Topology of Pollicott-Ruelle resonant states

TL;DR

This paper establishes a deep connection between the topology of certain flows on manifolds and the spectral properties of associated dynamical resonances, leading to new insights into Morse inequalities and Reidemeister torsion.

Contribution

It proves an isomorphism between twisted De Rham cohomology and invariant Pollicott-Ruelle resonant states, extending Morse inequalities and relating resonances to Reidemeister torsion.

Findings

Isomorphism between twisted De Rham cohomology and Pollicott-Ruelle resonant states

Generalized Morse inequalities for flows with these resonances

Reconstruction of Reidemeister torsion from dynamical resonances on the imaginary axis

Abstract

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott--Ruelleresonant states associated with Anosov and Morse--Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse--Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

Equations

$$\forall 0\le k\le n,\sum _{j=0}^k(-1{)}^{k-j}{b}_j(M)\le \sum _{j=0}^k(-1{)}^{k-j}{c}_j(f),$$

$$0\stackrel{d^{\nabla }}{\to }{\Omega }^0(M,\mathcal{E})\stackrel{d^{\nabla }}{\to }{\Omega }^1(M,\mathcal{E})\stackrel{d^{\nabla }}{\to }\dots \stackrel{d^{\nabla }}{\to }{\Omega }^n(M,\mathcal{E})\stackrel{d^{\nabla }}{\to }0.$$

$${\mathcal{L}}_{V,\nabla }^{(k)}={\left(d^{\nabla }+{\iota }_V\right)}^2:{\Omega }^k(M,\mathcal{E})\to {\Omega }^k(M,\mathcal{E}),$$

$${\partial }_t\psi =-{\mathcal{L}}_{V,\nabla }^{(k)}\psi ,\psi (t=0)={\psi }_0\in {\Omega }^k(M,\mathcal{E}).$$

$$\forall ({\psi }_1,{\psi }_2)\in {\Omega }^{n-k}(M,{\mathcal{E}}^{\prime })\times {\Omega }^k(M,\mathcal{E}),C_{{\psi }_1,{\psi }_2}(t):={\int }_M{\psi }_1\wedge {\Phi }_k^{-t\ast }({\psi }_2),$$

$$\forall ({\psi }_1,{\psi }_2)\in {\Omega }^{n-k}(M,{\mathcal{E}}^{\prime })\times {\Omega }^k(M,\mathcal{E}),{\hat{C}}_{{\psi }_1,{\psi }_2}(z):={\int }_0^{+\infty }{e}^{-zt}\left({\int }_M{\psi }_1\wedge {\Phi }_k^{-t\ast }({\psi }_2)\right)dt,$$

$${\pi }_{z_0}^{(k)}:{\Omega }^k(M,\mathcal{E})\to {\mathcal{D}}^{\prime k}(M,\mathcal{E})$$

$${\hat{C}}_{{\psi }_1,{\psi }_2}(z)=\sum _{l=1}^{m_k(z_0)}(-1{)}^{l-1}\frac{⟨({\mathcal{L}}_{V,\nabla }^{(k)}+z_0{)}^{l-1}{\pi }_{z_0}^{(k)}({\psi }_2),{\psi }_1⟩}{(z-z_0{)}^l}+R_{{\psi }_1,{\psi }_2}(z),$$

$$0\stackrel{d^{\nabla }}{\to }{C}_{V,\nabla }^0(z_0)\stackrel{d^{\nabla }}{\to }{C}_{V,\nabla }^1(z_0)\stackrel{d^{\nabla }}{\to }\dots \stackrel{d^{\nabla }}{\to }{C}_{V,\nabla }^n(z_0)\stackrel{d^{\nabla }}{\to }0$$

$${\pi }_0^{(k)}:{\Omega }^k(M,\mathcal{E})\to C_{V,\nabla }^k(0)$$

$$\forall 0\le k\le n,\sum _{j=0}^k(-1{)}^{k-j}\dim C_{V,\nabla }^j(0)\ge \sum _{j=0}^k(-1{)}^{k-j}{b}_j(M,\mathcal{E}),$$

$$\boxed{\sum _{\Lambda :\dim W^s(\Lambda )=k+1}{m}_{\Lambda }+N\sum _{j=0}^k(-1{)}^{k-j}{c}_k(V)\ge \sum _{j=0}^k(-1{)}^{k-j}{b}_j(M,\mathcal{E}),}$$

$$0\stackrel{{\iota }_V}{\to }{C}_{V,\nabla }^n(0)\stackrel{{\iota }_V}{\to }{C}_{V,\nabla }^{n-1}(0)\stackrel{{\iota }_V}{\to }\dots \stackrel{{\iota }_V}{\to }{C}_{V,\nabla }^0(0)\stackrel{{\iota }_V}{\to }0,$$

$${\mu }_{V,\nabla }(t)$$

$${\mu }_{V,\nabla }(t)=\sum _{k=0}^n(-1{)}^{n-k+1}\sum _{z_0\in {\mathcal{R}}_k(V,\nabla )\cap i\mathbb{R}}\dim \left(C_{V,\nabla }^k(z_0)\cap \mathrm{Ker}({\iota }_V)\right)\frac{e^{t{z}_0}}{t}$$

$$t{\mu }_{V,\nabla }(t)$$

$${\zeta }_{V,\nabla }^{♭}(s,z):=\frac{1}{\Gamma (s)}{\int }_0^{+\infty }{e}^{-tz}{\mu }_{V,\nabla }(t)t^{s}dt$$

$$Z_{V,\nabla }(z)=\exp \left(-\frac{d}{ds}{\zeta }_{V,\nabla }^{♭}(s,z){|}_{s=0}\right)$$

$${\partial }_s{\zeta }_{V,\nabla }^{♭}(0,z)=\chi (M,\mathcal{E})\ln z+\sum _{\Lambda \text{closed orbit}}(-1{)}^{\text{dim}{W}^u(\Lambda )}\ln \text{det}\left(\text{Id}-e^{-{\mathcal{P}}_{\Lambda }z}{\Delta }_{\Lambda }{M}_{\mathcal{E}}(\Lambda )\right).$$

$$\boxed{Z_{V,\nabla }(z)=z^{-\chi (M,\mathcal{E})}\prod _{\Lambda \text{closed orbit}}\text{det}{\left(\text{Id}-e^{-{\mathcal{P}}_{\Lambda }z}{\Delta }_{\Lambda }{M}_{\mathcal{E}}(\Lambda )\right)}^{-(-1{)}^{\text{dim}{W}^u(\Lambda )}}}$$

$${\zeta }_{RS}(s,z):=\sum _{k=0}^n(-1{)}^{n-k}k\sum _{z_0\in {\mathcal{R}}_k(V,\nabla )\cap i{\mathbb{R}}^{\ast }}\text{dim}\left(C_{V,\nabla }^k(z_0)\right){\left(z-z_0\right)}^{-s}$$

$$\boxed{Z_{RS}(z):=\exp \left(-\frac{d}{ds}{\zeta }_{RS}(s,z){|}_{s=0}\right)}.$$

$$\boxed{{\zeta }_{V,\nabla }^{♭}(s,z)=-\left(\chi (M,\mathcal{E})+\sum _{\Lambda \text{closed orbit}}(-1{)}^{\text{dim}{W}^u(\Lambda )}{m}_{\Lambda }\right)z^{-s}+{\zeta }_{RS}(s,z).}$$

$$\boxed{Z_{RS}(z)=\prod _{\Lambda \text{closed orbit}}{\left(z^{-m_{\Lambda }}\det \left(\mathrm{Id}-e^{-{\mathcal{P}}_{\Lambda }z}{\Delta }_{\Lambda }{M}_{\mathcal{E}}(\Lambda )\right)\right)}^{-(-1{)}^{\dim W^u(\Lambda )}}.}$$

$$C_{V,\nabla }^{\bullet }(i{\mathbb{R}}^{\ast })=\underset{z_0\in {\mathcal{R}}_{\bullet }(V,\nabla )\cap i{\mathbb{R}}^{\ast }}{⨁}{C}_{V,\nabla }^{\bullet }(z_0),$$

$$\forall (x,\xi )\in T^{\ast }M,H_V(x,\xi ):=\xi (V(x)).$$

$$\forall t\in \mathbb{R},{\Phi }_V^t(x,\xi ):=\left({\phi }^t(x,\xi ),{\left(d{\phi }^t(x{)}^T\right)}^{-1}\xi \right),$$

$$G(x,\xi ):=m(x,\xi )\log \sqrt{1+f(x,\xi {)}^2},$$

$${\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }m(x,\xi )=\mathcal{O}\left((1+\Vert \xi {\Vert }_x^2{)}^{-\frac{|\beta |}{2}}\right).$$

$$|H_V(x,\xi )|\ge \frac{1}{C}(1+\Vert \xi {\Vert }_x),$$

Full text

1. Introduction

Consider $M$ a smooth ($\mathcal{C}^{\infty}$), compact and oriented manifold which has no boundary and which is of dimension $n\geq 1$. We say that $f\in\mathcal{C}^{\infty}(M,\mathbb{R})$ is a Morse function if it has finitely many critical points all of them being nondegenerate. Denote by $c_{k}(f)$ the number of critical points of index $k$ and by $b_{k}(M)$ the $k$-th Betti number of the manifold. A famous result of Morse states that the following inequalities hold [32]:

[TABLE]

with equality when $k=n$. These inequalities are usually called Morse inequalities. Later on, Thom [46] and Smale [44] observed that, if we are given a Riemannian metric on $M$, then we can define a cell decomposition of the manifold by considering the stable manifolds associated with each critical point by the induced gradient flow. Then, $c_{j}(f)$ is thought as the number of stable manifolds of dimension $j$ and Smale gave a new proof of these inequalities [44] which had in some sense a more dynamical flavour. This dynamical approach to Morse inequalities was then pursued by many people and generalized to much more general dynamical systems. Among others, we refer the reader to the books of Conley [9] and Franks [21] for examples of such generalizations. This dynamical approach to differential topology also leads to dynamical interpretation of other topological invariants such as the Reidemeister torsion by Fried [22]. Inspired by the theory of currents, but still from a dynamical perspective, we can also mention the works of Laudenbach [29] and Harvey–Lawson [26] who showed how to realize the cell decomposition of Thom and Smale in terms of De Rham currents.

More than fifty years after Morse’s seminal work, Witten introduced in the context of Hodge theory another approach to Morse inequalities [50]. This point of view was further developped by Helffer and Sjöstrand using tools from semiclassical analysis [27] and brought a new spectral perspective on these results. In particular, the coefficients $c_{j}(f)$ are interpreted by Witten as the number of small eigenvalues (counted with multiplicity) of a certain deformation of the Hodge Laplacian acting on forms of degree $j$. This also lead to many developments in analysis and topology that would be again hard to describe in details. We can for instance quote the works of Bismut-Zhang [6] and Burghelea-Friedlander-Kappeler [7] who developped the relationship between Witten’s approach and Reidemeister torsion.

If we come back to the dynamical perspective, a spectral approach slightly different from the ones arising from Hodge-Witten theory was recently developped by several authors to study dynamical systems with hyperbolic behaviour. We shall describe these results more precisely below. In particular, this theory applies to the dynamical systems studied by Smale [45]: Anosov flows [8, 18, 14, 20], Axiom A flows [13] and Morse-Smale flows [10, 11]. These different works were initially motivated by the study of the correlation function in dynamical systems and by the analytic properties of the so-called Ruelle zeta function [42]. In the present work, we aim at showing how these results can be also used to obtain classical results from differential topology in a manner which is somewhat intermediate between the spectral approach of Witten and the more dynamical one of Thom and Smale. As a by-product, it will also unveil the topological informations contained in the correlation spectrum (also called Pollicott-Ruelle spectrum), resonances on the imaginary axis in particular, studied in the above works.

2. Statement of the main results

2.1. Dynamical framework

Let $M$ be a smooth, compact, oriented, boundaryless manifold of dimension $n\geq 1$ and let $\mathbf{p}:\mathcal{E}\rightarrow M$ be a smooth complex vector bundle of rank $N$. Suppose that $\mathcal{E}$ is endowed with a flat connection $\nabla:\Omega^{0}(M,\mathcal{E})\rightarrow\Omega^{1}(M,\mathcal{E})$ [30, Ch. 12]. This allows us to define an exterior derivative $d^{\nabla}:\Omega^{\bullet}(M,\mathcal{E})\rightarrow\Omega^{\bullet+1}(M,\mathcal{E})$ which satisfies $d^{\nabla}\circ d^{\nabla}=0$. In particular, one can introduce the twisted De Rham complex $(\Omega^{*}(M,\mathcal{E}),d^{\nabla})$ associated with $d^{\nabla}$:

[TABLE]

The $k$th-cohomology group of this complex is denoted by $\mathbf{H}^{k}(M,\mathcal{E})$. This kind of complex appears for instance naturally in Hodge theory [5, 31].

Fix now $V$ a smooth vector field on $M$ and denote by $\varphi^{t}$ the induced flow on $M$. Using the flat connection $\nabla$, one can lift this flow into a flow $\Phi_{k}^{t}$ on the vector bundle $\mathbf{p}_{k}:\Lambda^{k}(T^{*}M)\otimes\mathcal{E}\mapsto M$ for every $0\leq k\leq n$. For every $t\in\mathbb{R}$, one has $\mathbf{p}_{k}\circ\Phi_{k}^{t}=\varphi^{t}\circ\mathbf{p}_{k}$. If we define the corresponding Lie derivative on $\mathcal{E}$

[TABLE]

where $\iota_{V}$ is the contraction by the vector field $V$, then, for any section $\psi_{0}$ in $\Omega^{k}(M,\mathcal{E})$ and for any $t\geq 0$, $\Phi_{k}^{-t*}(\psi_{0})$ is the solution of the following partial differential equation:

[TABLE]

As a tool to describe the long time behaviour of this equation, it is natural to form the following function

[TABLE]

which we will call the correlation function.

