Ruelle-Taylor resonances of Anosov actions

Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^\kappa$-Anosov actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle-Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of $\mathbb{C}^\kappa$, with $\lambda=0$ being always a leading resonance. The joint resonant states at $0$ give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows.


Introduction
If P is a differential operator on a manifold M that has purely discrete spectrum as an unbounded operator acting on L 2 (M ) (e.g. an elliptic operator on a closed Riemannian manifold M ), then the eigenvalues and eigenfunctions carry a huge amount of information about the dynamics generated by P . Furthermore, if P is a geometric differential operator (e.g. Laplace-Beltrami operator, Hodge-Laplacian or Dirac operators) the discrete spectrum encodes important topological and geometric invariants of the manifold M .
Unfortunately, in many cases (e.g. if the manifold M is not compact or if P is non-elliptic) the L 2 -spectrum of P is not discrete anymore but consists mainly of essential spectrum. Still, there are certain cases where the essential spectrum of P is non-empty, but where there is a hidden intrinsic discrete spectrum attached to P , called the resonance spectrum. To be more concrete, let us give a couple of examples of this theory: • Quantum resonances of Schrödinger operators P = ∆ + V with V ∈ C ∞ c (R n ) on M = R n with n odd (see for example [DZ19, Chapter 3] for a textbook account to this classical theory). The definition of the resonances can be stated in different ways (using meromorphically continued resolvents, scattering operators or discrete spectra on auxiliary function spaces), and also the mathematical techniques used to establish the existence of resonances in the above examples are quite diverse (ranging from asymptotics of special functions to microlocal analysis). Nevertheless, all three examples above share the common point that the existence of a discrete resonance spectrum can be proven via a parametrix construction, i.e. one constructs a meromorphic family of operators Q(λ) (with λ ∈ C) such that (P − λ)Q(λ) = Id +K(λ), where K(λ) is a meromorphic family of compact operators on some suitable Banach or Hilbert space. Once such a parametrix is established, the resonances are the λ where Id +K(λ) is not invertible and the discreteness of the resonance spectrum follows directly from analytic Fredholm theory. In general, being able to construct such a parametrix and define a theory of resonances involve non-trivial analysis and pretty strong assumptions, but they lead to powerful results on the long time dynamics of the propagator e itP , for example in the study of dynamical systems [Liv04,NZ13,FT17b] or on evolution equations in relativity [HV18]. Furthermore, resonances form an important spectral invariant that can be related to a large variety of other mathematical quantities such as geometric invariants [GZ97,SZ07], topological invariants [DR17, DZ17, DGRS20, KW20] or arithmetic quantities [BGS11]. They also appear in trace formulas and are the divisors of dynamical Ruelle and Selberg zeta functions [BO99,PP01,GLP13,DZ16,FT17b].
The purpose of this work is to use analytic and microlocal methods to construct a theory of joint resonance spectrum for the generating vector fields of R κ -Anosov actions. In terms of PDE and spectral theory, this can be viewed as the construction of a good notion of joint spectrum for a family of κ commuting vector fields X 1 , . . . , X κ , generating a rank-κ subbundle E 0 ⊂ T M , when their flow is transversally hyperbolic with respect to that subbundle. These operators do not form an elliptic family and finding a good notion of joint spectrum is thus highly non-trivial. Our strategy is to work on anisotropic Sobolev spaces to make the non-elliptic region "small" and then obtain Fredholmness properties.
However, this involves working in a non self-adjoint setting, even if the X k 's were to preserve a Lebesgue type measure. We are then using Koszul complexes and a cohomological theory developed by Taylor [Tay70b,Tay70a] in order to define a proper notion of joint spectrum in these anisotropic spaces, and we will show that this spectrum is discrete.
We emphasize that, in terms of PDE and spectral theory, there are important new aspects to be considered and the results are far from being a direct extension of the κ = 1 case (the Anosov flows). But also outside the spectral theory of linear partial differential operator the developed theory might be helpful: the classical examples of such R κ -Anosov actions are Weyl chamber flows for compact locally symmetric spaces of rank κ ≥ 2, and it is conjectured by Katok-Spatzier [KS94] that essentially all non-product R κ actions are smoothly conjugate to homogeneous cases. Despite important recent advances [SV19], the conjecture is still widely open and it is important to extract as much information as possible on a general Anosov R κ action in order to address this conjecture: for example, having an ergodic invariant measure with full support plays an important role in the direction of this conjecture (see e.g. [KS07] where the existence of such a measure is a central assumption on which the results are base; see also the discussions in the recent preprint [SV19]). Based on the spectral theory developed in this article we show in a following paper [GBGW] the existence of such ergodic measures of full support for any positively transitive 1 Anosov action.
Let us summarize the main novelties of this work and its first applications: (1) we construct a new theory of joint resonance spectrum for a family of commuting differential operators by combining the theory of Taylor [Tay70b] with the use of anisotropic Sobolev spaces for the study of resonances; as far as we know, this is the first result of joint spectrum in the theory of classical or quantum resonances. (2) All the Weyl chamber flows on locally symmetric spaces and the standard actions of Katok-Spatzier [KS94] are included in our setting, and our results are completely new in that setting where representation theory is usually one of the main tools. This gives a new, analytic, way of studying homogeneous dynamics and spectral theory in higher rank. (3) We show that the leading joint resonance provides a construction of a new Sinai-Ruelle-Bowen (SRB) invariant measure µ for all R κ actions. In a companion paper [GBGW] based on this work, we show that our measure µ has all the properties of SRB measures of Anosov flows (rank 1 case), and it has full support if the Weyl chamber is positively transitive, an important step in the direction of the rigidity conjecture. (4) We show in [GBGW] that the periodic tori of the R κ action are equidistributed in the support of µ and that µ can be written as an infinite sum over Dirac measures on the periodic tori, in a way similar to Bowen's formula in rank 1. These results are new even in the case of locally symmetric spaces and give a new way to study periodic tori (also called flats) in higher rank. (5) Based on the present paper, the two last named authors together with L. Wolf prove a classical-quantum correspondence between the joint resonant states of Weyl chamber flows on compact locally symmetric spaces Γ\G/M of rank κ and the joint eigenfunctions of the commuting algebra of invariant differential operators on the locally symmetric space Γ\G/K [HWW21]. This gives a higher rank version of [DFG15].
Another expected consequence of this construction would be a proof of the exponential decay of correlations for the action and a gap of Ruelle-Taylor resonances under appropriate assumptions, with application to the local rigidity and the regularity of the invariant measure µ. These questions will be addressed in a forthcoming work.
