By Lemma 7.3, the condition $\varphi_{l}(x\operatorname{SO}(n-1))=gx\operatorname{SO}(n-1)$ is equivalent to $x^{-1}g^{-1}x\exp(lX_{0})\in\{I_{2}\}\times\operatorname{SO}(n-1)$. So there is an $h\in\operatorname{SO}(n-1)$ such that

[TABLE]

In particular, the matrix on the right has the same eigenvalues as $g$. This implies that the eigenvalues of $\exp(lX_{0})$, which are $e^{\pm il}$, equal $e^{\pm i\theta_{j}}$ for some $j$. Hence $l\in\pm{\theta_{j}}+2\pi\mathbb{Z}$. ∎

Remark 7.5**.**

Lemma 7.4 implies that $L_{g}(\varphi)$ is countable. The set of $l\in\mathbb{R}\setminus\{0\}$ for which there exists a geodesic $\gamma$ on $S^{n}$ such that $\gamma(l)=g\gamma(l)$ is uncountable, because the set

[TABLE]

is uncountable. This shows that $L_{g}(\varphi)$ is different from the latter set. The difference is caused by the action by $g$ on unit tangent vectors.

7.3. The case $n=2$

In the rest of this section, we assume that the powers of $g$ are dense in $T$.

Suppose for this subsection that $n=2$. Then $M=G=\operatorname{SO}(3)$, and $H$ is trivial. Since $g$ is regular, it follows from Lemma 7.2 that for all $l\in\mathbb{R}\setminus\{0\}$ and $x\in G$ be such that $\varphi_{l}(xH)=gxH$,

[TABLE]

So the flow $\varphi$ is $g$-nondegenerate.

Lemma 7.6**.**

Suppose that $g$ is a rotation by an angle $\theta$. Then

[TABLE]

Proof.

By Lemma 7.4, it is enough to show that the right hand side of (7.8) is contained in the left hand side. If $l\in\mathbb{R}$ and $x\in\operatorname{SO}(3)$, then $\varphi_{l}(x)=gx$ if and only if $x^{-1}gx=\exp(lX_{0})$. If $l\in\theta+2\pi\mathbb{Z}$, then $\exp(lX_{0})=g$, so taking $x=e$ shows that $l\in L_{g}(\varphi)$. And if $l\in-\theta+2\pi\mathbb{Z}$, then $\exp(lX_{0})=g^{-1}$, and taking

[TABLE]

shows that $l\in L_{g}(\varphi)$. ∎

Lemma 7.7**.**

Let $g\in\operatorname{SO}(3)$ be a rotation over an angle $\theta$. For all $l\in L_{g}(\varphi)$, the set $\Gamma_{l}^{g}(\varphi)$ consists of a single flow curve.

Proof.

Let $x,x^{\prime}\in\operatorname{SO}(3)\cong M$, and suppose that $\varphi_{l}(x)=gx$ and $\varphi_{l}(x^{\prime})=gx^{\prime}$. Then $x^{-1}gx=\exp(lX_{0})=x^{\prime-1}gx^{\prime}$. Since the powers of $g$ are dense in $\operatorname{SO}(2)$, it follows that for all $y\in\operatorname{SO}(2)$,

[TABLE]

This implies that $x^{-1}x^{\prime}$ lies in the centraliser of $\operatorname{SO}(2)$, which is $\operatorname{SO}(2)$ itself. We therefore find that

[TABLE]

for some $t\in\mathbb{R}$, so $x^{\prime}$ and $x$ lie on the same flow curve of $\varphi$. ∎

Remark 7.8**.**

In Lemma 7.7 (and also in Lemma 7.13 below), it is crucial that the powers of $g$ are dense in $T$. In the opposite extreme case where $g$ is the identity element, the set $L_{e}(\varphi)=2\pi\mathbb{Z}\setminus\{0\}$ is still countable, but $\Gamma_{l}^{e}(\varphi)$ is uncountable for all $l\in 2\pi\mathbb{Z}\setminus\{0\}$ if $n\geq 2$. This is because there is a closed geodesic though every point in $S^{n}$. This is why the classical Ruelle $\zeta$-function is not defined for the geodesic flow on spheres, whereas the equivariant version is defined for $g$ with dense powers in $\operatorname{SO}(2)$, as we will see for $n=2$ and $n=3$.

Lemma 7.9**.**

The only $G$-invariant, flat, Hermitian connection on the trivial line bundle $F=M\times\mathbb{C}\to M$ is the trivial connection $d$.

Proof.