2.2. Pollicott-Ruelle resonances

Set $\mathcal{E}^{\prime}$ to be the dual bundle to $\mathcal{E}$. Following the works of Pollicott [39] and Ruelle [43], it is sometimes simpler to consider first the Laplace transform of this quantity,

[TABLE]

which is well defined for $\text{Re}(z)>0$ large enough. In the above references, Pollicott and Ruelle showed that, for Axiom $A$ vector fields [45] and for $\psi_{1}$ and $\psi_{2}$ compactly supported near the basic sets of the flow, this function admits a meromorphic extension to some half plane $\text{Re}(z)>-\delta$ with $\delta>0$ small enough. The poles and residues of this meromorphic continuation describe in some sense the fine structure of the flow $\Phi_{k}^{-t*}=e^{-t\mathcal{L}_{V,\nabla}^{(k)}}$ as $t\rightarrow+\infty$. More recently, it was proved by Butterley and Liverani that, for Anosov vector fields, the Laplace transformed correlators have meromorphic extensions to the entire complex plane without any restrictions on the supports of $\psi_{1}$ and $\psi_{2}$ [8]. A different proof based on microlocal techniques was given by Faure and Sjöstrand [18] – see also [14, 13] for related results111We refer to section 3 for a more detailed account.. In [10, 11], we also proved this meromorphic extension in the case of Morse-Smale vector fields which are $\mathcal{C}^{1}$-linearizable – see appendix A for the precise definition.

Fix now $0\leq k\leq n$. To summarize, these recent developments showed that, for vector fields $V$ which are either Anosov or ($\mathcal{C}^{1}$-linearizable) Morse-Smale, there exists a minimal discrete222We mean that it has no accumulation point. In particular, it is at most countable. set $\mathcal{R}_{k}(V,\nabla)\subset\mathbb{C}$ such that, given any $(\psi_{1},\psi_{2})\in\Omega^{n-k}(M,\mathcal{E}^{\prime})\times\Omega^{k}(M,\mathcal{E})$, the map $z\mapsto\hat{C}_{\psi_{1},\psi_{2}}(z)$ has a meromorphic extension whose poles are contained inside $\mathcal{R}_{k}(V,\nabla)$. These poles are called the Pollicott-Ruelle resonances. Moreover, given any such $z_{0}\in\mathcal{R}_{k}(V,\nabla)$, there exists an integer $m_{k}(z_{0})$ and a linear map of finite rank

[TABLE]

such that, given any $(\psi_{1},\psi_{2})\in\Omega^{n-k}(M,\mathcal{E}^{\prime})\times\Omega^{k}(M,\mathcal{E})$, one has, in a small neighborhood of $z_{0}$,

[TABLE]

where $R_{\psi_{1},\psi_{2}}(z)$ is holomorphic. Here, we use the convention that $\mathcal{D}^{\prime k}(M,\mathcal{E})$ represents the currents of degree $k$ with values in $\mathcal{E}$. Elements inside the range of $\pi_{z_{0}}^{(k)}$ are called Pollicott-Ruelle resonant states and, as we shall explain it below, they can be interpreted as the generalized eigenvectors of the operator $-\mathcal{L}_{V,\nabla}^{(k)}$ acting on some appropriate Sobolev space. For any resonance $z_{0}$, we denote by $C_{V,\nabla}^{k}(z_{0})$ the range of the operator $\pi_{z_{0}}^{(k)}$ which is a finite dimensional space.

2.3. Realization of De Rham cohomology via Pollicott-Ruelle resonant states

From the spectral interpretation of $C^{\bullet}_{V,\nabla}(z_{0})$ – see section 3 for details, one can deduce that, for any333Whenever $z_{0}$ is not a resonance, we use the convention $C^{k}_{V,\nabla}(z_{0})=\{0\}$. $z_{0}\in\mathbb{C}$,

[TABLE]

defines a cohomological complex. We will denote by $\mathbf{H}^{k}(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ the cohomology of that complex. Our first main result states that this complex is isomorphic to the twisted De Rham complex when $z_{0}=0$:

Theorem 2.1**.**

Let $\mathcal{E}\rightarrow M$ be a smooth complex vector bundle of dimension $N$ which is endowed with a flat connection $\nabla$. Suppose that $V$ is a vector field which is either Morse-Smale and $\mathcal{C}^{1}$-linearizable or Anosov. Then, for every $0\leq k\leq n$, the maps

$\pi_{0}^{(k)}:\Omega^{k}(M,\mathcal{E})\rightarrow C_{V,\nabla}^{k}(0)$

induce isomorphisms between $\mathbf{H}^{k}(M,\mathcal{E})$ and $\mathbf{H}^{k}(C_{V,\nabla}^{\bullet}(0),d^{\nabla})$.

Observe that a direct corollary of this result is that the twisted De Rham cohomology is finite dimensional. Its dimension is usually denoted by $b_{k}(M,\mathcal{E})$ and it is called the $k$-th Betti number of $(\mathcal{E},\nabla)$. We will also explain in section 4.6 how to recover from our analysis that $b_{k}(M,\mathcal{E})=b_{n-k}(M,\mathcal{E}^{\prime})$ for any $0\leq k\leq n$. In the statement of our Theorem, the assumption that the vector field is either Anosov or Morse-Smale comes from the fact that we have a good spectral theory for such systems thanks to [8, 18, 14, 11]. Yet, as we shall see, our proof is rather robust and this result should hold for any type of vector fields which has a good spectral behaviour yielding in particular an isolated eigenvalue at $z=0$ of finite multiplicity. Thanks to a classical result of Peixoto [38] and to the Sternberg-Chen Theorem [34, 48], the assumption of being a $\mathcal{C}^{1}$-linearizable Morse-Smale vector field is in fact generic when $\text{dim}(M)=2$ – see appendix A.

In the case of Morse-Smale gradient flows, this theorem was proved in [10] under the assumption that $\mathcal{E}$ is the trivial bundle $M\times\mathbb{C}$ and it provided a new (spectral) proof of earlier results due to Laudenbach [29] and Harvey-Lawson [25, 26]. For other flows, this result seems to be new and it gives a realization of the twisted De Rham cohomology as the cohomology of Pollicott-Ruelle resonant states. Recall that in the case of Morse-Smale flows [12], we described explicitely a family of twisted currents generating the Pollicott-Ruelle resonant states and whose support are given by the stable manifolds. In a work related to the Anosov case [15], Dyatlov and Zworski showed how to identify the Pollicott-Ruelle resonant states $u\in\mathcal{D}^{\prime k}(M,\mathbb{C})$ satisfying $\iota_{V}(u)=0$ with the De Rham cohomology of $X$ when $V$ is the geodesic vector field on $M:=SX$ (with $X$ negatively curved and $\text{dim}(X)=2$). Finally, in the case of Hodge-De Rham theory [5, 31], the cohomology of the De Rham complex can be represented by the kernel of the Laplace operator $(d^{\nabla}+(d^{\nabla})^{*})^{2}$, which is also finite dimensional.

2.4. Generalized Morse inequalities

The gain compared with Hodge theory is that we do not have to go through Witten’s deformation procedure to derive the Morse inequalities and that we can deal directly with the limit operator $-\mathcal{L}_{V,\nabla}$. This is of course at the expense of the spectral analysis of $-\mathcal{L}_{V,\nabla}$. Let us now draw some generalizations of the Morse-Smale inequalities from this first result. By standard arguments on exact sequences – see e.g. [10, Par. 8.3] for a brief reminder, one can verify that the following holds:

Corollary 2.2** (Spectral Morse inequalities).**

Suppose the assumptions of Theorem 2.1 are satisfied. Then, the following holds:

$\forall 0\leq k\leq n,\ \sum_{j=0}^{k}(-1)^{k-j}\operatorname{dim}C_{V,\nabla}^{j}(0)\geq\sum_{j=0}^{k}(-1)^{k-j}b_{j}(M,\mathcal{E}),$

with equality in the case $k=n$.

Recall that, for $k=n$, one recovers the Euler characteristic $\chi(M,\mathcal{E})$. If $V$ is a Morse-Smale gradient vector field (which is $\mathcal{C}^{1}$-linearizable) and if $\mathcal{E}=M\times\mathbb{C}$ is the trivial bundle, we have shown in [10] that elements inside $C_{V,\nabla}^{j}(0)$ are generated by the currents of integration on unstable manifolds associated with critical points of index $k$. In particular, the dimension of that space is equal to the number of critical points of index $j$. Hence, this corollary applied to Morse-Smale gradient flows recovers in that case the classical Morse inequalities. Let us now apply this result in the more general framework of Morse-Smale vector fields. By definition [44], the nonwandering set of a Morse-Smale flow is composed of finitely many critical points and closed orbits, each of them being hyperbolic. Closed orbits $\Lambda$ can be divided into two categories: untwisted (if the corresponding stable manifold 444See the appendix of [11] for a brief reminder. $W^{s}(\Lambda)$ is orientable) and twisted (otherwise). We define the twisting index $\Delta_{\Lambda}$ of a closed orbit $\Lambda$ to be equal to $+1$ in the untwisted case and to $-1$ in the twisted case. Suppose now for simplicity of exposition that $\nabla$ preserves a Hermitian structure on $\mathcal{E}$ i.e. parallel transport by the flat connection $\nabla$ preserves the fiber metric of $\mathcal{E}$. It implies that, for every closed orbit, the monodromy matrix $M_{\mathcal{E}}(\Lambda)$ for the parallel transport is a unitary matrix. In particular, its spectrum is included in $\mathbb{S}^{1}$. For every closed orbit $\Lambda$, we then define $m_{\Lambda}$ to be the multiplicity of $\Delta_{\Lambda}$ as an eigenvalue of $M_{\mathcal{E}}(\Lambda)$ (this multiplicity can vanish and depends on $\nabla$). Using these conventions, we can state the following :

Corollary 2.3** (Generalized Morse-Smale inequalities).**

Let $\mathcal{E}\rightarrow M$ be a smooth, complex, hermitian vector bundle of rank $N$ endowed with a flat unitary connection $\nabla$. Suppose that $V$ is a Morse-Smale vector field. Denote by $c_{k}(V)$ the number of critical points of index555This means that the stable manifold of the point is of dimension $k$. $k$. Then, the following holds, for every $0\leq k\leq n$,

[TABLE]

with equality in the case $k=n$.

We will give the proof of this result in paragraph 4.4 and the main additional ingredient compared with Corollary 2.2 is that we make use of the complete description of the Pollicott-Ruelle resonances for Morse-Smale flows that we gave in [12]. The fact that we require our flat connection to preserve a Hermitian structure is not optimal but makes the statement simpler to state. Recall in fact that our description of the fine structure of Pollicott-Ruelle resonances holds under weaker assumptions – see [12, Sec. 4] for related discussion. We emphasize that this assumption is often made in Hodge theory [5, 31] especially for problems related to analytic torsion even if it is not completely necessary there too [33]. Compared to Theorem 2.1 and Corollary 2.2, we do not need to make the linearization assumption in that statement.

Again, when $V$ is a gradient flow, we recover the classical Morse inequalities. If $\mathcal{E}=M\times\mathbb{C}$ is the trivial bundle, then $m_{\Lambda}=1$ for untwisted orbits and $m_{\Lambda}=0$ otherwise. In other words, the sum over the closed orbits is exactly the number of untwisted closed orbits whose stable manifold has dimension $k+1$. This inequality is slightly stronger than the original result of Smale [44] whose upper bound involved also the number of twisted closed orbits. This stronger version of Morse-Smale inequalities was in fact proved by Franks in [21, Ch. 8] by a completely different approach than ours. In the case of more general vector bundles, we obtain Morse type inequalities which seem to be new.

2.5. Koszul homological complexes

As for the case of the coboundary operator, we can verify from the spectral definition of the spaces $C_{V,\nabla}^{\bullet}(0)$ that the sequence of linear maps :

[TABLE]

defines a complex. We shall refer to this complex as the Morse-Smale-Koszul complex. In [10], we showed that, for Morse-Smale gradient flows and for the trivial vector bundle $M\times\mathbb{C}$, the Euler characteristic of the manifold is equal to the Euler characteristic of the Morse-Koszul complex. The same remains in fact true for Morse-Smale flows :

Theorem 2.4**.**

*Let $\mathcal{E}\rightarrow M$ be a smooth, complex, hermitian vector bundle of rank $N$ endowed with a flat unitary connection $\nabla$. Suppose that $V$ is a vector field which is Morse-Smale and $\mathcal{C}^{\infty}$-diagonalizable.

Then, the Euler characteristic $\chi(M,\mathcal{E})$ of $\mathcal{E}$ is equal to the Euler characteristic of the Morse-Smale-Koszul complex $(C_{V,\nabla}^{\bullet}(0),\iota_{V})$.*

We will prove this Theorem in section 5 and we will in fact show something slightly stronger as we will compute exactly the homology of that complex in each degree in terms of the critical points of $V$. The assumption of being $\mathcal{C}^{\infty}$ diagonalizable is defined in appendix A and it holds as soon as certain (generic) nonresonance assumptions are satisfied by the Lyapunov exponents. This hypothesis appeared in our previous work [12] in order to simplify the exposition and it could probably be removed if we are only interested in the part of the spectrum corresponding to resonances having a small enough real part. Yet, this would be at the expense of some extra (probably technical) work that was beyond the scope of [12] and of the present article.