1.1. Statement of the main results. Let us now introduce the setting in more details and state the main results. Let M be a closed manifold, A R κ be an abelian group and let τ : A → Diffeo(M) be a smooth locally free group action. If a := Lie(A) ∼ = R κ , we can define a generating map X : a → C ∞ (M; T M) A → X A := d dt |t=0 τ (exp(tA)), so that for each basis A 1 , . . . , A κ of a, [X A j , X A k ] = 0 for all j, k. For A ∈ a we denote by ϕ X A t the flow of the vector field X A . Notice that, as a differential operator, we can view X as a map It is customary to call the action Anosov if there is an A ∈ a such that there is a continuous dϕ X A t -invariant splitting where E 0 = span(X A 1 , . . . , X Aκ ), and there exists a C > 0, ν > 0 such that for each x ∈ M ∀w ∈ E s (x), ∀t ≥ 0, dϕ X A t (x)w ≤ Ce −νt w , ∀w ∈ E u (x), ∀t ≤ 0, dϕ X A t (x)w ≤ Ce −ν|t| w . Here the norm on T M is fixed by choosing any smooth Riemannian metric g on M. We say that such an A is transversely hyperbolic. It can be easily proved that the splitting is invariant by the whole action. However, we do not assume that all A ∈ a * have this transversely hyperbolic behavior. In fact, there is a maximal open convex cone W ⊂ a containing A such that for all A ∈ W, X A is also transversely hyperbolic with the same splitting as A (see Lemma 2.2); W is called a positive Weyl chamber. This name is motivated by the classical examples of such Anosov actions that are the Weyl chamber flows for locally symmetric spaces of rank κ (see Example 2.3). There are also several other classes of examples (see e.g. [KS94,SV19]).
Since we have now a family of commuting vector fields, it is natural to consider a joint spectrum for the family X A 1 , . . . , X Aκ of first order operators if the A j 's are chosen transversely hyperbolic with the same splitting. Guided by the case of a single Anosov flow (done in [BL07,FS11,DZ16]), we define E * u ⊂ T * M to be the subbundle such that E * u (E u ⊕E 0 ) = 0. We shall say that λ = (λ 1 . . . , λ κ ) ∈ C κ is a joint Ruelle resonance for the Anosov action if there is a non-zero distribution u ∈ C −∞ (M) with wavefront set WF(u) ⊂ E * u such that 2 ∀j = 1, . . . , κ, (X A j + λ j )u = 0. (1. 2) The distribution u is called a joint Ruelle resonant state (from now on we will denote C −∞ E * u (M) the space of distributions u with WF(u) ⊂ E * u ). In an equivalent but more invariant way (i.e. independently of the choice of basis (A j ) j of a), we can define a joint Ruelle resonance as an element λ ∈ a * C of the complexified dual Lie algebra such that there is a non-zero u ∈ C −∞ E * u (M) with ∀A ∈ a, (X A + λ(A))u = 0. We notice that we also define a notion of generalized joint Ruelle resonant states and Jordan blocks in our analysis (see Proposition 4.17). It is a priori not clear that the set of joint Ruelle resonances is discrete -or non empty for that matter -nor that the dimension of joint resonant states is finite, but this is a consequence of our work: Theorem 1. Let τ be a smooth abelian Anosov action on a closed manifold M with positive Weyl chamber W. Then the set of joint Ruelle resonances λ ∈ a * C is a discrete set contained in A∈W {λ ∈ a * C | Re(λ(A)) ≤ 0}. (1.3) Moreover, for each joint Ruelle resonance λ ∈ a * C the space of joint Ruelle resonant states is finite dimensional.
We remark that this spectrum always contains λ = 0 (with u = 1 being the joint eigenfunction) and that for locally symmetric spaces it contains infinitely many joint Ruelle resonances, as is shown in [HWW21, Theorem 1.1].
We also emphasize that this theorem is definitely not a straightforward extension of the case of a single Anosov flow. It relies on a deeper result based on the theory of joint spectrum and joint functional calculus developed by Taylor [Tay70b,Tay70a]. This theory allows us to set up a good Fredholm problem on certain functional spaces by using Koszul complexes, as we now explain.
Let us define X + λ, for λ ∈ a * C , as an operator X + λ : C ∞ (M) → C ∞ (M; a * C ), ((X + λ)u)(A) := (X A + λ(A))u. We can then define for each λ ∈ a * C the differential operators d (X+λ) : Due to the commutativity of the family of vector fields X A for A ∈ a, it can be easily checked that d (X+λ) • d (X+λ) = 0 (see Lemma 3.2). Moreover, as a differential operator, it extends to a continuous map and defines an associated Koszul complex We prove the following results on the cohomologies of this complex: Theorem 2. Let τ be a smooth abelian Anosov action 3 on a closed manifold M with generating map X. Then for each λ ∈ a * C and j = 0, . . . , κ, the cohomology We want to remark that the statement about the cohomologies in Theorem 2 is not only a stronger statement than Theorem 1, but that the cohomological setting is in fact a fundamental ingredient in proving the discreteness of the resonance spectrum and its finite multiplicity. Our proof relies on the theory of joint Taylor spectrum (developed by J. Taylor in [Tay70b,Tay70a]), defined using such Koszul complexes carrying a suitable notion of Fredholmness. In our proof of Theorem 2 we show that the Koszul complex furthermore provides a good framework for a parametrix construction via microlocal methods. More precisely, the parametrix construction is not done on the topological vector spaces C −∞ E * u (M) but on a scale of Hilbert spaces H N G , depending on the choice of an escape function G ∈ C ∞ (T * M) and a parameter N ∈ R + , by which one can in some sense approximate C −∞ E * u (M). The spaces H N G are anisotropic Sobolev spaces which roughly speaking allow H N (M) Sobolev regularity in all directions except in E * u where we allow for H −N (M) Sobolev regularity. They can be rigorously defined using microlocal analysis, following the techniques of Faure-Sjöstrand [FS11]. By further use of pseudodifferential and Fourier integral operator theory we can then construct a parametrix Q(λ), which is a family of bounded operators on H N G ⊗ Λa * C depending holomorphically on λ ∈ a * C and fulfilling d (X+λ) Q(λ) + Q(λ)d (X+λ) = Id +K(λ). (1.5) Here K(λ) is a holomorphic family of compact operators on H N G ⊗ Λa * C for λ in a suitable domain of a * C that can be made arbitrarily large letting N → ∞. Even after having this parametrix construction, the fact that the joint spectrum is discrete and intrinsic (i.e. independent of the precise construction of the Sobolev spaces) is more difficult than for an Anosov flow (the rank 1 case): this is because holomorphic functions in C κ do not have discrete zeros when κ ≥ 2 and we are lacking a good notion of resolvent, while for one operator the resolvent is an important tool. Due to the link with the theory of the Taylor spectrum, we call λ ∈ a * C a Ruelle-Taylor resonance for the Anosov action if for some j = 0, . . . , κ the j-th cohomology is non-trivial and we call the non-trivial cohomology classes Ruelle-Taylor resonant states. Note that the definition of joint Ruelle resonances precisely means that the 0-th cohomology is non-trivial. Thus, any joint Ruelle resonance is a Ruelle-Taylor resonance. The converse statement is not obvious but turns out to be true, as we will prove in Proposition 4.15: if the cohomology of degree j > 0 is not 0, then the cohomology of degree 0 is not trivial. We continue with the discussion of the leading resonances. In view of (1.3) and Figure 1, a resonance is called a leading resonance when its real part vanishes. We show that this spectrum carries important information about the dynamics: it is related to a special type of invariant measures as well as to mixing properties of these measures. 3 We actually prove Theorem 1 and Theorem 2 in the more general setting of admissible lifts to vector bundles, as defined in Section 2.2.