Let $\nabla^{F}$ be a $G$-invariant, Hermitian connection on $F$. Then $\nabla^{F}=d+i\omega$ for an element $\omega\in\mathfrak{so}(3)^{*}$, viewed as a left-invariant one-form on $M\cong\operatorname{SO}(3)$. For $\nabla^{F}$ to be flat, one needs $d\omega=0$. This is the case if and only if $\omega$ is zero on $[\mathfrak{so}(3),\mathfrak{so}(3)]=\mathfrak{so}(3)$. ∎

Proposition 7.10**.**

Let $F=M\times\mathbb{C}\to M$ be the trivial line bundle, and $\nabla^{F}=d$ the trivial connection. The flow $\varphi$ is $g$-nondegenerate, the set $L_{g}(\varphi)$ is countable, and for all $\sigma\in\mathbb{C}$ with positive real parts,

[TABLE]

Proof.

In this case, $\operatorname{sgn}(\det(1-P_{\gamma}^{g}))=1$ for all $l\in L_{g}(\varphi)$ and $\gamma\in\Gamma_{g}^{l}(\varphi)$, because $S^{2}$ is oriented. The constant function $1$ is a cutoff function because $G$ is compact, while one can take $I_{\gamma}=[0,2\pi]$ for any $g$-periodic flow curve $\gamma$. So for all such curves $\gamma$ and $x\in\operatorname{SO}(3)$,

[TABLE]

Normalising the measure on $G/Z$ to give it unit volume, and applying Lemmas 2.19, 7.6 and 7.7, we conclude that (7.9) holds.

∎

In this example, the kernel of $\Delta_{F}$ is the cohomology of $\operatorname{SO}(3)$, which is not zero. Furthermore, the equivariant Ruelle $\zeta$-function does not extend to $\sigma=0$; Lemma 5.4 does not apply, since $\alpha=0$ in the current case. So we are not in the situation of the equivariant Fried conjecture.

7.4. The case $n=3$

In this subsection, we suppose that $n=3$, and remind the reader that we are assuming the powers of $g$ are dense in $T$. Then $H=\{I_{2}\}\times\operatorname{SO}(2)$ is the copy of $\operatorname{SO}(2)$ embedded into $\operatorname{SO}(4)$ in the bottom right corner.

Consider the matrices

[TABLE]

Then for all $\varepsilon\in\{\pm 1\}$ and $\alpha\in\mathbb{R}$,

[TABLE]

Lemma 7.11**.**

We have

[TABLE]

Let $\varepsilon\in\{\pm 1\}$. If $l\in\varepsilon\theta_{1}+2\pi\mathbb{Z}$ and $x\in\operatorname{SO}(4)$, then the condition $\varphi_{l}(xH)=gxH$ is equivalent to

[TABLE]

for $a,d\in\operatorname{SO}(2)$. If $l\in\varepsilon\theta_{2}+2\pi\mathbb{Z}$ and $x\in\operatorname{SO}(4)$, then the condition $\varphi_{l}(xH)=gxH$ is equivalent to

[TABLE]

for $b,c\in\operatorname{SO}(2)$.

Proof.

By Lemma 7.4, the equality (7.11) follows if the right hand side is contained in the left hand side. This follows from the characterisations (7.12) and (7.13) of the elements $x\in\operatorname{SO}(4)$ such that $\varphi_{l}(xH)=gxH$, for $l$ in the right hand side of (7.11). We will prove these characterisations.

Let $l\in\mathbb{R}\setminus\{0\}$ and $x\in\operatorname{SO}(4)$ be such that $\varphi_{l}(xH)=xgH$. As in the proof of Lemma 7.4, this is equivalent to

[TABLE]

where $l=\pm\theta_{1}$ and $\alpha=\pm\theta_{2}$, or $l=\pm\theta_{2}$ and $\alpha=\pm\theta_{1}$, where all equalities (here and also below) are modulo $2\pi\mathbb{Z}$. It is immediate that if $x$ is of the form (7.12) and (7.13) in the respective cases, then (7.14) holds. So it remains to prove the converse implication.

Since, by assumption, the powers of $g$ are dense in $T$, the equality (7.14) implies that for all $\alpha_{1},\alpha_{2}\in\mathbb{R}$,

[TABLE]

for $l=\pm\alpha_{1}$ and $\alpha=\pm\alpha_{2}$, or $l=\pm\alpha_{2}$ and $\alpha=\pm\alpha_{1}$.

Write $x=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, for $a,b,c,d\in M_{2}(\mathbb{R})$. Then the left hand side of (7.15) equals

[TABLE]

We first consider the case where $l=\pm\alpha_{1}$ and $\alpha=\pm\alpha_{2}$. Let $\varepsilon_{1},\varepsilon_{2}\in\{\pm 1\}$ be such that $l=\varepsilon_{1}\alpha_{1}$ and $\alpha=\varepsilon_{2}\alpha_{2}$. Then by (7.10), we have $r(l)=w_{\varepsilon_{1}}r(\alpha_{1})w_{\varepsilon_{1}}$ and $r(\alpha)=w_{\varepsilon_{2}}r(\alpha_{2})w_{\varepsilon_{2}}$. Since the diagonal entries of (7.16) equal the diagonal entries of the right hand side of (7.15), we have for all $\alpha_{1},\alpha_{2}\in\mathbb{R}$,