2.6. Resonances on the imaginary axis and Reidemeister torsion

We conclude by relating the Pollicott-Ruelle resonances to the results of Fried in [22, p. 44–53]. Recall that, in this reference, Fried introduced the twisted Fuller measure of $(V,\nabla)$ :

Definition 2.5** (Twisted Fuller measure).**

In the notations of the previous paragraphs, the twisted Fuller measure $\mu_{V,\nabla}$ is a distribution in $\mathcal{D}^{\prime}(\mathbb{R}_{+}^{*})$ defined by the formula :

[TABLE]

where $P_{\Lambda}$ is a linearized Poincaré map associated with the closed orbit $\Lambda$ and where $\mathcal{P}_{\Lambda}$ is the minimal period of $\Lambda$.

In fact, Fried only defined this quantity when there are no critical points and he observed [22, p. 49] that, in this case, $t\mu_{V,\nabla}(t)$ coincides with the distributional traces of Guillemin and Sternberg [24, Ch. VI] – see also [14] for a brief and self-contained account. We refer to the recent survey of Gouëzel [23, Sect. 2.4] for a brief account on the role of these distributional traces in the study of transfer operators and Ruelle zeta functions [42]. Here, we adopt a slightly more general definition than Fried’s one which encompasses the case of critical points. If there are no closed orbits, then our definition also coincides with the distributional traces of [24]. In the mixed case, the formula would be slightly more subtle to justify from the functional perspective as we would have to deal with vector bundles having some “jumps”. Recall we denote by $C_{V,\nabla}^{k}(z_{0})$ the range of the spectral projector $\pi^{(k)}_{z_{0}}$ on the eigenspace of eigenvalue $z_{0}$ acting on $k$–forms. Our last Theorem relates these twisted Fuller measures with the correlation spectrum on the imaginary axis $i\mathbb{R}$:

Theorem 2.6** (Spectral interpretation of Fuller measures).**

Let $\mathcal{E}\rightarrow M$ be a smooth, complex, hermitian vector bundle of rank $N$ endowed with a flat unitary connection $\nabla$. Suppose that $V$ is a vector field which is Morse-Smale and $\mathcal{C}^{\infty}$-diagonalizable.

Then, one has

$\mu_{V,\nabla}(t)=\sum_{k=0}^{n}(-1)^{n-k+1}\sum_{z_{0}\in\mathcal{R}_{k}(V,\nabla)\cap i\mathbb{R}}\operatorname{dim}\left(C_{V,\nabla}^{k}(z_{0})\cap\operatorname{Ker}(\iota_{V})\right)\frac{e^{tz_{0}}}{t}$

in the sense of distributions in $\mathbb{R}_{+}^{*}$.

We will prove this Theorem in section 6.4. In the particular case where $V$ is non singular, the Fuller measure can be related to the Guillemin trace as follows. We denote by $V^{\perp}\subset T^{*}M$ the vector bundle $\ker(\iota_{V})$ of invariant $1$-forms by the flow and by $\Lambda^{k}V^{\perp}$ the bundle of invariant $k$-forms. The Guillemin trace on twisted invariant forms of degree $k$ is denoted by $TR^{\flat}_{\Lambda^{k}V^{\perp}\otimes\mathcal{E}}$. Then, following Fried [22, p. 49–50 in particular equation VII], a direct application of the Guillemin trace formula yields an identity relating twisted Fuller measures, Guillemin traces and imaginary resonances as follows :

[TABLE]

in the sense of distributions in $\mathbb{R}_{+}^{*}$.

Let us now briefly recall the topological content of these twisted Fuller measures (and thus of the Pollicott-Ruelle spectrum). Fried observed that the twisted Fuller measure shares a lot of properties with some topological invariants such as the Euler characteristic or the Reidemeister torsion [22, p. 46-49]. In fact, we can observe that the coefficients defining $\mu_{V,\nabla}(t)$ are of purely topological nature even if the support of the measure depends on the parametrization of the flow and hence is not topological. In order to get rid of these nontopological informations, Fried evaluated the “total mass” of these Fuller measures and showed that in certain cases this defines indeed a topological invariant. For that purpose, one can introduce the following zeta function :

[TABLE]

and following Fried [22, p. 51], we define the corresponding torsion function

[TABLE]

as the exponential of $-\partial_{s}\zeta^{\flat}_{V,\nabla}(0,z)$. We emphasize that Fried directly used the Laplace transform to define his twisted zeta function as he only considered nonsingular vector fields. Here, due to the fact that we allow critical points, we have to make this extra regularization involving a Mellin transform. In our slightly more general framework, the terminology “torsion function” may sound a little bit abusive as we do not recover the Reidemeister torsion in general. Computing the derivative of $\zeta_{V,\nabla}^{\flat}(s,z)$ w.r.t. $s$ at $s=0$, one can verify (see paragraph 6.5) that, for $\text{Re}(z)$ large enough,

[TABLE]

Therefore, the torsion function $Z_{V,\nabla}(z)$ satisfies the identity :

[TABLE]

for $\text{Re}(z)$ large enough. The right hand side of the above equality is a weighted Ruelle zeta function [23, equation (13) p. 19], the weight being given by the monodromy of the flat connection. If the vector field is nonsingular which means the Euler characteristic term vanishes and if $m_{\Lambda}=0$ for every closed orbit666In particular, $(\mathcal{E},\nabla)$ is acyclic., then $Z_{V,\nabla}$ converges as $z\rightarrow 0^{+}$ and Fried showed that the modulus of the limit is equal to the Reidemeister torsion [22, Sect. 3].

Following the works of Ray–Singer on analytic torsion [40], we now introduce the following:

Definition 2.7** (Spectral zeta determinant of the nonzero resonances on the imaginary axis).**

We define the spectral zeta function:

[TABLE]

and the corresponding spectral zeta determinant777This regularizes the infinite product $\prod_{z_{0}\in\mathcal{R}_{k}(V,\nabla)\cap i\mathbb{R}^{*}}\left(z-z_{0}\right)^{(-1)^{k}k\text{dim}\left(C_{V,\nabla}^{k}(\lambda)\right)}$ of nonzero resonances on the imaginary axis.

[TABLE]

Then, for $\text{Re}(z)$ large enough, as a consequence of Theorem 2.6 and of standard arguments from Hodge theory (see paragraph 6.5), both the flat zeta function $\zeta_{V,\nabla}^{\flat}(s,z)$ and the spectral zeta function $\zeta_{RS}(s,z)$ admit analytic continuations in $s\in\mathbb{C}$ and are related by the equation :

[TABLE]

Hence, as a direct corollary of our Theorem and of Fried’s Theorem, we have:

Corollary 2.8**.**

Suppose that the assumptions of Theorem 2.6 are satisfied. Then, one has:

[TABLE]

In particular if $V$ is non singular and $m_{\Lambda}=0$ for every periodic orbit $\Lambda$ then $|Z_{RS}(0)|$ coincides with the Reidemeister torsion.

Informally, the above discussion means that we can recover the Euler characteristic and the Reidemeister torsion (under proper assumptions) from the Pollicott-Ruelle spectrum lying on the imaginary axis. These different results illustrate that the Pollicott-Ruelle spectrum on the imaginary axis has a nice topological interpretation in the case of Morse-Smale flows. More specifically, the topological content seems to be contained in the first band of resonances in the terminology of Faure and Tsujii [19, 20]. Recall that, for Anosov contact flows, they showed that the Pollicott-Ruelle spectrum exhibits a band structure. We proved in [12] that the same band structure remains true in the case of Morse-Smale flows and that the first band of resonances is in that case contained inside the imaginary axis. Hence, the main observation of these results is that, at least in certain acyclic cases, we can keep track of topological invariants inside the first band of the correlation spectrum and not only inside the kernel which is here reduced to $\{0\}$.

In section 6, we will be even more precise on the topological features of the first band of resonant states for Morse-Smale flows. In fact, for every $z_{0}\in i\mathbb{R}^{*}$, $(C^{\bullet}_{V,\nabla}(z_{0}),d^{\nabla})$ defines an acyclic cohomological complex of finite dimensional spaces – see paragraph 4.2. Thus, once we have specified a preferred basis of that space, we can compute the torsion of that complex [22, 31] which is a certain determinant related to the coboundary operator. In the case of Morse-Smale flows, such a preferred basis of Pollicott-Ruelle resonant states can be naturally introduced following our previous work [12] – see paragraphs 5.1.1 and 5.1.2 for a brief reminder. Then, we will introduce the infinite dimensional space

[TABLE]

which still defines an acyclic complex. We will define a notion of regularized torsion of this infinite dimensional “complex” by regularizing the infinite product of torsions of every individual finite dimensional complex of Pollicott-Ruelle resonant states associated to the spectrum on the imaginary axis. In paragraph 6.4, we will see that this regularized torsion is related to the behaviour of the torsion function $Z_{V,\nabla}(z)$ at $z=0$ and to the spectral zeta function $\zeta_{RS}$. Thus, it is also linked to the Reidemeister torsion whenever $V$ has no critical points and where $m_{\Lambda}$ is equal to [math] for every closed orbit.

2.7. Conventions

All along the article, $M$ will denote a smooth, compact, oriented, boundaryless manifold of dimension $n$. We shall denote by $V$ a smooth vector field on $M$ and by $\mathcal{E}\rightarrow M$ a complex vector bundle of dimension $N$ which is endowed with a flat connection $\nabla$. Except mention of the contrary, $\mathcal{E}$ is not necessarly endowed with a Hermitian structure compatible with $\nabla$.

Acknowledgements

In the spring 2016, Viviane Baladi asked us if there were some relations between our work [10] and the works of Fried on Reidemeister torsion: we warmly thank her for pointing to us this series of works which motivated part of the results presented here. We also thank Frédéric Faure for many explanations on his works with Johannes Sjöstrand and Masato Tsujii. We also acknowledge useful discussions related to this article and its companion articles [11, 12] with Livio Flaminio, Colin Guillarmou, Benoit Merlet, Frédéric Naud and Patrick Popescu Pampu. The second author is partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR project GERASIC (ANR-13-BS01-0007-01).

3. Spectral theory of Anosov and Morse-Smale vector fields

We begin with a brief account on the existence of Pollicott-Ruelle resonant states following the microlocal approach of Faure and Sjöstrand [18] – see also [14, 10] for extensions of these results. In the Anosov case, another approach would be to use the earlier results of Butterley and Liverani via spaces of anisotropic Hölder distributions [8]. We could probably also proceed via coherent states like in the works of Faure and Tsujii [47, 20]. We also refer to [4] for a recent account by Baladi concerning the related case of hyperbolic diffeomorphisms.

3.1. Escape function

The key ingredient in Faure-Sjöstrand’s analysis is the construction of a so-called escape function, or equivalently a Lyapunov function for the Hamiltonian flow on $T^{*}M$ induced by

[TABLE]

Recall that the corresponding Hamiltonian flow is given by

[TABLE]

where $\varphi^{t}$ is the flow induced by the vector field $V$ on $M$. We shall denote by $X_{V}(x,\xi)$ the Hamiltonian vector field induced by $\Phi_{V}^{t}$. Using the terminology of [18], an escape function for the flow $\Phi_{V}^{t}$ on $T^{*}M$ is a function of the form

[TABLE]

meeting the following requirements:

• •

$m$** is a symbol of order [math]**. This means $m(x,\xi)$ belongs to $\mathcal{C}^{\infty}(T^{*}M)$ and, for all multi-indices $(\alpha,\beta)\in\mathbb{N}^{2n}$,

[TABLE]

• •

$f$** has the correct homogeneity in $\xi$**. $f(x,\xi)$ belongs to $\mathcal{C}^{\infty}(T^{*}M)$, and, for $\|\xi\|_{x}\geq 1$, $f(x,\xi)>0$ is positively homegeneous of degree $1$.

• •

The symbol $H_{V}$ is micro elliptic in some conical open set $\mathcal{N}_{0}$. There exist a conical open set $\mathcal{N}_{0}$ and a constant $C>0$ such that, for every $(x,\xi)\in\mathcal{N}_{0}$ satisfying $\|\xi\|_{x}\geq C$, one has

[TABLE]

and $f(x,\xi)=|H_{V}(x,\xi)|$. Moreover, outside some slightly bigger conical open set $(\mathcal{N}_{0}\subset)\mathcal{N}_{1}$, one has $f(x,\xi)=\|\xi\|_{x}$ for $\|\xi\|_{x}\geq 1$.

• •

Outside $\mathcal{N}_{0}$, $G$ has uniform decrease along the Hamiltonian flow. There exist $R\geq 1$ and $c>0$ such that, for every $(x,\xi)\in T^{*}M$ satisfying $\|\xi\|_{x}\geq R$,

[TABLE]

In other words, the escape function is strictly decreasing except in the directions where the Hamiltonian $H_{V}$ is elliptic. Observe that $\mathcal{N}_{0}$ and $\mathcal{N}_{1}$ can be even chosen to be empty – this was for instance the case for gradient flows [10]. Note that compared with Lemma 1.2 in [18], we do not require anything on the value of the order function $m$ in certain directions of phase space. The reason for this is that the above axioms are the key ingredients to build a convenient spectral theory for the vector field $V$. The other requirements on $m$ are additional informations on the structure Sobolev space which are specific to the Anosov framework. According to [18, 11], examples of vector fields possessing such an escape function are given by Anosov or Morse-Smale vector fields which are $\mathcal{C}^{1}$-linearizable. Motivated by the main results from [18] and in order to alleviate the notations in the upcoming sections, we shall say that a flow $\varphi^{t}$ possessing such an escape function is microlocally tame. We shall make this assumption in the following and we keep in mind that the only known examples of such flows are the ones mentionned above.