First, let v g be the Riemannian measure of a fixed metric g on M. We call a τ -invariant probability measure µ on M, a physical measure if there is v ∈ C ∞ (M) non-negative such that for any continuous function f and any open cone C ⊂ W, where C T := {A ∈ C | |A| ≤ T }, and here | · | denotes a fixed Euclidean norm on a. In other words, µ is the weak Cesaro limit of a Lebesgue type measure under the dynamics. We prove the following result: Theorem 3. Let τ be a smooth abelian Anosov action with generating map X and let W be a positive Weyl chamber.
(i) The linear span over C of the physical measures is isomorphic (as a C vector space) to , the space of joint Ruelle resonant states at λ = 0 ∈ a * C ; in particular, it is finite dimensional. 4 (ii) A probability measure µ is a physical measure if and only if it is τ -invariant and µ has wavefront set WF Assume that there is a unique physical measure µ (or by (i) equivalently that the space of joint resonant states at 0 is one dimensional). Then the following are equivalent: • The only Ruelle-Taylor resonance on ia * is zero.
• There exists A ∈ a such that ϕ X A t is weakly mixing with respect to µ. • For any A ∈ W, ϕ X A t is strongly mixing with respect to µ. (iv) λ ∈ ia * is a joint Ruelle resonance if and only if there is a complex measure µ λ with WF(µ λ ) ⊂ E * s satisfying for all A ∈ W, t ∈ R the following equivariance under pushforwards of the action: (ϕ X A t ) * µ λ = e −λ(A)t µ λ . Moreover, such measures are absolutely continuous with respect to the physical measure obtained by taking v = 1 in (1.6).
(v) If M is connected and if there exists a smooth invariant measure µ with supp(µ) = M, we have for any j = 0, . . . , κ We show that the isomorphism stated in (i) and the existence of the complex measures in (iv) can be constructed explicitly in terms of spectral projectors built from the parametrix (1.5). We refer to Propositions 5.4 and 5.10 for these constructions and for slightly more complete statements.
In the case of a single Anosov flow, physical measures are known to coincide with SRB measures (see e.g. [You02] and references therein). The latter are usually defined as invariant measures that can locally be disintegrated along the stable or unstable foliation of the flow with absolutely continuous conditional densities.
We prove in a subsequent article [GBGW] that the microlocal characterization Theorem 3(ii) of physical measures via their wavefront set implies that the physical measures of an Anosov action are exactly those invariant measures that allow an absolutely continuous disintegration along the stable manifolds. We show in [GBGW, Theorem 2] that for each physical/SRB measure, there is a basin B ⊂ M of positive Lebesgue measure such that for all f ∈ C 0 (M), all proper open subcones C ⊂ W and all x ∈ B, we have the convergence Moreover, we prove in [GBGW,Theorem 3] that the measure µ can be written as an infinite weighted sums over Dirac measures on the periodic tori of the action, showing an equidistribution of periodic tori in the support of µ. Finally, we show that this measure has full support in M if the action is positively transitive in the sense that there is a dense orbit ∪ A∈W ϕ X A 1 (x) for some x ∈ M. As mentioned before, the existence of such a measure is considered as one important step towards the resolution of the [KS94] rigidity conjecture.
1.2. Relation to previous results. The notion of resonances for certain particular Anosov flows appeared in the work of Ruelle [Rue76], and was later extended by Pollicott [Pol85]. The introduction of a spectral approach based on anisotropic Banach and Hilbert spaces came later and allowed to the definition of resonances in the general setting, first for Anosov/Axiom A diffeomorphisms [BKL02,GL06,BT07,FRS08], then for general Anosov/Axiom A flows [Liv04,BL07,FS11,GLP13,DZ16,DG16,Med]. It was also applied to the case of pseudo Anosov maps [FGL19], Morse-Smale flows [DR18], geodesic flows for manifolds with cusps [GBW21] and billiards [BDL18]. This spectral approach has been used to study SRB measures [BKL02,BL07] but it led also to several important consequences on dynamical zeta function [GLP13,DZ16,FT17a,DG16] of flows, and links with topological invariants [DZ17,DR17,DGRS20].
Concerning the notion of joint spectrum in dynamics, there are several cases that have been considered but they correspond to a different context of systems with symmetries (e.g. [BR01]).
Higher rank R κ -Anosov actions have in particular been studied mostly for their rigidity: they are conjectured to be always smoothly conjugated to several models, mostly of algebraic nature (see e.g. the introduction of [SV19] for a precise statement and a state of the art on this question). The local rigidity of R κ -Anosov actions near standard Anosov actions 5 was proved in [KS94], and an important step of the proof relies on showing The main tools are based on representation theory to prove fast mixing with respect to the canonical invariant (Haar) measure. It is also conjectured in [KK95] that, more generally, for such standard actions, one has for j = 1, . . . , κ − 1 This can be compared to (v) in Theorem 3, except that there the functional space is different. Having a notion of Ruelle-Taylor resonances provides an approach to obtain exponential mixing for more general Anosov actions by generalizing microlocal techniques for spectral gaps [NZ13,Tsu10] to a suitable class of higher rank Anosov action, and by using the functional calculus of Taylor [Tay70a,Vas79]. We believe that such tools might be very useful to obtain new results on the rigidity conjecture.
We would like to conclude by pointing out a different direction: on rank κ > 1 locally symmetric spaces Γ\G/K, there is a commuting algebra of invariant differential operators that can be considered as a quantum analog to the Weyl chamber flows. If the locally symmetric space is compact, this algebra always has a discrete joint spectrum of L 2 -eigenvalues. Its joint spectrum and relations to trace formulae have been studied in [DKV79]. In [HWW21], it is shown that a subset of the Ruelle-Taylor resonances for the Weyl chamber flow are in correspondence with the joint discrete spectrum of the invariant differential operators on Γ\G/K, giving a generalization of the classical/quantum correspondence of [DFG15, GHW21, KW21] to higher rank.
1.3. Outline of the article. In Section 2 we introduce the geometric setting of Anosov actions and the admissible lifts that we study. In Section 3 we explain how to define the Taylor spectrum for a certain class of unbounded operators and discuss some properties of this Taylor spectrum. In Section 4 we prove Theorem 1 and Theorem 2, using microlocal analysis. A sketch of the central techniques is given at the beginning of Section 4. The last Section 5 is devoted to the proof of Theorem 3. In Appendix A, we recall some classical results of microlocal analysis needed in the paper.

Geometric preliminaries
2.1. Anosov actions. We first want to explain the geometric setting of Anosov actions and the admissible lifts that we will study.