[TABLE]

So $b=c=0$, and $x=\begin{pmatrix}a&0\\ 0&d\end{pmatrix}$, for $a,d\in\operatorname{O}(2)$. The products $\tilde{a}:=aw_{\varepsilon_{1}}$ and $\tilde{d}:=dw_{\varepsilon_{2}}$ both centralise $\operatorname{SO}(2)$ by (7.17), so they lie in $\operatorname{SO}(2)$. We find that $x=\begin{pmatrix}\tilde{a}w_{\varepsilon_{1}}&0\\ 0&\tilde{d}w_{\varepsilon_{2}}\end{pmatrix}$, for $\tilde{a},\tilde{d}\in\operatorname{SO}(2)$. Since $x$ has determinant $1$, we have $\varepsilon_{1}=\varepsilon_{2}$.

In the case where $l=\varepsilon_{1}\alpha_{2}$ and $\alpha=\varepsilon_{2}\alpha_{1}$ for $\varepsilon_{1},\varepsilon_{2}\in\{\pm 1\}$, we can argue similarly. Then the equality between (7.16) and the right hand side of (7.15) is equivalent to $a=d=0$, and $x=\begin{pmatrix}0&\tilde{b}w_{\varepsilon_{2}}\\ \tilde{c}w_{\varepsilon_{1}}&0\end{pmatrix}$ with $\tilde{b}:=bw_{\varepsilon_{2}}$ and $\tilde{c}:=cw_{\varepsilon_{1}}$ lie in $\operatorname{SO}(2)$. The fact that $x$ has determinant $1$ now implies that $\varepsilon_{1}=-\varepsilon_{2}$. ∎

Lemma 7.12**.**

The flow $\varphi_{l}$ is $g$-nondegenerate.

Proof.

Let $l\in L_{g}(\varphi)$ and $x\in\operatorname{SO}(4)$ be such that $\varphi_{l}(xH)=gxH$. If $x$ is of the form (7.12), then $\operatorname{Ad}(x)\mathfrak{h}=\mathfrak{h}$, so it follows from Lemma 7.2 and regularity of $g$ that the kernel of $T_{xH}(\varphi_{l}\circ g^{-1})$ is one-dimensional.

If $x$ is of the form (7.13), then $\operatorname{Ad}(x)\mathfrak{h}=\mathfrak{so}(2)\oplus\{0\}$, and a similar argument applies. ∎

Lemma 7.13**.**

For all $l\in L_{g}(\varphi)$, the set $\Gamma_{l}^{g}(\varphi)$ consists of a single flow curve.

Proof.

Let $l\in L_{g}(\varphi)$. Let $x,x^{\prime}\in\operatorname{SO}(4)$ be such that $\varphi_{l}(xH)=gxH$ and $\varphi_{l}(x^{\prime}H)=gx^{\prime}H$.

First , suppose that $l\in\varepsilon\theta_{1}+2\pi\mathbb{Z}$, for $\varepsilon\in\{\pm 1\}$. Then by Lemma 7.11, there are $a,d,a^{\prime},d^{\prime}\in\operatorname{SO}(2)$ such that

[TABLE]

so that

[TABLE]

As $w_{\varepsilon}a^{T}aw_{\varepsilon}\in\operatorname{SO}(2)$, it follows that $x^{\prime}H=x\exp(tX_{0})H$ for some $t\in\mathbb{R}$.

Next, suppose that $l\in\varepsilon\theta_{2}+2\pi\mathbb{Z}$, for $\varepsilon\in\{\pm 1\}$. Then by Lemma 7.11, there are $b,c,b^{\prime},c^{\prime}\in\operatorname{SO}(2)$ such that

[TABLE]

which implies that

[TABLE]

As in the previous case, it follows that $x^{\prime}H=x\exp(tX_{0})H$ for some $t\in\mathbb{R}$. ∎

Proposition 7.14**.**

The flow $\varphi$ is $g$-nondegenerate, the set $L_{g}(\varphi)$ is countable, and for all $\sigma\in\mathbb{C}$ with positive real parts,

[TABLE]

Proof.

As in the proof of Proposition 7.10, $\operatorname{sgn}(\det(1-P_{\gamma}^{g}))=1$. We can choose $\chi$ to be the constant $1$, so that

[TABLE]

for all $l\in L_{g}(\varphi)$, $\gamma\in\Gamma_{g}^{l}(\varphi)$ and $x\in\operatorname{SO}(4)$. Using Lemmas 7.11, 7.13, we obtain (7.18). ∎

As in the case $n=2$, the equivariant Ruelle $\zeta$-function does not have a meromorphic continuation to $\sigma=0$ in this case, and the kernel of $\Delta_{F}$ is nonzero. This implies that the hypotheses of the equivariant Fried conjecture are not satisfied.