3.2. Anisotropic Sobolev spaces

Let us now briefly recall the construction of Faure-Sjöstrand based on the existence of such a function [18]. Note that, in this reference, the authors only treated the case of the trivial bundle $M\times\mathbb{C}$. The extension of this microlocal approach to more general bundles was made by Dyatlov and Zworski in [14] by some slightly different point of view. Fix $s>0$ some large parameter (corresponding to the Sobolev regularity we shall require) and $0\leq k\leq n$ (the degree of the forms). We also fix some Riemannian metric $g$ on $M$ and some hermitian structure $\langle.,\rangle_{\mathcal{E}}$ on $\mathcal{E}$, none of them being a priori related to $V$ and $\nabla$. We can fix the inner product $\langle,\rangle_{g^{*}}^{(k)}$ on $\Lambda^{k}(T^{*}M)$ which is induced by the metric $g$ on $M$. Then, the the Hodge star operator is the unique isomorphism $\star_{k}:\Lambda^{k}(T^{*}M)\rightarrow\Lambda^{n-k}(T^{*}M)$ such that, for every $\psi_{1}$ in $\Omega^{k}(M)\simeq\Omega^{k}(M,\mathbb{C})$ and $\psi_{2}$ in $\Omega^{n-k}(M)\simeq\Omega^{n-k}(M,\mathbb{C})$,

[TABLE]

where $\omega_{g}$ is the volume form induced by the Riemannian metric on $M$. This induces a map $\star_{k}:\Omega^{k}(M,\mathcal{E})\rightarrow\Omega^{n-k}(M,\mathcal{E})$ which acts trivially on the $\mathcal{E}$-coefficients. Using the metric $g_{\mathcal{E}}$ on $\mathcal{E}$, we can also introduce the following pairing, for every $0\leq k,l\leq n$,

[TABLE]

Combining these, we can define the positive definite Hodge inner product on $\Omega^{k}(M,\mathcal{E})$ as

[TABLE]

In particular, we can define $L^{2}(M,\Lambda^{k}(T^{*}M)\otimes\mathcal{E})$ as the completion of $\Omega^{k}(M,\mathcal{E})$ for this scalar product.

Set now

[TABLE]

where $G(x,\xi)=m(x,\xi)\log\left(1+f(x,\xi)^{2})^{\frac{1}{2}}\right)$ is the escape function. We can define

[TABLE]

belonging to $\text{Hom}(\Lambda^{k}(T^{*}M)\otimes\mathcal{E})$ and introduce an anisotropic Sobolev space of currents by setting

[TABLE]

where $\operatorname{Op}(\mathbf{A}_{s}^{(k)})$ is a (essentially selfadjoint) pseudodifferential operator with principal symbol $\mathbf{A}_{s}^{(k)}$.

Remark 3.1*.*

Note that this requires to deal with symbols of variable order $m(x,\xi)$ whose symbolic calculus was described in Appendix A of [17]. This can be done as the symbol $m(x,\xi)$ belongs to the standard class of symbols $S^{0}(T^{*}M)$. We also refer to [14, App. C.1] for a brief reminder of pseudodifferential operators with values in vector bundles. In particular, adapting the proof of [17, Cor. 4] to the vector bundle valued framework, one can verify that $\mathbf{A}_{s}^{(k)}$ is an elliptic symbol, and thus $\operatorname{Op}(\mathbf{A}_{s}^{(k)})$ can be chosen to be invertible.

We observe from the composition rule for pseudodifferential operators that

[TABLE]

Mimicking the proofs of [17], we can also deduce some properties of these spaces of currents. First of all, they are endowed with a Hilbert structure inherited from the $L^{2}$-structure on $M$. The space

[TABLE]

is the topological dual of $\mathcal{H}^{m}_{k}(M,\mathcal{E})$ which is in fact reflexive. The following injections holds and they are continuous:

[TABLE]

The Riemannian metric on $\mathcal{E}$ allows to define a canonical isomorphism $\tau_{\mathcal{E}}:\mathcal{E}\rightarrow\mathcal{E}^{\prime}$ by setting, for every $\psi$ in $\mathcal{E}$, $\tau_{\mathcal{E}}(\psi)=\langle\psi,.\rangle_{\mathcal{E}}$. Then, combined with the Hodge star map, it induces an isomorphism from $\mathcal{H}^{m}_{k}(M,\mathcal{E})^{\prime}$ to $\mathcal{H}^{-m}_{n-k}(M,\mathcal{E}^{\prime})$, whose Hilbert structure is given by the scalar product

[TABLE]

Thus, the topological dual of $\mathcal{H}_{k}^{m}(M,\mathcal{E})$ can be identified with $\mathcal{H}^{-m}_{n-k}(M,\mathcal{E}^{\prime})$, where, for every $\psi_{1}$ in $\Omega^{k}(M,\mathcal{E})$ and $\psi_{2}$ in $\Omega^{n-k}(M,\mathcal{E}^{\prime})$, one has the following duality relation:

[TABLE]

3.3. Pollicott-Ruelle resonances

Now that we have settled the functional framework for our microlocally tame vector field, one can establish the existence of Pollicott-Ruelle resonances. More precisely, one can show that, for every $T_{0}>0$, there exists some $s>0$ large enough such that, for every $0\leq k\leq n$,

[TABLE]

has a discrete spectrum on the half plane $\text{Im}(z)>-T_{0}$, consisting of eigenvalues of finite algebraic multiplicity. We refer to [18, Th. 1.4] for a more precise statement. This result essentially follows from the fact that Faure and Sjöstrand proved that $(\mathcal{L}_{V,\nabla}+z)$ is a Fredholm depending analytically on $z$. The eigenvalues are the so-called Pollicott-Ruelle resonances and the corresponding generalized eigenmodes are called resonant states. These objects correspond exactly to the poles and the residues of the function $\hat{C}_{\psi_{1},\psi_{2}}(z)$ defined in (4).

Remark 3.2*.*

In [18, Sect. 3], the authors only treated the case $k=0$ and $\mathcal{E}=M\times\mathbb{C}$. Yet, their proof can be adapted almost verbatim to our framework except that we have to deal with pseudodifferential operators with values in $\Lambda^{k}(T^{*}M)\otimes\mathcal{E}$ – see [5, Part I.3] [14, App. C] for a brief review. As was already observed in [14], the main point to extend directly Faure-Sjöstrand’s proof to more general bundles is that the (pseudodifferential) operators under consideration have a diagonal symbol. In fact, given any local basis $(e_{j})_{j=1,\ldots J_{k}}$ of $\Lambda^{k}(T^{*}M)\otimes\mathcal{E}$ and any family $(u_{j})_{j=1,\ldots J_{k}}$ of smooth functions $\mathcal{C}^{\infty}(M)$, one has

[TABLE]

where the second part of the sum in the right-hand side is a lower order term (of order [math]). In other words, the principal symbol of $\mathcal{L}_{V,\nabla}^{(k)}$ is $\xi(V(x))\mathbf{Id}_{\Lambda^{k}(T^{*}M)\otimes\mathcal{E}}$. This diagonal form allows to adapt the proofs of [18] to this vector bundle framework.

We conclude this preliminary paragraph with some properties of this spectrum:

• •

Faure and Sjöstrand implicitely show [18, Lemma 3.3] that, for every $z$ in $\mathbb{C}$ satisfying $\text{Im}z>C_{0}$ (for some $C_{0}>0$ large enough), one has

[TABLE]

In particular, combining with the Hille-Yosida Theorem [16, Cor. 3.6, p. 76], it implies that

[TABLE]

generates a strongly continuous semigroup which is defined for every $t\geq 0$ and whose norm is bounded by $e^{tC_{0}}.$

• •

As in [18, Th. 1.5], we can show that the eigenvalues (counted with their algebraic multiplicity) and the eigenspaces of $-\mathcal{L}_{V,\nabla}^{(k)}:\mathcal{H}_{k}^{sm+n-k}(M,\mathcal{E})\rightarrow\mathcal{H}_{k}^{sm+n-k}(M,\mathcal{E})$ are in fact independent of the choice of escape function and of the parameter $s$.

• •

By duality, the same spectral properties holds for the dual operator

[TABLE]

with $\nabla^{\dagger}$ the dual connection [11, Sect. 5].

• •

Given any $z_{0}$ in $\mathbb{C}$, the corresponding spectral projector $\pi_{\lambda}^{(k)}$ is given by [28, Appendix]:

[TABLE]

where $\gamma_{z_{0}}$ is a small contour around $z_{0}$ which contains at most the eigenvalue $z_{0}$ in its interior.

• •

Given any $z_{0}$ in $\mathbb{C}$ with $\operatorname{Re}(\lambda)>-T_{0}$, there exists $m_{k}(z_{0})\geq 1$ such that, in a small neighborhood of $z_{0}$, one has

[TABLE]

with $R_{z_{0},k}(z)$ an holomorphic function.

The last two statements allow to make the connection between this spectral framework and the functions $\hat{C}_{\psi_{1},\psi_{2}}(z)$ appearing in the introduction.

4. Morse-Smale complex and generalized Morse inequalities

In this section, we will give the proof of Theorem 2.1 and draw some consequences in terms of Morse-Smale inequalities and of spectral trace asymptotics. In all this section, $V$ is a smooth vector field which is microlocally tame in the sense of section 3 and $\nabla$ is a flat connection on $\mathcal{E}$.

Remark 4.1*.*

Except mention of the contrary, we will not suppose that $\nabla$ preserves some hermitian structure on $\mathcal{E}$ (or equivalently that $\mathcal{E}$ is endowed with a flat unitary connection $\nabla$). In that sense, it is slightly different from the case of Hodge theory [5, 31]. Recall that preserving a hermitian structure would mean that, for any $(\psi_{1},\psi_{2})$ in $\Omega^{0}(M,\mathcal{E})\times\Omega^{0}(M,\mathcal{E})$, one has

[TABLE]

This last property ensures that, for any $\psi_{1}$ in $\Omega^{k-1}(M)$ and for any $\psi_{2}$ in $\Omega^{n-k}(M)$,

[TABLE]

It also imposes that the monodromy matrices for the parallel transport are unitary – see [12, Sect. 4].

4.1. De Rham cohomology

We start with a brief reminder on de Rham cohomology [41, 5]. An element $\omega$ in $\Omega^{*}(M,\mathcal{E})$ such that $d^{\nabla}\omega=0$ is called a cocycle while an element $\omega$ which is equal to $d^{\nabla}\alpha$ for some $\alpha\in\Omega^{*}(M,\mathcal{E})$ is called a coboundary. We define then

[TABLE]

Obviously, $B^{k}(M,\mathcal{E})\subset Z^{k}(M,\mathcal{E})$, and the quotient space $\mathbf{H}^{k}(M,\mathcal{E})=Z^{k}(M,\mathcal{E})/B^{k}(M,\mathcal{E})$ is called the $k$-th de Rham cohomology. The complex is said to be acyclic if all the cohomology is reduced to $\{0\}$.

The coboundary operator $d^{\nabla}$ can be extended into a map acting on the space of currents $\mathcal{D}^{\prime,*}(M,\mathcal{E})$. This allows to define another cohomological complex $(\mathcal{D}^{\prime,*}(M,\mathcal{E}),d^{\nabla})$:

[TABLE]

where we recall that $\mathcal{D}^{\prime,k}(M,\mathcal{E})$ is the topological dual of $\Omega^{n-k}(M,\mathcal{E})$. One can similarly define the $k$-th cohomology of that complex. A remarkable result of de Rham is that these two cohomologies coincide [41, Ch. 4]:

Theorem 4.2** (de Rham).**

Let $u$ be an element in $\mathcal{D}^{\prime,k}(M,\mathcal{E})$ satisfying $d^{\nabla}u=0$.

1. (1)

There exists $\omega$ in $\Omega^{k}(M,\mathcal{E})$ such that $u-\omega$ belongs to $d^{\nabla}\left(\mathcal{D}^{\prime,k-1}(M,\mathcal{E})\right)$. 2. (2)

If $u=d^{\nabla}v$ with $(u,v)$ in $\Omega^{k}(M,\mathcal{E})\times\mathcal{D}^{\prime,k-1}(M,\mathcal{E})$, then there exists $\omega$ in $\Omega^{k-1}(M,\mathcal{E})$ such that $u=d^{\nabla}\omega$.

Remark 4.3*.*

We will in fact need something slightly more precise involving the anisotropic Sobolev spaces we have introduced – see also [15, Lemma 2.1] for related statements in terms of wavefront sets. More precisely, if $u$ belongs to $\mathcal{H}_{k}^{sm+n-k}(M,\mathcal{E})$, then $u-\omega$ belongs to $d^{\nabla}\mathcal{H}_{k-1}^{sm+n-k+1}(M,\mathcal{E})$. This observation will be crucial in our proof of Theorem 2.1.

Let us for instance recall the argument from [15] which is based on elliptic regularity in order to verify that Remark 4.3 indeed holds.

Proof.

First of all, fix a Riemannian metric $g$ on $M$ and a hermitian structure $\langle.,.\rangle_{\mathcal{E}}$ on $\mathcal{E}$. Following [5, Th. 4.11], we can compute the formal adjoint for the Hilbert structure on $L^{2}(M,\Lambda^{k}(T^{*}M)\otimes\mathcal{E})$. More precisely, for $\psi_{1}$ in $\Omega^{k}(M,\mathcal{E})$ and for $\psi_{2}$ in $\Omega^{k+1}(M,\mathcal{E})$, one has

[TABLE]

where the second equality follows from the Stokes Theorem. This tells us that888Under the additional assumptions that $\nabla$ preserves the Hermitian structure, we would have $\tau_{\mathcal{E}}^{-1}d^{\nabla^{\dagger}}\tau_{\mathcal{E}}=d^{\nabla}$. $(d^{\nabla})^{*}=(-1)^{k}\star_{k}^{-1}\tau_{\mathcal{E}}^{-1}d^{\nabla^{\dagger}}\tau_{\mathcal{E}}\star_{k+1}.$ We then form the corresponding Laplace Beltrami operator $\Delta_{g,\mathcal{E}}=d^{\nabla}(d^{\nabla})^{*}+(d^{\nabla})^{*}d^{\nabla},$ which is formally selfadjoint. According to [5, p.19], the principal symbol of this pseudodifferential operator is $-\|\xi\|_{x}^{2}\mathbf{Id}_{\Lambda^{k}(T^{*}M)\otimes\mathcal{E}}$. In particular, this defines an elliptic operator. We can now make use of elliptic regularity to prove De Rham Theorem.