Let (M, g) be a closed, smooth Riemannian manifold (normalized with volume 1) equipped with a smooth locally free action τ : A → Diffeo(M) for an abelian Lie group A ∼ = R κ . Let a := Lie(A) ∼ = R κ be the associated commutative Lie algebra and exp : a → A the Lie group exponential map. After identifying A ∼ = a ∼ = R κ , this exponential map is simply the identity, but it will be quite useful to have a notation that distinguishes between transformations and infinitesimal transformations. Taking the derivative of the A-action one obtains the infinitesimal action, called an a action, which is an injective Lie algebra homomorphism (2.1) Note that X can alternatively be seen as a Lie algebra morphism into the space Diff 1 (M) of first order differential operators. By commutativity of a, ran(X) ⊂ C ∞ (M; T M) is a κ-dimensional subspace of commuting vector fields which span a κ-dimensional smooth subbundle which we call the neutral subbundle E 0 ⊂ T M. Note that this subbundle is tangent to the A-orbits on M. It is often useful to study the one-parameter flow generated by a vector field X A which we denote by ϕ X A t . One has the obvious identity ϕ X A t = τ (exp(At)) for t ∈ R. The Riemannian metric on M induces norms on T M and T * M, both denoted by · .
Definition 2.1. An element A ∈ a and its corresponding vector field X A are called transversely hyperbolic if there is a continuous splitting that is invariant under the flow ϕ X A t and such that there are ν > 0, C > 0 with (2.4) We say that the A-action is Anosov if there exists an A 0 ∈ a such that X A 0 is transversely hyperbolic.
Given a transversely hyperbolic element A 0 ∈ a we define the positive Weyl chamber W ⊂ a to be the set of A ∈ a which are transversely hyperbolic with the same stable/unstable bundle as A 0 .
Lemma 2.2. Given an Anosov action and a transversely hyperbolic element A 0 ∈ a, the positive Weyl chamber W ⊂ a is an open convex cone.
Proof. Let us first take the ϕ (2.5) In particular, dϕ decays exponentially fast as t → +∞. This implies that dϕ X A t 0 v ∈ E u and the same argument works with E s . Next, we choose an arbitrary norm on a. There exist C, C > 0 such that for each v ∈ E u we have for t ≥ 0 This implies that by choosing A − A 0 small enough, E u is an unstable bundle for A as well.
The same construction works for E s and we have thus shown that W is open. By re-parametrization, it is clear that W is a cone, so that only the convexity is left to be proved. Now, take A 1 , A 2 ∈ W and let C 1 , ν 1 , C 2 , ν 2 be the corresponding constants for the transversal hyperbolicity estimates (2.3) and (2.4). Then for s ∈ [0, 1] and v ∈ E u we can again use the commutativity and obtain, dϕ and this shows that Here we emphasize that the Weyl chamber W only depends on the Anosov splitting associated to A 0 but not on A 0 itself. Notice also that in general there are other Weyl chambers W associated to a different Anosov splitting. In the standard example of Weyl chamber flows they are images of W by the Weyl group of the higher rank locally symmetric space, explaining the terminology Weyl chambers (see for example [HWW21] for details). In general the structure of Weyl chambers can be quite complicated (see for example the example of non total Anosov actions given in [SV19, Section 6.3.4.]). In that case, the Ruelle-Taylor spectrum that we shall define has no reason to be the same for W and for W .
There is an important class of examples given by the Weyl chamber flow on Riemannian locally symmetric spaces. Example 2.3. Consider a real semi-simple Lie group G, connected and of non-compact type, and let G = KAN be an Iwasawa decomposition with A abelian, K the compact maximal subgroup and N nilpotent. Then A ∼ = R κ and κ is called the real rank of G. Let a be the Lie algebra of A and consider the adjoint action of a on g which leads to the definition of a finite set of restricted roots ∆ ⊂ a * . For α ∈ ∆ let g α be the associated root space. It is then possible to choose a set of positive roots ∆ + ⊂ ∆ and with respect to this choice there is an algebraic definition of a positive Weyl chamber If one now considers Γ < G a torsion free, discrete, co-compact subgroup one can define the biquotient M := Γ\G/M where M ⊂ K is the centralizer of A in K. As A commutes with M, the space M carries a right A-action. Using the definition of roots, it is direct to see that this is an Anosov action: all elements of the positive Weyl chamber W are transversely hyperbolic elements sharing the same stable/unstable distributions given by the associated vector bundles: Here n := α∈∆ + g α and n := −α∈∆ + g α are the sums of all positive, respectively negative root spaces, and n coincides with the Lie algebra of the nilpotent group N.
Note that there are various other constructions of Anosov actions and we refer to [KS94, Section 2.2] for further examples.
2.2. Admissible lifts. We want to establish the spectral theory not only for the commuting vector fields X A that act as first order differential operators on C ∞ (M) but also for first order differential operators on Riemannian vector bundles E → M which lift the Anosov action.
Definition 2.4. Let M be a closed manifold with an Anosov action of A ∼ = R κ and generating map X. Let E → M be the complexification of a smooth Riemannian vector bundle over M. Denote by Diff 1 (M; E) the Lie-algebra of first order differential operators with smooth coefficients and scalar principal symbol, acting on sections of E. Then we call a Lie algebra homomorphism X : a → Diff 1 (M; E), an admissible lift of the Anosov action if it satisfies the following Leibniz rule: for any section s ∈ C ∞ (M; E) and any function f ∈ C ∞ (M) one has for all A ∈ a (2.7) A typical example to have in mind would be when E is a tensor bundle, (e.g. exterior power of the cotangent bundle E = Λ m T * M or symmetric tensors E = ⊗ m S T * M), and X A s := L X A s where L denotes the Lie derivative. This admissible lift can be restricted to any subbundle that is invariant under the differentials dϕ X A t for all A ∈ a, t > 0. Another class of examples comes from flat connections. More generally, the above examples can be seen as a special case where the A-action τ on M lifts to an action τ on E which is fiberwise linear. Then one can define an infinitesimal action which is an admissible lift.

Taylor spectrum and Fredholm complex
The Taylor spectrum was introduced by Taylor in [Tay70b,Tay70a] as a joint spectrum for commuting bounded operators, using the theory of Koszul complexes. While there are different competing notions of joint spectra (see e.g. the lecture notes [Cur88]), the Taylor spectrum is from many perspectives the most natural notion. Its attractive feature is that it is defined in terms of operators acting on Hilbert spaces and does not depend on a choice of an ambient commutative Banach algebra. Furthermore, it comes with a satisfactory analytic functional calculus developed by Taylor and Vasilescu [Tay70a,Vas79].
3.1. Taylor spectrum for unbounded operators. Most references introduce the Taylor spectrum for tuples of bounded operators. In our case, we need to deal with unbounded operators. Additionally, working with a tuple implies choosing a basis, which should not be necessary. Let us thus explain how the notion of Taylor spectrum can easily be extended to an important class of abelian actions by unbounded operators.