Fix $u$ in $\mathcal{D}^{\prime,k}(M,\mathcal{E})$ satisfying $d^{\nabla}u=0$. According to [5, Part. I.3], there exists a pseudodifferential operator $A_{k}$ of order $-2$ such that $u-\Delta_{g,\mathcal{E}}A_{k}u$ belongs to $\Omega^{k}(M,\mathcal{E})$. Using the fact that $u$ is a cocycle, we find that $d^{\nabla}\Delta_{g,\mathcal{E}}A_{k}u=\Delta_{g,\mathcal{E}}d^{\nabla}A_{k}u$ belongs to $\Omega^{k+1}(M)$. Thus, by elliptic regularity, $d^{\nabla}A_{k}u$ belongs to $\Omega^{k+1}(M)$ and, by composing with $(d^{\nabla})^{*}$, $(d^{\nabla})^{*}d^{\nabla}A_{k}u$ belongs to $\Omega^{k}(M)$. Hence, we can find $\omega$ in $\Omega^{k}(M,\mathcal{E})$ such that $u-\omega=d^{\nabla}(d^{\nabla})^{*}A_{k}u$ which proves point $(2)$. Note that, if $u$ belongs to $\mathcal{H}_{k}^{sm+n-k}(M,\mathcal{E})$, then we can verify from the composition rules of pseudodifferential operators that $(d^{\nabla})^{*}A_{k}u$ belongs to $\mathcal{H}_{k-1}^{sm+n-k+1}(M,\mathcal{E})$ as pointed out in Remark 4.3.

For the proof of point $(2)$, we proceed similarly except that we replace $u$ by $v$ which is not a priori a cocyle. Using the fact that $u=d^{\nabla}v$ is smooth, we can still conclude that there exists $\omega\in\Omega^{k-1}(M,\mathcal{E})$ such that $v-\omega=d^{\nabla}(d^{\nabla})^{*}A_{k}v$. Applying $d^{\nabla}$, we get the expected conclusion. ∎

4.2. Chain homotopy equation

Thanks to (9) and to the fact that $d^{\nabla}\circ\mathcal{L}_{V,\nabla}^{(k)}=\mathcal{L}_{V,\nabla}^{(k+1)}\circ d^{\nabla}$, we can define, for every $z_{0}\in\mathbb{C}$ the cohomological complex $(\text{Ker}(\mathcal{L}_{V,\nabla}^{(*)}+z_{0})^{m_{k}(z_{0})},d^{\nabla})$

[TABLE]

By construction, this cohomological complex coincide with the complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ defined by (6). Observe that, for $z_{0}\neq 0$, $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ is acyclic and we shall come back to this case later on. Before that, we focus on the case $z_{0}=0$ and we start with a few preliminary observations. First, as $(\mathcal{L}_{V,\nabla}+z)$ commutes with $d^{\nabla}$ and $\iota_{V}$, we can verify from (13) that, for every $0\leq k\leq n$,

[TABLE]

Hence, one has

[TABLE]

According to [28, Appendix A.18], the operator $\mathcal{L}_{V,\nabla}^{(k)}$ is invertible on the space

[TABLE]

The same holds in degree $k+1$. Hence, thanks to the relation

[TABLE]

we can deduce that

[TABLE]

Now that we have settled these equalities, we write, for any $\psi_{1}$ in $\mathcal{H}_{k}^{sm+n-k}(M,\mathcal{E})$,

[TABLE]

where the last equality follows from (16) and (18). Thus, if we set

[TABLE]

we obtain the following chain homotopy equation:

[TABLE]

Remark 4.4*.*

Note that the map $R$ will play a crucial in our calculations related to the torsion. By a similar argument as the one used to prove (18), we can verify that

[TABLE]

In particular, for every $z_{0}\neq 0$, $R$ is a chain contraction map for the acyclic complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ [31], in the sense that $\text{Id}=(d^{\nabla}+R)^{2}$ and $R^{2}=0$ for that complex.

4.3. Proof of Theorem 2.1

We can now give the proof of Theorem 2.1. Thanks to (16), the induced maps on the cohomology are well defined. Suppose now that $\psi$ is a cocycle in $\Omega^{k}(M,\mathcal{E})$ which verifies $\pi_{0}^{(k)}(\psi)=0$. Thanks to (20), we deduce that

[TABLE]

From the second part of de Rham’s Theorem, we deduce that there exists $\omega\in\Omega^{k-1}(M,\mathcal{E})$ such that $\psi=d\omega$. Hence, $\psi$ is a coboundary. This shows the injectivity of the map. Consider now surjectivity and fix $\psi$ a cocycle in $\operatorname{Ker}(\mathcal{L}_{V}^{(k)})^{m_{k}(0)}$. From the first of de Rham’s Theorem (and more precisely Remark 4.3), there exists a cocycle $\omega$ in $\Omega^{k}(M,\mathcal{E})$ such that $\psi-\omega$ belongs to $d^{\nabla}\mathcal{H}_{k-1}^{sm+n-(k-1)}(M,\mathcal{E})$. Using (20) one more time, we can write

[TABLE]

Hence, we deduce that $\psi-\pi_{0}^{(k)}(\omega)$ is in the range of $\mathcal{H}^{sm+n-k}_{k-1}(M,\mathcal{E})$ under $d^{\nabla}$. Using the second part of de Rham’s Theorem, there exists $\tilde{\omega}$ in $\mathcal{H}^{sm+n-k}_{k-1}(M,\mathcal{E})$ such that

[TABLE]

Recall that the Pollicott-Ruelle spectrum is independent of the choice of the anisotropic Sobolev space [18, Th. 1.5]. In particular, the resonant state associated with $z_{0}=0$ are the same for the spaces $\mathcal{H}_{\bullet}^{sm+n-\bullet}$ and $\mathcal{H}_{\bullet}^{sm+n-\bullet-1}$. Hence, applying the spectral projector $\pi_{0}^{(k)}$ and implementing relation (16), we find that

[TABLE]

which concludes the proof of the surjectivity.

4.4. Morse type inequalities

In this paragraph, we will explain how to prove Corollary 2.3. The proof of Corollary 2.2 follows from a standard argument on finite dimensional cohomological complexes and we omit it – see [10, Sect. 8] for a brief reminder. In the case of Corollary 2.3, we remind that, in the case where $V$ is a Morse-Smale vector field which is $\mathcal{C}^{\infty}$-diagonalizable, the main result of [12] (see paragraphs 5.1.1 and 5.1.2 for a brief reminder) implies that the dimension of $C^{k}_{\nabla,V}(0)$ is equal to

[TABLE]

Thus, under the additional assumption that $V$ is $\mathcal{C}^{\infty}$-diagonalizable, we can conclude the proof of the Corollary 2.3 thanks to Corollary 2.2. Fix now a general Morse-Smale vector field $V_{0}$ on $M$. Knowing that the set of Morse-Smale vector fields is open inside the set of smooth vector fields, we can deduce that, for every $V$ in a small neighborhood of $V_{0}$, the vector field is still Morse-Smale. Thanks to the Sternberg-Chen Theorem [34, 48] (see also [11, Appendix] for a brief reminder), we know that the assumption of being $\mathcal{C}^{\infty}$-diagonalizable is satisfied as soon as all the eigenvalues of the linearized systems satisfies certain nonresonance assumptions and as soon as they are distinct. In particular, all these conditions are satisfied by a dense subset of vector fields and we can find $V$ arbitrarily close to $V_{0}$ which is Morse-Smale and $\mathcal{C}^{\infty}$-diagonalizable. In order to conclude, we observe that, for a small enough perturbation, we do not modify the number of critical elements of the flow of index $k$ and every closed orbit stays in the same homotopy class. In particular, all the terms in the sum defining the upper bound of corollary 2.3 are equal for $V$ and $V_{0}$.

4.5. Trace asymptotics and spectral determinants

Let us now draw some nice spectral interpretation of these results. Fix $U$ a bounded open set in $\mathbb{C}$. We define

[TABLE]

Note that, as the spectrum is discrete, this defines a finite sum. We now write

[TABLE]

Thanks to Remark 4.4, we can note that $Q=d^{\nabla}+R$ defines an operator which exchanges the chiralities and that, for $z_{0}\neq 0$, $Q$ is an isomorphism on $C_{V,\nabla}^{\text{even}}(z_{0})\oplus C_{V,\nabla}^{\text{odd}}(z_{0})$ (as $Q^{2}=\text{Id}$). In particular,

[TABLE]

Aguing as in [10, Sect. 7], we can then derive from the case of equality in Corollary 2.2:

Corollary 4.5**.**

Suppose the assumptions of Theorem 2.1 are satisfied. Then, for every bounded open set $U$ in $\mathbb{C}$ containing [math] and for every $z\in\mathbb{C}^{*}$,

[TABLE]

and, for all $t>0$,

[TABLE]

4.6. Poincaré duality

Consider now the dual complex associated with the vector field $-V$ and the connection $\nabla^{\dagger}$, i.e.

[TABLE]

From (12) and from section 3, we know that the operator $-\mathcal{L}_{-V,\nabla^{\dagger}}^{(k)}$ can be identified with the dual of $-\mathcal{L}_{V,\nabla}^{(n-k)}.$ These two complexes are dual to each other via the duality bracket between $\mathcal{H}^{m+k}_{k}(M,\mathcal{E})$ and $\mathcal{H}_{n-k}^{-(m+k)}(M,\mathcal{E}^{\prime})$. In other words, we have, for every $\psi_{1}$ in $\text{Ker}(\mathcal{L}_{V,\nabla}^{(k)})^{m_{k}(0)}$ and every $\psi_{2}$ in $\text{Ker}(\mathcal{L}_{-V,\nabla^{\dagger}}^{(n-k)})^{m_{k}(0)}$,

[TABLE]

We can then define a Poincaré map:

[TABLE]

By a classical argument for complex which are dual to each other – see [10, Paragraph 8.5] for a brief reminder, we can then deduce that $\mathcal{P}_{0}^{(k)}$ induces an isomorphism between the $k$-th cohomological group of $(\operatorname{Ker}(\mathcal{L}_{V,\nabla}^{(*)})^{m_{*}(0)},d^{\nabla})$ and the dual of the $(n-k)$-th cohomological group of $(\operatorname{Ker}(\mathcal{L}_{-V,\nabla^{\dagger}}^{(*)})^{m_{*}(0)},d^{\nabla^{\dagger}})$. Combining this with Theorem 2.1, one can recover the following version of Poincaré duality:

[TABLE]

5. Koszul homological complex

As a warm-up for the calculation of the spectral torsion, we start with some consideration on the following homological complex:

[TABLE]

This complex will be refered as the Morse-Smale-Koszul complex and we will verify that, in the case of Morse-Smale vector fields, its homology counts the number of critical points in each degree. More precisely, we will prove:

Proposition 5.1**.**

Suppose that $V$ is a Morse-Smale vector field which is $\mathcal{C}^{\infty}$-diagonalizable and that $\nabla$ preserves a hermitian structure on $\mathcal{E}$ (which is of dimension $N$).

Then, the dimension of the homology of degree $k$ of $(C^{\bullet}_{V,\nabla},\iota_{V})$ is equal to $Nc_{k}(V)$, where $c_{k}(V)$ is the number of critical points such that $\operatorname{dim}W^{s}(\Lambda)=k$.

This proposition proves999This follows from Corollary 2.3. Theorem 2.4 and it generalizes the results of [10, Sect. 8.6] which were only valid for gradient flows and for the trivial bundle $M\times\mathbb{C}$. We refer to [10] for more details and examples on Koszul complexes. Recall in particular that, for any smooth vector field $V$ with nondegenerate critical points, the homology of degree [math] of $(\Omega^{\bullet}(M,\mathcal{E}),\iota_{V})$ is equal to the total number of critical points while the homology in higher degree vanishes. Here, the Morse-Smale-Koszul complex allows to compute something slightly more precise as its homology allows to count the critical points of any degree.

In all this section, we will suppose that $V$ is a Morse-Smale vector field which is $\mathcal{C}^{\infty}$-diagonalizable and that $\nabla$ preserves a Hermitian structure on $\mathcal{E}$.

5.1. Spectrum of Morse-Smale flows on the imaginary axis

Let us first use our assumptions that the flow is smoothly diagonalizable in order to put it into a normal form near the critical elements $\Lambda$ of the vector field $V$, namely its closed orbits and critical points. Thanks to our results from [12, Sect. 7], we can write down the local expression of the Pollicott-Ruelle resonant states in this system of normal coordinates.