We start with E → M a smooth complex vector bundle over a smooth manifold M (not necessarily compact), a ∼ = R κ an abelian Lie algebra and X : a → Diff 1 (M; E) a Lie algebra morphism. For the moment we do not have to assume that M possesses an Anosov action. Note that X extends by linearity to X : a C → Diff 1 (M; E) and for the definition of the spectra we will need to work with this complexified version. Using the map X we define where we have set (Xu)(A) := X A u for each A ∈ a C . This will be the central ingredient to define the Koszul complex which will lead to the definition of the Taylor spectrum. In order to do this we need some more notation: we denote by Λa * C := κ =0 Λ a * C the exterior algebra of a * C -this is just a coordinate-free version of ΛC κ . Given a topological vector space V we use the shorthand notation V Λ := V ⊗ Λ a * C and V Λ := V ⊗ Λa * C . As Λa * C is finite dimensional V Λ is again a topological vector space. We notice that since Λa * C is a finite dimensional vector space, we can view it as a trivial bundle M × Λa * C → M, and when V = C ∞ c (M; E), . We shall freely make this identitifcation as this will sometime be useful when we use pseudo-differential operators.
We have the contraction and exterior product maps and ∧ : We can then extend d X to a continuous map on the spaces C ∞ Similarly, for each A ∈ a we will also extend X A on these spaces by setting Remark 3.1. Choosing a basis A 1 , . . . , A κ ∈ a provides an isomorphism Λa * ∼ = ΛR κ . One checks that under this isomorphism the coordinate free version d X : V ⊗ Λ a * → V ⊗ Λ +1 a * of the Taylor differential transforms to the Taylor differential d X : V ⊗ Λ R κ → V ⊗ Λ +1 R κ of the operator tuple X = (X A 1 , . . . , X Aκ ) defined as if the basis (e j ) j of R κ is identified to the dual basis of (A j ) j in a * .
Lemma 3.2. For each A ∈ a C one has the following identities as continuous operators on C ∞ c Λ and C −∞ Λ: which yields (i). In order to prove (ii) it suffices, by definition of d X , to prove the identity as Take arbitrary A, A ∈ a C and note that by definition which proves the statement. Note that we crucially use the commutativity of the differential operators X A in this step.
For (iii) we first conclude from (i) and As a direct consequence of Lemma 3.2(iii) we conclude that We now want to construct a complex of bounded operators on Hilbert spaces which lies between the complexes on C ∞ c Λ and C −∞ Λ. For this, we consider H a Hilbert space with con- is a dense subspace of H. If we fix a non-degenerate Hermitian inner product ·, · a * C , then this induces a scalar product ·, · HΛ and gives a Hilbert space structure on HΛ. While the precise value of ·, · HΛ obviously depends on the choice of the Hermitian product on a * C , the finite dimensionality of a * C implies that all Hilbert space structures on HΛ obtained in this way are equivalent. Note that on the Hilbert spaces HΛ the operators d X will in general be unbounded operators. However, we have: Lemma 3.3. For any choice of a non-degenerate Hermitian product on a * C , the vector space D(d X ) := {u ∈ HΛ | d X u ∈ HΛ} becomes a Hilbert space when endowed with the scalar product ·, · D(d X ) := ·, · HΛ + d X ·, d X · HΛ .
(3.4) Furthermore, all scalar products obtained this way are equivalent and induce the same topology Proof. We have to check that D(d X ) is complete with respect to the topology of ·, · D(d X ) : suppose u n is a Cauchy sequence in D(d X ), then u n and d X u n are Cauchy sequences in H and we denote by v 0 , v 1 ∈ HΛ their respective limits. By the continuous embedding which proves the completeness. For the boundedness, we take u ∈ D(d X ) and we compute To be able to use the usual techniques, it is crucial that C ∞ c (M; E) is not only dense in H but also in D(d X ) -on this level of generality, this is not a priori guaranteed. For this reason, we say the a-action X has a unique extension to H if and there is only one closed extension for d X .
In order to finally define the Taylor spectrum in an invariant way, we consider λ ∈ a * C as a Lie algebra morphism In this way we can define X − λ : a C → Diff 1 (M; E) and the associated operator d X−λ on C ∞ c Λ and C −∞ Λ. Since d X−λ = d X − d λ , and d λ is bounded on HΛ, D(d X−λ ) does not depend on λ. Furthermore, note that from Lemma 3.2 we know that d 2 ∩ HΛ k and we gather the results above in the following: defines a complex of bounded operators, and the operators d X−λ depend holomorphically on λ ∈ a * C . Recall from the discussion above that the unique extension property was crucially used to get d X−λ • d X−λ = 0, thus to have a well defined complex of bounded operators.
We introduce the notation (3.7) Now, following the previous discussion of the Taylor spectrum, we can define Definition 3.5. Let X : a → Diff 1 (M; E) be a Lie algebra morphism and H a Hilbert space such that the a action X has a unique extension to H. Then we define the Taylor spectrum This is equivalent to saying that the sequence (3.6) is not an exact sequence. The complex is said to be Fredholm if ran HΛ (d X−λ ) is closed and the cohomology ker HΛ (d X−λ )/ ran HΛ (d X−λ ) has finite dimension. In this case we say that λ is not in the essential Taylor spectrum σ ess T,H (X) of X and define the index by (3.8) As the usual Fredholm index for a single operator, the Fredholm index in the Taylor complex is also a locally constant function of λ (see Theorem 6.6 in [Cur88]).
Note that the non-vanishing of the 0-th cohomology ker HΛ 0 d X−λ of the complex is equiv- which corresponds to (λ 1 , . . . , λ κ ) being a joint eigenvalue of (X A 1 , . . . , X Aκ ). Obviously, on infinite dimensional vector spaces the joint eigenvalues do not provide a satisfactory notion of joint spectrum. Recall that for a single operator, λ ∈ C is in its spectrum if X − λ is either not injective or not surjective. In terms of the Taylor complex for a single operator (κ = 1) the non-injectivity corresponds to the vanishing of the zeroth cohomology group whereas the surjectivity corresponds to the vanishing of the first cohomology group.
Remark 3.6. So far we always started with a Lie algebra morphism X : a → Diff 1 (M; E), then considered the action of Diff 1 (M; E) on some topological vector space V (e.g. C ∞ c (M)) in order to define Taylor complex and Taylor spectrum. This will also be our main case of interest. However we notice that the construction of the operator d X and the complex associated to d X works exactly the same if we take instead any Lie algebra morphism where V is a topological vector space and L(V ) denotes the Lie algebra of continuous linear operators on V with Lie bracket [A, B] := AB −BA. We shall call the complex induced by d X on V Λ the Taylor complex of X on V . If V is a Hilbert space we define the Taylor spectrum of X on V by Such Lie algebra morphism that are not directly coming from differential operators will occasionally show up within the parametrix constructions in Sections 4 and 5.