5.1.1. Critical points

Suppose that $\Lambda$ is a critical point of $V$ of index $k$, i.e. such that $\text{dim}\ W^{s}(\Lambda)=k$. Thanks to the fact that the flow is supposed to be smoothly diagonalizable, we can find a system of smooth coordinates $(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ near $\Lambda$ such that the flow $\varphi^{t}$ induced by $V$ can be written as follows:

[TABLE]

where $A_{s}^{\Lambda}$ (resp. $A_{u}^{\Lambda}$) belongs to $M_{k}(\mathbb{R})$ (resp. $M_{n-k}(\mathbb{R})$), is diagonalizable (in $\mathbb{C}$) and has all its eigenvalues on the half plane $\text{Re}(z)<0$ (resp. $\text{Re}(z)>0$). In particular, the local stable manifold of $\Lambda$ corresponds to the set $y=0$ while the unstable manifold is given by $x=0$. Near $\Lambda$, we can also construct a moving frame $(\mathbf{c}_{1}^{\Lambda},\ldots,\mathbf{c}_{N}^{\Lambda})$ of $\mathcal{E}$ such that $\nabla c_{j}^{\Lambda}=0$ for every $0\leq j\leq N$ – see e.g. [30, Th. 12.25]. According to [12, Sect. 7], one can associate to every such critical element of $V$ a family of $N$ Pollicott-Ruelle resonant states of degree $k$ associated with the resonance $z_{0}=0$. We denote them by

[TABLE]

and their local form near $\Lambda$ is given, for every $1\leq j\leq N$ by

[TABLE]

Moreover, one knows that the support of $U_{j}^{\Lambda}$ is equal to $\overline{W^{u}(\Lambda)}$

5.1.2. Closed orbits

Suppose that $\Lambda$ is a closed orbit of $V$ of index $k$, i.e. such that $\text{dim}\ W^{s}(\Lambda)=k$. The situation is now slightly more complicated. We denote by $\mathcal{P}_{\Lambda}$ the minimal period of the closed orbit. Thanks to the hypothesis that $V$ is $\mathcal{C}^{\infty}$-diagonalizable, we can find a system of smooth coordinates $(x,y,\theta)\in\mathbb{R}^{k-1}\times\mathbb{R}^{n-k}\times(\mathbb{R}/\mathcal{P}_{\Lambda}\mathbb{Z})$ near $\Lambda$ such that the flow can be put under the following normal form

[TABLE]

where

• •

$A$ is of the form $\text{diag}(A_{s}^{\Lambda},A_{u}^{\Lambda})$ with $A_{s}^{\Lambda}$ (resp. $A_{u}^{\Lambda}$) belonging to $M_{k-1}(\mathbb{R})$ (resp. $M_{n-k}(\mathbb{R})$) and diagonalizable (in $\mathbb{C}$). Moreover, $A_{s}^{\Lambda}$ (resp. $A_{u}^{\Lambda}$) has all its eigenvalues on the half plane $\text{Re}(z)<0$ (resp. $\text{Re}(z)>0$).

• •

$P(\theta)$ is $2\mathcal{P}_{\Lambda}$-periodic. Moreover, it satisfies $P(0)=\text{Id}_{\mathbb{R}^{n-1}}$, $J_{\Lambda}:=P(\mathcal{P}_{\Lambda})=\text{diag}(\pm 1)$ and $P(\theta+\mathcal{P}_{\Lambda})=J_{\Lambda}P(\theta)$.

The manifold $W^{s}(\Lambda)$ (and thus $W^{u}(\Lambda)$) is orientable if and only if the determinant of $J_{\Lambda}$ restricted to $\mathbb{R}^{k-1}\oplus\{0\}$ is equal to $1$. We set $\varepsilon_{\Lambda}=0$ whenever $\Lambda$ is orientable and $\varepsilon_{\Lambda}=1/2$ otherwise. It is related to the twisting index by the identity $\Delta_{\Lambda}=e^{2i\pi\varepsilon_{\Lambda}}$. In this system of coordinates, $\Lambda$ is given by the set $\{(0,0),\theta)\}$, the stable manifold by $\{P(\theta)(x,0),\theta)\}$ and the unstable one by $\{P(\theta)(0,y),\theta)\}$. As for critical points, we can construct a moving frame $(\mathbf{c}_{1}^{\Lambda},\ldots,\mathbf{c}_{N}^{\Lambda})$ of $\mathcal{E}$ such that there exists $\gamma_{1}^{\Lambda},\ldots,\gamma_{N}^{\Lambda}$ in $[0,1)$ for which one has $\nabla\mathbf{c}_{j}^{\Lambda}=\frac{2i\pi\gamma_{j}^{\Lambda}}{\mathcal{P}_{\Lambda}}\mathbf{c}_{j}^{\Lambda}d\theta$ for every $0\leq j\leq N$ – see e.g. [12, Sect. 4] for details. Here, we note that $\left(e^{2i\pi\gamma_{j}^{\Lambda}}\right)_{j=1}^{N}$ are the eigenvalues of the monodromy matrix $M_{\mathcal{E}}(\Lambda)$ for the parallel transport around $\Lambda$.

Following one more time our previous work [12, Sect. 7], we can associate families of Pollicott-Ruelle resonant states of degree $k-1$ and $k$. They are now indexed by $(p,j)\in\mathbb{Z}\times\{1,\ldots,N\}$ and we denote them by $U_{j,p}^{\Lambda}$ and $\tilde{U}_{j,p}^{\Lambda}$. Their local form near $\Lambda$ is given by

[TABLE]

and

[TABLE]

For a fixed $(j,p)$, the resonant states $U_{j,p}^{\Lambda}$ and $\tilde{U}_{j,p}^{\Lambda}$ are associated with the resonance

[TABLE]

and the support of $U_{j,p}^{\Lambda}$ is equal to $\overline{W^{u}(\Lambda)}$

5.1.3. Resonant states on the imaginary axis

Finally, the main result of [12, Sect. 7] states that the sets

[TABLE]

[TABLE]

and

[TABLE]

generate all the Pollicott-Ruelle resonances of degree $k$ associated with a resonance lying on the imaginary axis. Moreover, all these states are linearly independent. We also recall from [12] that each of these states verifies $(\mathcal{L}_{V,\nabla}-z_{0})U^{\Lambda}=0$ on $W^{u}(\Lambda)$ but that $U^{\Lambda}$ is only a generalized eigenvalue equation $(\mathcal{L}_{V,\nabla}-z_{0})^{m}U^{\Lambda}=0$ on $M$.

Remark 5.2*.*

We note that there is no resonant state of degree $k$ associated with $z_{0}=0$ if the Morse-Smale flow is nonsingular and if $\Delta_{\Lambda}$ is not an eigenvalue of $M_{\mathcal{E}}(\Lambda)$ for every closed orbit $\Lambda$.

5.2. Proof of Proposition 5.1

In order to prove Proposition 5.1, we start by fixing some critical point $\Lambda$ of index $k$ and $1\leq j\leq N$. Applying the contraction operator $\iota_{V}$, we find that $\iota_{V}(U_{j}^{\Lambda})$ is equal to [math] near $\Lambda$. Thus, as $\iota_{V}(U_{j}^{\Lambda})$ verifies $\mathcal{L}_{V,\nabla}\circ\iota_{V}(U_{j}^{\Lambda})=0$ on $W^{u}(\Lambda)$ and as it is supported on $\overline{W^{u}(\Lambda)}$, we find that $\iota_{V}(U_{j}^{\Lambda})$ is supported on $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$. As $(C^{\bullet}_{V,\nabla},\iota_{V})$ is an homological complex, we know that $\iota_{V}(U_{j}^{\Lambda})$ will be a linear combination of elements inside $C_{V,\nabla}^{k-1}(0)$. According to the above discussion, such elements are supported either by the closure of unstable manifolds of dimension $n-k+1$ (in the case of a critical point) or by the closure of unstable manifolds of dimension $n-k+1$ or $n-k+2$ (in the case of closed orbits). Yet, according to Smale [44, Lemma 3.1] – see [11] for a brief reminder, such unstable submanifolds cannot be contained in the closure of $W^{u}(\Lambda)$. It means that the linear combination is equal to [math], i.e. $\iota_{V}(U_{j}^{\Lambda})=0$.

Fix now a closed orbit $\Lambda$ such that101010Hence, $\text{dim}\ W^{u}(\Lambda)=n-k+1$. $\text{dim}\ W^{s}(\Lambda)=k$ and $(j,p)$ such that $U_{j,p}^{\Lambda}$ and $\tilde{U}_{j,p}^{\Lambda}$ are associated with the resonance [math]. Let us first compute $\iota_{V}(U_{j,p}^{\Lambda})$. As before, we find, by using the local form of the resonant state and the eigenvalue equation on $W^{u}(\Lambda)$, that $\iota_{V}(U_{j,p}^{\Lambda})$ is supported on $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$ which is the union of unstable manifolds whose dimension is $\leq n-k+1$ thanks to Smale’s Lemma [44, Lemma 3.1]. Recall also that if $W^{u}(\Lambda^{\prime})\subset\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$ with $\text{dim}\ W^{u}(\Lambda^{\prime})=n-k+1$, then $\Lambda^{\prime}$ is a closed orbit. As we have a homological complex, $\iota_{V}(U_{j,p}^{\Lambda})$ is an element in $C^{k-2}_{V,\nabla}(0)$. From the description of the resonant state we just gave, we can verify as above that this subspace is generated by critical elements associated with unstable manifolds of dimension $\geq n-k+2$. This implies that $\iota_{V}(U_{j,p}^{\Lambda})=0$. It remains to compute $\iota_{V}(\tilde{U}_{j,p}^{\Lambda})$ which is now an element in $C^{k-1}_{V,\nabla}(0)$. Contrary to the previous cases, $\iota_{V}(\tilde{U}_{j,p}^{\Lambda})$ is supported by $\overline{W^{u}(\Lambda)}$ and it is equal to $U_{j,p}^{\Lambda}$ near $\Lambda$. Hence, from the eigenvalue equation, $\iota_{V}(\tilde{U}_{j,p}^{\Lambda})-U_{j,p}^{\Lambda}$ is now supported by $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$ which is a union of unstable manifold of dimension $\leq n-k+1$. Thus, using Smale’s result one more time, we can conclude by similar arguments that

[TABLE]

In any degree, we can finally conclude that

[TABLE]

where the kernel and the range are understood as for operators acting on $C^{\bullet}_{V,\nabla}(0)$.

5.3. The case $z_{0}\neq 0$

As for the case of the coboundary operator, we can verify that the homology of the complex $(C^{\bullet}_{V,\nabla}(z_{0}),\iota_{V})$ is trivial for any $z_{0}\neq 0$. If we restrict to the case where $z_{0}\in i\mathbb{R}^{*}$, we can in fact be slightly more precise. In fact, we should emphasize that, in the second part of the argument, we did not use the fact that $z_{0}$ is equal to [math] and the argument is in fact valid for any resonance $z_{0}$ lying on the imaginary axis $i\mathbb{R}^{*}$. More precisely, for every $(\Lambda,j,p)$ such that

[TABLE]

we will have $\iota_{V}(U_{j,p}^{\Lambda})=0$ and

[TABLE]

In particular, we have, for $z_{0}\in i\mathbb{R}^{*}$ and for all $0\leq k\leq n$,

[TABLE]

6. Torsion of Pollicott-Ruelle resonant states

In this section, we will discuss the topological properties of the resonant states lying on the imaginary axis together with the proof of Theorem 2.6 which is related to the spectral zeta functions associated with the Pollicott-Ruelle resonances on the imaginary axis.

In the present section, we first present a new definition of torsion for the infinite dimensional chain complex of Pollicott–Ruelle resonances states corresponding by the spectrum on the imaginary axis which is directly related to Theorem 2.6. More specifically, we know from paragraph 4.2 that, for every $z_{0}\in i\mathbb{R}^{*}$,

[TABLE]

defines an acyclic cohomological complex. Moreover, we constructed in [12] a natural basis for this complex – see paragraph 5.1.2 for a brief reminder. Recall now that, as soon as we are given a cohomological complex with a preferred basis, we can compute its torsion which is just a certain determinant associated with $d^{\nabla}$ and computed in this basis. Hence, we will first compute the torsion of each individual complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ for a purely imaginary resonance $z_{0}$ in the basis of paragraph 5.1.2. Then, mimicking standard procedures used in spectral theory and QFT, we will define the torsion of the infinite dimensional complex $\bigoplus_{z_{0}\in i\mathbb{R}}(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ as the zeta regularized infinite product of the individual torsions in order to compute the torsion of the complex carried by the entire imaginary axis. Finally, we will prove Theorem 2.6 which is in fact related to the above regularized torsion.

From this point on, we will suppose that $V$ is a Morse-Smale vector field which is $\mathcal{C}^{\infty}$-diagonalizable and that $\nabla$ preserves a Hermitian structure on $\mathcal{E}$.

6.1. Torsion of individual complexes $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$

As was already pointed out, the finite dimensional complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ is acyclic as soon as $z_{0}\neq 0$. Recall also from paragraph 5.1.2 that it has a preferred basis which is associated to the unstable manifolds of the closed orbits of the vector field $V$. Thus, given $z_{0}\neq 0$ and its preferred basis of Pollicott-Ruelle resonant states, it is natural to compute the torsion of the complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ [22, 31] and we shall start with this preliminary calculation when $z_{0}\in i\mathbb{R}^{*}$.

Using the conventions of paragraph 5.1.2, we have, for every $0\leq k\leq n$,

[TABLE]

which can be splitted as the direct sum

[TABLE]

Recall from remark 4.4 that $R$ is a chain contraction map for the complex $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$. Then, according to [22, p. 30], the torsion of this finite dimensional complex (with respect to its preferred basis) is equal to

[TABLE]

where the determinant is computed in the basis $(U_{j,p}^{\Lambda},\tilde{U}_{j,p}^{\Lambda})_{j,p,\Lambda}$. In order to make this computation, we will to prove a preliminary result:

Lemma 6.1**.**

Let $(\Lambda,j,p)$ be such that

[TABLE]

Then, the supports of $\mathcal{L}_{V,\nabla}^{-1}(U_{j,p}^{\Lambda})$ and $\mathcal{L}_{V,\nabla}^{-1}(\tilde{U}_{j,p}^{\Lambda})$ are equal to $\overline{W^{u}(\Lambda)}$.

Proof.

Recall from paragraph 5.1.2 that the supports of $U_{j,p}^{\Lambda}$ and $\tilde{U}_{j,p}^{\Lambda}$ are equal to $\overline{W^{u}(\Lambda)}$. According to Smale’s Theorem A.1, $\overline{W^{u}(\Lambda)}$ is the union of unstable manifolds $W^{u}(\Lambda^{\prime})$ with $\text{dim}W^{u}(\Lambda^{\prime})\leq\text{dim}W^{u}(\Lambda)$. As $\mathcal{L}_{V,\nabla}^{(k)}$ preserves $C^{k}_{V,\nabla}(z_{0})$ and as the support of $\mathcal{L}_{V,\nabla}U_{j,p}^{\Lambda}$ is contained in $\overline{W^{u}(\Lambda)}$, we can write

[TABLE]

where $\leqq$ is the partial order relation of Smale’s Theorem A.1 and where $\alpha_{*}$ are complex numbers. Similarly,

[TABLE]

where $\alpha_{*}$ and $\tilde{\alpha}_{*}$ are complex numbers. From the explicit expressions of $U_{j,p}^{\Lambda}$ and $\tilde{U}_{j,p}^{\Lambda}$, we can already observe that $\alpha_{j,p,\Lambda}=\tilde{\alpha}_{j,p,\Lambda}=-z_{0}$. As $z_{0}\neq 0$, we can also apply $\mathcal{L}_{V,\nabla}^{-1}$ to both sides of these equalities and we find that

[TABLE]

and

[TABLE]

Thus, determining the support of $\mathcal{L}_{V,\nabla}^{-1}(U_{j,p}^{\Lambda})$ and of $\mathcal{L}_{V,\nabla}^{-1}(\tilde{U}_{j,p}^{\Lambda})$ follows from the fact that we are able to compute the support of their analogues for $\Lambda^{\prime}\leqq\Lambda$ with $\text{dim}W^{s}(\Lambda^{\prime})=k$ or $k+1$. Recall that the partial order relation $\leqq$ is defined in Theorem A.1 and that all the sums are implicitely over indices $(\Lambda^{\prime},j^{\prime},p^{\prime})$ such that

[TABLE]

If we consider the smallest such $\Lambda^{\prime}$ and if we are able to prove that the corresponding supports are equal to $\overline{W^{u}(\Lambda^{\prime})}$, then the lemma will follow from an induction argument on $\text{dim}W^{u}(\Lambda^{\prime})$. Fix now $\Lambda^{\prime}\leqq\Lambda$ which is minimal and $(j^{\prime},p^{\prime})$ such that (23) is satisfied. As $\Lambda^{\prime}$ is minimal, we can decompose $\mathcal{L}_{V,\nabla}U_{j^{\prime},p^{\prime}}^{\Lambda^{\prime}}$ and $\mathcal{L}_{V,\nabla}\tilde{U}_{j^{\prime},p^{\prime}}^{\Lambda^{\prime}}$ but there is now no remainder term, i.e. one has

[TABLE]

Applying $\mathcal{L}_{V,\nabla}^{-1}$ and using the fact $z_{0}\neq 0$, we can conclude that the supports of $\mathcal{L}_{V,\nabla}^{-1}U_{j^{\prime},p^{\prime}}^{\Lambda^{\prime}}$ and $\mathcal{L}_{V,\nabla}^{-1}\tilde{U}_{j^{\prime},p^{\prime}}^{\Lambda^{\prime}}$ are equal to $\overline{W^{u}(\Lambda^{\prime})}$. By induction, we can then recover that the supports of $\mathcal{L}_{V,\nabla}^{-1}(U_{j,p}^{\Lambda})$ and of $\mathcal{L}_{V,\nabla}^{-1}(\tilde{U}_{j,p}^{\Lambda})$ are equal to $\overline{W^{u}(\Lambda)}$. ∎

We can now compute the torsion $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ using formula (22). For that purpose, we fix $\Lambda$ to be a closed orbit of the flow such that $\text{dim}\ W^{s}(\Lambda)=k$. Suppose also that $(j,p)$ verifies the equality

[TABLE]

On the one hand, if $k$ is even, we would like to express $(d^{\nabla}+R)(\tilde{U}^{\Lambda}_{j,p})$ in the preferred basis of $C^{\text{odd}}_{V,\nabla}(z_{0})$. On the other hand, if $k$ is odd, we are interested in the expression of $(d^{\nabla}+R)(U^{\Lambda}_{j,p})$. In order to compute these determinants, it will be convenient to order these basis which are indexed by $(\Lambda,j,p)$ according to Smale’s partial order on the unstable manifolds. More precisely, we will order them in such a way that $(\Lambda^{\prime},j^{\prime},p^{\prime})$ is less than $(\Lambda,j,p)$ whenever $\Lambda^{\prime}\leqq\Lambda$.

6.1.1. The case $k$ even

Let us start with the case $k\equiv 0\ \text{mod}\ 2$. Recall that near $\Lambda$, $\tilde{U}^{\Lambda}_{j,p}$ is of the form

[TABLE]

First, from this expression, we can verify that $d^{\nabla}\tilde{U}_{j,p}^{\Lambda}$ is equal to [math] in a neighborhood of $\Lambda$. Moreover, as it satisfies $\mathcal{L}_{V,\nabla}d^{\nabla}\tilde{U}_{j,p}^{\Lambda}$ is equal to [math] on $W^{u}(\Lambda)$ and as $d^{\nabla}\tilde{U}_{j,p}^{\Lambda}$ is supported on $\overline{W^{u}(\Lambda)}$, we can deduce that $d^{\nabla}\tilde{U}_{j,p}^{\Lambda}$ is supported on $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$. From Smale’s Theorem [44] and from the fact that $(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla})$ is a cohomological complex, we can deduce that

[TABLE]

where we only sum over the $(\Lambda^{\prime},j^{\prime},p^{\prime})$ satisfying

[TABLE]

Now, we can compute $R\tilde{U}_{j,p}^{\Lambda}$. First, we write that $\mathcal{L}_{V,\nabla}(\tilde{U}_{j,p}^{\Lambda})$ belongs to $C^{k}_{V,\nabla}(z_{0})$ and that its support is contained in $\overline{W^{u}(\Lambda)}$. Hence, one can write

[TABLE]

From the expression of $\tilde{U}_{j,p}^{\Lambda}$ near $\Lambda$, we know that $\alpha_{\Lambda,j,p}=-z_{0}$. Hence, we find

[TABLE]

Recall now that $R=\iota_{V}\mathcal{L}_{V,\nabla}^{-1}$ when it acts on $C^{\bullet}_{V,\nabla}(z_{0})$. Hence, one has

[TABLE]

Arguing as in the proof of Proposition 5.1 – see paragraph 5.3, we can verify that $\iota_{V}(\tilde{U}_{j,p}^{\Lambda})$ is equal to $U_{j,p}^{\Lambda}$ plus some remainder term which is carried on $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$. Hence to summarize, one finds that

[TABLE]

with $T_{j,p}^{\Lambda}$ belonging to $C^{\text{odd}}_{V,\nabla}(z_{0})$ and supported in $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$.

6.1.2. The case $k$ odd

Suppose now that $k\equiv 1\ \text{mod}\ 2$. Near $\Lambda$, $U^{\Lambda}_{j,p}$ is of the form

[TABLE]

From paragraph 5.3, we know that $\iota_{V}(U_{j,p}^{\Lambda})=0$. Hence, from the definition of $R$, one finds that $(d^{\nabla}+R)(U_{j,p}^{\Lambda})=d^{\nabla}(U_{j,p}^{\Lambda})$. Applying $d^{\nabla}$ to $U_{j,p}^{\Lambda}$, we find that, in a neighborhood of $\Lambda$, it is equal to $z_{0}\tilde{U}_{j,p}^{\Lambda}$ (recall from paragraph 5.1.2 that $\nabla\mathbf{c}_{j}^{\Lambda}=\frac{2i\pi\gamma_{j}^{\Lambda}}{\mathcal{P}_{\Lambda}}\mathbf{c}_{j}^{\Lambda}$). The current $d^{\nabla}(U_{j,p}^{\Lambda})-z_{0}\tilde{U}_{j,p}^{\Lambda}$ belongs to $C^{k}_{V,\nabla}(z_{0})$, it is supported inside $\overline{W^{u}(\Lambda)}$ and it identically vanishes near $\Lambda$. Hence, by propagation, it is supported inside $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$. Equivalently, one has

[TABLE]

with $T_{j,p}^{\Lambda}$ belonging to $C^{\text{odd}}_{V,\nabla}(z_{0})$ and supported in $\overline{W^{u}(\Lambda)}-W^{u}(\Lambda)$.

6.1.3. Conclusion

Combining (22) with (24) and 25, one finally finds the following expression for the torsion:

[TABLE]

where $(*)$ means that we take the product over the triples $(\Lambda,j,p)$ satisfying

[TABLE]

In other words, the torsion $\tau\left(C_{V,\nabla}^{\bullet}(z_{0}),d^{\nabla}\right)$ is of the form $|z_{0}|^{(-1)^{n}m(z_{0})}$ with

[TABLE]

In a more spectral manner, we have, using the results of paragraphs 5.3,

[TABLE]

Observe that, if we set

[TABLE]

then one easily finds that

[TABLE]

which motivates the upcoming definitions.

6.2. Another zeta function associated to the flow.

For every $0\leq k\leq n$, we define the infinite dimensional vector space of Pollicott-Ruelle resonant states:

[TABLE]

This induces an infinite dimensional complex $(C^{\bullet}_{V,\nabla}(i\mathbb{R}^{*}),d^{\nabla})$ and we define the following zeta function:

[TABLE]

where the sum runs over the closed orbits $\Lambda$ of the flow, $1\leq j\leq N$ and $p\in\mathbb{Z}$ with the assumption that $p+\varepsilon_{\Lambda}+\gamma_{j}^{\Lambda}\neq 0$. In a more spectral manner, this can be written

[TABLE]

We shall explain below by classical arguments from Hodge theory [31] that, up to the modulus, this zeta function is related to the spectral zeta function $\zeta_{RS}$ from corollary 2.8. Yet, we emphasize that it appears here really from the computation of the torsion of an acyclic complex in a preferred basis and not just as a reproduction of Ray-Singer’s definition for the Laplace operator. In any case, this function is well defined for $\text{Re}(s)>1$ and we aim at describing its meromorphic extension to $\mathbb{C}$. In particular, motivated by (28), we would like to define the torsion of the cohomological complex $(C^{\bullet}_{V,\nabla}(i\mathbb{R}^{*}),d^{\nabla})$ as $e^{-\zeta_{V,\nabla}^{\prime}(0)}$ provided that it makes sense.

Let us now study the meromorphic continuation of the zeta functions we have just defined. For that purpose, we write

[TABLE]

Hence, equivalently, it amounts to understand the meromorphic continuation of

[TABLE]

for every closed orbit and for every $1\leq j\leq N$. This can in fact easily be rewritten in terms of Hurwitz zeta functions and of Riemann zeta functions. Recall that $\gamma_{j}^{\Lambda}\in[0,1)$ and that $\varepsilon_{\Lambda}\in\{0,1/2\}$ (the value depending on the orientability of $W^{u}(\Lambda)$). In particular, there exists a unique $q_{j}^{\Lambda}\in(0,1]$ such that $\varepsilon_{\Lambda}+\gamma_{j}^{\Lambda}$ is equal to $q_{j}^{\Lambda}$ modulo $1$. Introduce now the so-called Hurwitz zeta function [3, Ch. 12]:

[TABLE]

Hence, for $q_{j}^{\Lambda}\neq 1$, one has

[TABLE]

while, for $q_{j}^{\Lambda}=1$,

[TABLE]

which is nothing else than twice the Riemann zeta function. From [3, Th. 12.4-5], we can conclude that, for every $1\leq j\leq N$ and for every closed orbit $\Lambda$, $\xi_{j,\Lambda}(s)$ has a meromorphic extension to $\mathbb{C}$ which is analytic except for a simple pole at $s=1$ (with residue $2$). In particular, $\zeta_{V,\nabla}(s)$ extends meromorphically to $\mathbb{C}$ with a unique pole at $s=1$ which is simple and whose residue is given by

[TABLE]

6.3. The regularized torsion of the infinite dimensional complex $(C^{\bullet}_{V,\nabla}(i\mathbb{R}^{*}),d^{\nabla})$.

We define the regularized torsion of the infinite dimensional complex $(C^{\bullet}_{V,\nabla}(i\mathbb{R}^{*}),d^{\nabla})$ as follows:

[TABLE]

which depends implicitely on the choice of the basis from paragraph 5.1.2. Let us compute the contribution coming from the derivative of the zeta function. We have

[TABLE]

In order to compute the derivative at $s=0$, we shall come back to the expression in terms of Hurwitz zeta function. According to [3, Th. 12.13], we know that $\zeta(0,q)=\frac{1}{2}-q$. Hence, for $q_{j}^{\Lambda}=1$, one has $\xi_{j,\Lambda}(0)=-1$ while, for $q_{j}^{\Lambda}\neq 1$, $\xi_{j,\Lambda}(0)=0$. For the derivatives, one has $\xi_{j,\Lambda}^{\prime}(0)=-\ln(2\pi)$ if $q_{j}^{\Lambda}=1$ and $\xi_{j,\Lambda}^{\prime}(0)=-\ln(2\sin(\pi q_{j}^{\Lambda}))$ otherwise [49, p. 271] [35, p. 195].