3.2. Useful observations. For the reader not familiar with the Taylor spectrum, and for our own use, we have gathered in this section several observations that are helpful when manipulating these objects. First, we shall say that an operator P : , ω ∈ Λa * C , P (u ⊗ ω) = (P u) ⊗ ω. As usual with differential complexes, we have a dual notion of divergence complex. For this, we need a way to identify a with a * , i.e. a scalar product ·, · on a, extended to a C-bilinear two form. If one chooses a basis, the implicit scalar product is given by the standard one in that basis. In any case, A → A := A, · is an isomorphism between a and a * . If In this fashion, We can thus define the divergence operator associated to Y In an orthonormal basis (e j ) j of a for ·, · and (e j ) j the dual basis in a * , we get for u ∈

We get directly that for
It follows from similar arguments as before that We have the following If we fix an inner product ·, · on a and a corresponding orthonormal basis (e j ) j and if X i := X e i and Y j := Y e j , we then have as continuous operators on The sum does not depend on the choice of basis, because it is the trace of the matrix representing XY with ·, · .
Proof. Let e i be the dual basis to the chosen orthogonal basis e i . For I = (i 1 , . . . , i ) let Using the commutation of [X i , Y j ] = 0, we obtain the result.
As an illustration, let us recall the following classical fact: Lemma 3.8. Let X 1 , . . . , X κ be commuting operators on a finite dimensional vector space V .
Proof. By the basic theory of weight spaces (see e.g. [Kna02][Proposition 2.4]) V can be decomposed into generalized weight spaces, i.e. there are finitely many λ (j) = (λ κ ) ∈ C κ and a direct sum decomposition V = ⊕ j V j which is invariant under all X 1 , . . . , X κ and there are n j such that Commutativity and the Jordan normal form then imply that the λ (j) are precisely the joint eigenvalues of the tuple X. Now let µ = λ (j) for all j. We have to prove that µ / ∈ σ T,V (X). By µ = λ (j) we deduce that for any j there is at least Then the Y k satisfy all the assumptions of Lemma 3.7 and Consequently the Taylor complex (3.6) is exact.
In the particular case that X = (X 1 , . . . , X κ ) are symmetric matrices, using the spectral theorem, we can reduce the problem to the case that X 1 , . . . , X κ are scalars acting on some R m . From this we deduce that for λ ∈ σ T,R m (X), and we check that Our next step is to give a criterion for d X−λ to be Fredholm. We first notice that since We shall use the following criterion for the d X -complex to be Fredholm.
Lemma 3.9. Let X be an a-action with unique extension to H. Assume that there are bounded operators Q, R and K on HΛ, acting continuously on C −∞ (M; E)Λ, such that K is compact, R L(HΛ) < 1, and Then the complex defined by d X is Fredholm. Denote by Π 0 the projector on the eigenvalue 0 of the Fredholm operator Id +R + K; it is bounded on D(d X ) and commutes with d X . Then the map u → Π 0 u from ker d X ∩ D(d X ) to ker d X ∩ ran Π 0 factors to an isomorphism (3.10) Proof. First, since Q, R and K are continuous on distributions, it makes sense to write d X Q + Qd X = Id +R + K in the distribution sense. Further, from this relation, we deduce that Q is bounded on D(d X ). Additionally, without loss of generality (by modifying R) we can assume that K is a finite rank operator. Let us prove that the range of d X is closed. (3.11) We get that there is Since K is of finite rank, we obtain by a standard argument that d X has closed range (both in HΛ and D(d X )). The operator F := Id +R + K is Fredholm of index 0 and, since In that case, the spectral projector Π 0 of F for the eigenvalue 0 commutes with d X , is bounded on D(d X ), and since D(d X ) is dense in HΛ and Π 0 has finite rank, its image is contained in D(d X ). Further, we can write F = (F + Π 0 )(Id − Π 0 ), and F := F + Π 0 is invertible on HΛ and D(d X ), and commuting with d X , so that on D(d X ) (3.13) (3.14) Since Π 0 and d X commute, u → Π 0 u factors to a homomorphism between the cohomologies in (3.10). This map in cohomologies is obviously surjective since ran Π 0 ⊂ D(d X ). To prove that the map is injective, we need to prove that if . This fact actually follows directly from (3.14) by using that both F −1 and Q are bounded on D(d X ).
We can also deduce the following: Proof. From Lemma 3.9, we deduce that 0 ∈ σ T,H (X) if and only if the complex given by d X is not exact on ran Π 0 (recall that d X commutes with Π 0 ). However, if F is Λ-scalar, then Π 0 = Π 0 ⊗ Id with Π 0 the spectral projector at 0 of F on H, and ran Π 0 = (ran Π 0 ) ⊗ Λa * C . It follows that d X restricted to ran Π 0 is exactly the Taylor complex of the operator X on ran Π 0 in the sense of Remark 3.6. We are thus reduced to finite dimension and we can apply Lemma 3.8.
The version of the Analytic Fredholm Theorem for the Taylor spectrum is the following statement: Proposition 3.11. Let X be an a-action with unique extension to H. Then σ T,H (X)\σ ess T,H (X) is a complex analytic submanifold of C κ \ σ ess T,H (X). Proof. As the complex (3.6) is an analytic Fredholm complex of bounded operators on C κ \ σ ess T,H (X) the statement is classical and a proof can be found in [Mül00, Theorem 2.9].
In general, the question of whether the spectrum is discrete does not seem to have a very simple answer. For example, a characterization can be found in [AM09, Corollary 2.6 and Lemma 2.7]. Such a criterion is particularly adapted to microlocal methods and it can actually be used in our setting. However, it turns out that an even simpler criterion is sufficient for us: Lemma 3.12. Under the assumptions of Lemma 3.9, assume in addition that Q = δ Q for some Lie algebra morphism Q : a → L(H) such that Q A acts continuously on C −∞ (M; E) and [Q A , X B ] = 0 for all A, B ∈ a. Then, Lemma 3.10 applies, and the Taylor spectrum of X on H is discrete in a neighborhood of 0.
Proof. Let A 1 , . . . , A κ ∈ a be an orthonormal basis for ·, · and let Q j := Q A j . We observe from Lemma 3.7 that for λ ∈ a * C the following identity holds on D(d X ) Thus, denoting F (λ) := F + λ · Q on H and F (λ) := F (λ) ⊗ Id on HΛ, we see that Lemma 3.10 indeed applies.
Next, we observe two things. The first is that F and λ · Q commute. The second is that for λ small enough, F (λ) can still be decomposed in the form Id +R(λ) + K(λ) with R(λ) L(H) < 1 and K(λ) compact, because Q is bounded. It follows that d X−λ is Fredholm for λ close enough to 0.
From Lemma 3.9, we know that the cohomology of d X−λ on D(d X ) is isomorphic to and the isomorphism is given by In general, there is no reason for this map to be injective. However, if we further assume that Π 2 and Π 0 (λ) commute, and that ran(Π 0 (λ)) ⊂ ran(Π 2 ), then we can see Π 0 (λ) as a projector on ran(Π 2 ). The mapping [ by using ker Π 2 ⊂ ker Π 0 (λ), and it has to be surjective. Using this and the surjectivity of (3.15) we deduce the bounds Since we have proved above that the lower and upper bound are equal, then (3.15) is actually an isomorphism. Let us write, with F := F + Π 0 where Π 0 is the spectral projector of F at 0 and we have the following identity on H In particular, u ∈ ran Π 0 , so that ker F (λ) ⊂ ker F and ran Π 0 (λ) ⊂ ran Π 0 . But certainly, Π 0 and Π 0 (λ) commute. So we can apply the argument above with Π 0 playing the role of Π 2 , and deduce that for λ sufficiently small, Since ran Π 0 is a fixed finite dimensional space, the Taylor spectrum of X is discrete near 0 by Lemma 3.8.

Discrete Ruelle-Taylor resonances via microlocal analysis
In this section, M is a compact manifold, equipped with a vector bundle E → M and an admissible lift X of an Anosov action (see Definition 2.4). We have seen in Section 3.1 how to define the Taylor differential d X which acts in its coordinate free form on C ∞ (M; E) ⊗ Λa * . We have furthermore seen how d X can be used to define a Taylor spectrum σ T,H (X) ⊂ a * C . We take coordinates whenever it is convenient. In that case, we will use the notation d X to avoid confusion. In the sequel it will be convenient to pass back and forth between these versions and we will mostly use the shorthand notation C ∞ Λ, leaving open which version we currently consider.
The Ruelle-Taylor resonances that we will introduce will correspond to a discrete spectrum of −X on some anisotropic Sobolev spaces. From a spectral theoretic point of view this sign convention might seem unnatural. However, from a dynamical point of view this convention is very natural: given the flow ϕ X t of a vector field X, the one-parameter group that propagates probability densities with respect to an invariant measure is given by (ϕ X −t ) * and thus generated by the differential operator −X. We will therefore from now on consider the holomorphic family of complexes generated by d X+λ for λ ∈ a * C (respectively λ ∈ C κ after a choice of coordinates). Let us denote by e −tX A the 1-parameter family generated by X A , Since we work with spaces that are deformations of L 2 (M), we will compare our results with the growth rate of the action on L 2 (M), defined for A ∈ a as The goal of this section is to show the following: Theorem 4. Let τ be a smooth abelian Anosov action with generating map X and X an admissible lift. Let A 0 ∈ W be in the positive Weyl chamber. There exists c > 0, locally uniformly with respect to A 0 , such that for each N > 0, there is a Hilbert space H N containing C ∞ (M) and contained in C −∞ (M) such that the following holds true: 1) −X has no essential Taylor spectrum on the Hilbert space H N in the region 4) An element λ ∈ F N is in the Taylor spectrum of −X on H N if and only if λ is a joint Ruelle resonance of X.
The Hilbert space H N will be rather written H N G below, where G is a certain weight function on T * M giving the rate of Sobolev differentiability in phase space. We use this notation in order to emphasize the dependence of the space on G.
The central point of the proof will be a parametrix construction for the exterior differential d X+λ . We will prove in Proposition 4.7 that there are holomorphic families of operators The operators Q(λ) and F (λ) will be Fourier integral operators and independent of any Hilbert space on which the operators act. However, the crucial fact is that for these operators there exists a scale of Hilbert spaces are Fredholm and can be decomposed as F (λ) = Id + R(λ) + K(λ) with K(λ) compact and R(λ) L(H N G Λ) < 1/2. Then by Lemma 3.9 we directly conclude that the Taylor complex on H N G Λ is Fredholm on λ ∈ F N G . The fact that the construction of the operator family F (λ) : C ∞ Λ → C −∞ Λ is independent of the specific Hilbert spaces on which they act will be the key for proving in Section 4.3 that the Taylor spectrum of d X+λ is intrinsic to the Anosov action, i.e. independent of the constructed spaces H N G . The flexibility which we will have in the construction of the escape function G will furthermore allow to identify this intrinsic spectrum with the spectrum of d X+λ on the space C −∞ E * u Λ of distributions with wavefront set contained in the annihilator E * u ⊂ T * M of E u ⊕ E 0 (see Proposition 4.10). Finally, we will see that the choice of Q(λ) can be made more geometric, to enable the use of Lemma 3.12 and prove that this intrinsic spectrum is discrete in a * C . The construction of the parametrix Q(λ) and the Hilbert spaces H N G will be done using microlocal analysis. Appendix A contains a brief summary of the necessary microlocal tools. Section 4.1 will be devoted to the construction of the anisotropic Sobolev spaces. With these tools at hand we will construct the parametrix (Section 4.2), and prove that the spectrum is intrinsic (Section 4.3) as well as discrete (Section 4.4).
4.1. Escape function and anisotropic Sobolev space. In this section we define the anisotropic Sobolev spaces. Their construction will be based on the choice of a so-called escape function for the given Anosov action. We first give a definition for such an escape function and then prove the existence of escape functions with additional useful properties.
Given any smooth vector field X ∈ C ∞ (M; T M) with flow ϕ X t we define the symplectic lift of the flow and the corresponding vector field by Φ X t : and T is the transpose of the inverted differential (dϕ X t ) −1 . The notation X H is chosen because it is the Hamilton vector field of the principal symbol σ 1 Recall from Example A.2 that for an admissible lift of an Anosov action, the principal symbols of the lifted differential operator X A and that of the vector field X A tensorized with Id E coincide. This will turn out to be the reason why we do not have to care about the admissible lifts for the construction of the escape function. We will denote by {0} := {(x, 0) ∈ T * M | x ∈ M } the zero section.
.3), we will prove the existence of escape functions for Anosov actions. Before coming to this point let us explain how we can build the anisotropic Sobolev spaces based on the escape function: given an escape function G, Property (1) of Definition 4.1 implies that m ∈ S 0 1 (M) and for any N > 0, e N G ∈ S N m 1− (M) is a real elliptic symbol. According to [FRS08, Lemma 12 and Corollary 4] there exists a pseudodifferential operator We can now define the anisotropic Sobolev spaces Note that the scalar product u, v H N G depends not only on the choice of the escape function but also on the choice of its quantizationÂ N G . However, by L 2 -continuity (Proposition A.9), these different choices all yield equivalent scalar products on the given vector space H N G . For that reason we can suppress this dependence in our notation.