Recall that we used the notations $\Delta_{\Lambda}=e^{2i\varepsilon_{\Lambda}\pi}$ and $M_{\mathcal{E}}(\Lambda)$ for the monodromy matrix around $\Lambda$. Also, we have set $q_{j}^{\Lambda}=\varepsilon_{\Lambda}+\gamma_{j}^{\Lambda}\ \text{mod}\ 1$ where $e^{2i\pi\gamma_{j}^{\Lambda}}$ are the eigenvalues of the monodromy matrix $M_{\mathcal{E}}(\Lambda)$. With these notations, we finally find

[TABLE]

and thus

[TABLE]

Note that if $\Delta_{\Lambda}$ is not an eigenvalue of $M_{\mathcal{E}}(\Lambda)$ for every closed orbit111111This is the assumption made in [22]., then we find

[TABLE]

which is equal to the Reidemester torsion (if $V$ is also nonsingular) thanks to the work of Fried in [22, Sect. 3]. Recall that the left hand side can be interpreted either in terms of the nonzero Pollicott-Ruelle resonances on the imaginary axis or in terms of the torsion of the corresponding resonant states. To summarize, we have shown

Proposition 6.2**.**

Suppose that the assumptions of Theorem 2.6 are satisfied. Then, one has

• •

$\zeta_{V,\nabla}(s)$* has a meromorphic extension to $\mathbb{C}$ with a unique pole at $s=1$ which is simple and whose residue equals*

[TABLE]

• •

one has

[TABLE]

where $(e^{2i\pi\gamma_{j}^{\Lambda}})_{j=1,\ldots,N}$ are the eigenvalues of $M_{\mathcal{E}}(\Lambda)$.

• •

if $V$ is nonsingular and $m_{\Lambda}=0$ for every closed orbit, then $e^{(-1)^{n}\zeta_{V,\nabla}^{\prime}(0)}$ is equal to the Reidemeister torsion of $(\mathcal{E},\nabla)$.

6.4. Twisted Fuller measures

In this paragraph, we prove Theorem 2.6. First of all, we rewrite the Fuller measure in terms of the dimension of the unstable manifolds of the critical elements and the twisting index $\Delta_{\Lambda}$ of the closed orbits, i.e.

[TABLE]

The Morse inequality from Corollary 2.3 turns out to be an equality in the case $k=n$, thus we find that $\sum_{\Lambda\ \text{fixed point}}(-1)^{\text{dim}\ W^{u}(\Lambda)}=\sum_{k}(-1)^{k}b_{k}(M,\mathcal{E})=\chi(M,\mathcal{E})$. Therefore the above formula can be rewritten in a more compact way as :

[TABLE]

More explicitely, we can also write the sum on the r.h.s of the above identity in terms of the eigenvalues $\left(e^{2i\pi\gamma_{j}^{\Lambda}}\right)_{j=1}^{N}$ of the monodromy matrices and the coefficients $\varepsilon_{\Lambda}$ such that $\Delta_{\Lambda}=e^{2i\pi\varepsilon_{\Lambda}}$ which yields :

[TABLE]

For $T>0$ and $\gamma\in\mathbb{R}$, recall that the Poisson summation formula implies that :

[TABLE]

We can now apply the above formula for $T=\mathcal{P}_{\Lambda}$ and $\gamma=\frac{\gamma_{j}^{\Lambda}+\varepsilon_{\Lambda}}{\mathcal{P}_{\Lambda}}$, which gives us :

[TABLE]

in the sense of distributions in $\mathcal{D}^{\prime}(\mathbb{R}_{+}^{*})$.

Finally, this quantity can be rewritten in a more spectral manner thanks to the results of paragraphs 5.2 and 5.3. More precisely,

[TABLE]

which concludes the proof of Theorem 2.6.

6.5. Fried’s torsion functions

It now remains to prove Corollary 2.8 related to the torsion function. Recall that we introduced in subsection 2.6 the following zeta function :

[TABLE]

which, by definition of the twisted Fuller measure, can be rewritten thanks to (30) as

[TABLE]

Recall that $\Gamma(s)^{-1}=s+o(s)$. Hence, if we differentiate $\zeta_{V,\nabla}^{\flat}$ with respect to $s$, then we find that, for $\text{Re}(z)$ large enough,

[TABLE]

where one has

[TABLE]

as soon as $\text{Re}(z)>0$ since $\|e^{-\mathcal{P}_{\Lambda}z}\Delta_{\Lambda}M_{\mathcal{E}}(\Lambda)\|\leqslant e^{-\mathcal{P}_{\Lambda}Re(z)}<1$ by unitarity of $\Delta_{\Lambda}M_{\mathcal{E}}(\Lambda)$.

Equivalently, this can be rewritten as

[TABLE]

Note that this expression is well defined for $\text{Re}(z)>0$ and since the torsion function $Z_{V,\nabla}$ was defined as $Z_{V,\nabla}(z)=e^{-\partial_{s}\zeta_{V,\nabla}^{\flat}(0,z)}$, we deduce the following identity relating the torsion function and the weighted Ruelle zeta function :

[TABLE]

as was stated in paragraph 2.6.

In the case where $V$ is a nonsingular Morse-Smale vector field, we recognize the torsion function introduced by Fried in [22, p. 51-53], also called twisted Ruelle zeta function. In any case, we can already verify from the exact expressions of the eigenvalues and their multiplicities (see paragraphs 5.1.1 and 5.1.2) that the poles and zeros of $Z_{V,\nabla}(z)$ are completely determined by the resonances on the imaginary axis.

Let us now come back to Corollary 2.8 and perform the Mellin transform on the right hand side of (31). We find that

[TABLE]

This is also equal to

[TABLE]

Remark 6.3*.*

This expression can be formally differentiated at $s=0$ yielding :

[TABLE]

from which we deduce the formal expression relating Fried’s torsion and some infinite product indexed by the resonances lying on the imaginary axis :

[TABLE]

Using our expression for the kernel of $\mathcal{L}_{V,\nabla}$, we now find that

[TABLE]

Now, in the spirit of Ray-Singer definition of analytic torsion, we can rewrite the second term in the right-hand side in a slightly different manner. As in the case of Hodge theory, we can observe that

[TABLE]

and that

[TABLE]

As in [31, paragraph 8.2 p. 53], this implies that

[TABLE]

as expected.

Remark 6.4*.*

Following Dyatlov and Zworski [14], we could also have defined the (twisted) dynamical zeta function as :

[TABLE]

which can be rewritten thanks to (30) as

[TABLE]

or equivalently

[TABLE]

In that case, we would find that the poles of this zeta function are contained inside the intersection of the Pollicott-Ruelle resonances with the imaginary axis.

Appendix A A brief reminder on Anosov and Morse-Smale flows

In this appendix, we briefly review some classical definitions and results from dynamical systems.

A.1. Morse-Smale flows

We say that $\Lambda\subset M$ is an elementary critical element if $\Lambda$ is either a fixed point or a closed orbit of $\varphi^{t}$. Such an element is said to be hyperbolic if the fixed point or the closed orbit is hyperbolic – see [1] or the appendix of [11] for a brief reminder. Following [45, p. 798], $\varphi^{t}$ is a Morse-Smale flow if the following properties hold:

1. (1)

the non-wandering set $\operatorname{NW}(\varphi^{t})$ is the union of finitely many elementary critical element $\Lambda_{1},\ldots,\Lambda_{K}$ which are hyperbolic, 2. (2)

for every $i,j$ and for every $x$ in $W^{u}(\Lambda_{j})\cap W^{s}(\Lambda_{i})$, one has 121212See appendix of [11] for the precise definition of the stable/unstable manifolds $W^{s/u}(\Lambda)$. $T_{x}M=T_{x}W^{u}(\Lambda_{j})+T_{x}W^{s}(\Lambda_{i})$.

We now briefly expose some important properties of Morse-Smale flows and we refer to [21, 37, 11] for a more detailed exposition on the dynamical properties of these flows. Under such assumptions, one can show that, for every $x$ in $M$, there exists a unique couple $(i,j)$ such that $x\in W^{u}(\Lambda_{j})\cap W^{s}(\Lambda_{i})$ (see e.g. Lemma 3.1 in [11]). In particular, the unstable manifolds $(W^{u}(\Lambda_{j}))_{j=1,\ldots,K}$ form a partition of $M$, i.e.

[TABLE]

The same of course holds for stable manifolds. One of the main feature of such flows is the following result which is due to Smale [44, 45]:

Theorem A.1** (Smale).**

Suppose that $\varphi^{t}$ is a Morse-Smale flow. Then, for every $1\leq j\leq K$, the closure of $W^{u}(\Lambda_{j})$ is the union of certain $W^{u}(\Lambda_{j^{\prime}})$. Moreover, if we say that $W^{u}(\Lambda_{j^{\prime}})\leqq W^{u}(\Lambda_{j})$ if $W^{u}(\Lambda_{j^{\prime}})$ is contained in the closure of $W^{u}(\Lambda_{j})$, then, $\leqq$ is a partial ordering. Finally if $W^{u}(\Lambda_{j^{\prime}})\leqq W^{u}(\Lambda_{j})$, then $\operatorname{dim}W^{u}(\Lambda_{j^{\prime}})\leq\operatorname{dim}W^{u}(\Lambda_{j}).$

The partial order relation on the collection of subsets $W^{u}(\Lambda_{j})_{j=1}^{K}$ defined above is called Smale causality relation. Following Smale, we define an oriented graph131313This diagram is the Hasse diagram associated to the poset $\left(W^{u}(\Lambda_{j})_{j=1}^{K},\leqq\right)$. $D$ whose $K$ vertices are given by $W^{u}(\Lambda_{j})_{j=1}^{K}$. Two vertices $W^{u}(\Lambda_{j}),W^{u}(\Lambda_{i})$ are connected by an oriented path starting at $W^{u}(\Lambda_{j})$ and ending at $W^{u}(\Lambda_{i})$ iff $W^{u}(\Lambda_{j})\leqq W^{u}(\Lambda_{i})$. Recall from the works of Peixoto [38] that Morse-Smale flows form an open and dense of all smooth vector fields on surfaces while it is an open set in higher dimensions [36].

In the constructions from [11, 12], we needed to make extra assumptions on our flows, namely that they are linearizable near every critical element $\Lambda_{i}$. More precisely, we fix $1\leq l\leq\infty$ and we say that the Morse-Smale flow is $\mathcal{C}^{l}$-linearizable if for every $1\leq i\leq k$, the following hold:

• •

If $\Lambda_{i}$ is a fixed point, there exists a $\mathcal{C}^{l}$ diffeomorphism $h:B_{n}(0,r)\rightarrow W$ (where $W$ is a small open neighborhood of $\Lambda_{i}$ and $B_{n}(0,r)$ is a small ball of radius $r$ centered at [math] in $\mathbb{R}^{n}$) and a linear map $A_{i}$ on $\mathbb{R}^{n}$ such that $V\circ h=dh\circ L$ where $V$ is the vector field generating $\varphi^{t}$ and where

[TABLE]

• •

If $\Lambda_{i}$ is closed orbit of period $\mathcal{P}_{\Lambda_{i}}$ if there exists a $\mathcal{C}^{l}$ diffeomorphism $h:B_{n-1}(0,r)\times\mathbb{R}/(\mathcal{P}_{\Lambda_{i}}\mathbb{Z})\rightarrow W$ (where $W$ is a small open neighborhood of $\Lambda_{i}$ and $r>0$ is small) and a smooth map $A:\mathbb{R}/(\mathcal{P}_{\Lambda_{i}}\mathbb{Z})\rightarrow M_{n-1}(\mathbb{R})$ such that $V\circ h=dh\circ L$ with

[TABLE]

In other words, the flow can be put into a normal form in a certain chart of class $\mathcal{C}^{l}$. We shall say that a Morse-Smale flow is $\mathcal{C}^{l}$-diagonalizable if it is $\mathcal{C}^{k}$-linearizable and if, for every critical element $\Lambda$, either the linearized matrix $A\in GL_{n}(\mathbb{R})$ or the monodromy matrix $M$ associated with $A(\theta)$ is diagonalizable in $\mathbb{C}$. Such properties are satisfied as soon as certain (generic) non resonance assumptions are made on the Lyapunov exponents thanks to the Sternberg-Chen Theorem [34, 48]. We refer to the appendix of [11] for a detailed description of these nonresonant assumptions.

A.2. Anosov flows

We say that a flow $\varphi^{t}:M\rightarrow M$ $\Lambda$ is of Anosov type [2, 1] if there exist $C>0$ and $\chi>0$ and a family of spaces $E_{u}(\rho),E_{s}(\rho)\subset T_{\rho}\mathbb{M}$ (for every $\rho$ in $M$) satisfying the following properties, for every $\rho$ in $M$ and for every $t\geq 0$,

1. (1)

$T_{\rho}\mathbb{M}=\mathbb{R}V(\rho)\oplus E_{u}(\rho)\oplus E_{s}(\rho)$ with $V(\rho)=\frac{d}{d}(\varphi^{t}(\rho))\rceil_{t=0}$, 2. (2)

$d_{\rho}\varphi^{t}E_{u/s}(\rho)=E_{u/s}(\varphi^{t}(\rho))$, 3. (3)

for every $v$ in $E_{u}(\rho)$, $\|d_{\rho}\varphi^{-t}v\|\leq Ce^{-\chi t}\|v\|$, 4. (4)

for every $v$ in $E_{s}(\rho)$, $\|d_{\rho}\varphi^{t}v\|\leq Ce^{-\chi t}\|v\|$.

Again, it is known from the works of Anosov that such flows form an open set inside smooth vector fields [2].