We want to study the Taylor spectrum of the admissible lift of the Anosov action on these anisotropic Sobolev spaces. Recall from Section 3.1 that due to the unboundedness of the differential operators we have to verify the unique extension property: Then there is a c X > 0, an open conic neighborhood Γ E * 0 ⊂ Γ 0 of E * 0 , and R > 0 such that there is an escape function G for A 0 compatible with c X and Γ E * 0 with the additional property that the order function satisfies m(x, ξ) ≥ 1/2 for (x, ξ) ∈ Γ reg and |ξ| > R. (4.4) Proof. The proof follows from [DGRS20, Lemma 3.2]: indeed, first we note that the proof there only uses the continuity of the decomposition T * M = E * 0 ⊕ E * u ⊕ E * s and the contracting/expanding properties of E * s , E * u but not the fact that E * 0 is one dimensional. It suffices to take, in the notations of [DGRS20], N 1 = 4, N 0 = 1/4 and Γ reg = T * M \ C uu (α 0 ) with α 0 > 0 small enough. Although it is not explicitly written in the statement of [DGRS20, Lemma 3.3], the order function m constructed there satisfies X H A 0 m(x, ξ) ≤ 0 for |ξ| large enough and Γ E * 0 is arbitrarily small if α 0 > 0 is small (see [DGRS20, Section A.2]). In the proof of the fact that the Ruelle-Taylor spectrum is discrete, we shall also need an escape function that works for all A in a neighborhood U ⊂ W of a fixed element A 1 ∈ W.
Lemma 4.4. Let A 1 ∈ W be fixed. Then there is an escape function G for A 1 , a conic neighborhood , a constant c X > 0 and a neighborhood U ⊂ W of A 1 such that G is an escape function for all A ∈ U compatible with c X > 0 and Γ E * 0 . Moreover, G can be chosen to satisfy X H A G ≤ 0 in {|ξ| ≥ R} for some R ≥ 1.
Proof. In a first step we need to construct an order function m that fulfills all properties of Definition 4.11 but additionally X H A m ≤ 0 for |ξ| ≥ R for all A close enough to A 1 . To obtain this order function we can follow exactly the construction for Anosov flows given in [GB20, Section 2]. It works mutatis mutandis in our case as the proof simply uses the continuity of the decomposition T * M = E * 0 ⊕ E * s ⊕ E * u and the expanding/contracting properties of E * s and E * u , but not the fact that dim E * 0 = 1. We can then define the function G as in [FS11, Lemma 1.2] by setting G(x, ξ) = m(x, ξ) log (1 + f (x, ξ)), where f > 0 is positively homogeneous of degree 1 in ξ for |ξ| > R, in a conic neighborhood of E * s (resp. of E * u ) for some c 1 > 0. To construct such f near E * s , we can use the construction from [DZ16, Lemma C.1]: for (x, ξ) in a conic neighborhood N s of E * s , set for some C > 0 uniform with respect to A as above, the last term following from the classical estimate max |α|+|β|≤1 sup (x,ξ),|ξ|=1 ∂ α x ∂ β ξ |e −tX H A (x, ξ)| ≤ Ce C|t| and the homogeneity in ξ. Fix T large enough so that we have |ξ| − |e −T X H A (x, ξ)| ≤ −2|ξ| for all |ξ| > 1 in N s . Once T has been fixed, one can choose 0 < < e −CT so that for some c 1 > 0, we obtain (4.5). The same construction applies near E * u . We then extend f in a positively homogeneous function of degree 1 in {|ξ| ≥ R} in a smooth fashion (its value far from E * u ∪ E * s ∪ E * 0 will not matter). The proof of [FS11, Lemma 1.2] (using the fact that X H A |ξ(X A 1 )| = 0 as [X A , X A 1 ] = 0) shows that X H A G ≤ 0 for all |ξ| ≥ R if R is large enough and that G is an escape function for all A ∈ U := A 1 + { A ∈ a | |A | a ≤ 1} compatible with some c X > 0 and some Γ E * 0 .
4.2. Parametrix construction. The goal of this section is to construct an operator Q(λ) as in Lemma 3.9 for the complex d X+λ , and so that Q will be bounded on the anisotropic Sobolev spaces H N G Λ. The construction will be microlocal in the elliptic region and dynamical near the characteristic set. In Section 4.4 we will provide an alternative construction of a Q(λ) which is purely dynamical, i.e. which is a function of the operators X A j . Recall the notation E ⊗ Λ = E ⊗ Λa * . We will also freely identify operators P : Lemma 4.5. Let P ∈ Ψ 0 (M; E) be such that WF(Id − P ) does not intersect a conic neighborhood of E * u ⊕ E * s , and we make it act as a Λ-scalar operator. There exists a holomorphic family of pseudo-differential operators Q ell (λ) ∈ Ψ −1 (M; E ⊗ Λa * C ) for λ ∈ a * C such that Proof. We will use an arbitrary choice of basis A 1 , . . . , A κ in a and consider the commuting differential operators X A 1 , . . . , X Aκ . Recall that the corresponding divergence operator δ X+λ on C ∞ (M; E) ⊗ Λa * is defined by where λ j := λ(A j ) ∈ C (here (e j ) j is a dual basis to A j in a * ). Thus, using the commutations [X A j + λ j , X A k + λ k ] = 0 and Lemma 3.7 with Y j = X A j + λ j , we obtain that the operator ∆ X+λ := d X+λ δ X+λ + δ X+λ d X+λ is Λ-scalar and given for each ω ∈ Λa * by the expression This shows that ∆ X+λ ∈ Ψ 2 (M; E ⊗ Λ) with principal symbol given by (see Example A.2) It is an operator which is microlocally elliptic outside E * u ⊕E * s (i.e. ell 2 (∆ X+λ ) = T * M\(E * u ⊕ E * s )). Thus, by Proposition A.7, if P ∈ Ψ 0 (M, E ⊗ Λ) is Λ-scalar and has WF(P ) contained in a conic open set of T * M not intersecting E * u ⊕ E * s , then there exists a Λ-scalar pseudodifferential operator Q ∆ (λ) ∈ Ψ −2 (M; E ⊗ Λ) holomorphic in λ with WF(Q ∆ (λ)) ⊂ WF(P ) such that ∆ X+λ Q ∆ (λ) = P + S (λ) with S (λ) ∈ Ψ −∞ (M; E ⊗ Λ) holomorphic in λ and Λ-scalar. We now choose P so that WF(P ) ∩ (E * u ⊕ E * s ) = ∅ and WF(Id − P ) ∩ WF(Id − P ) = ∅; in other words, P = Id microlocally on WF(Id − P ). Note that d X+λ ∆ X+λ = ∆ X+λ d X+λ implies that Using microlocal ellipticity of ∆ X+λ outside E * u ⊕ E * s and the fact that has wavefront set contained in WF(P ) ∩ WF(Id − P ).
A second ingredient for the construction of the parametrix will be the following estimates of the essential spectral radius of the propagator on the anisotropic Sobolev spaces. We recall that if Y is a bounded operator on a Hilbert space H, The proof of the following Lemma is inspired by the argument in [FRS08] for Anosov diffeomorphisms.
Lemma 4.6. Let P ∈ Ψ 0 (M; E) be such that WF(P ) is disjoint from E * 0 , and choose an arbitrary constant C P > C P := lim sup |ξ|→∞ σ 0 p (P )(x, ξ) and some T > 0. Let A ∈ W ⊂ a, Γ reg be an open cone disjoint from E * u ⊂ T * M, and Γ 0 ⊂ T * M be a small conic neighborhood of E * 0 . By Proposition 4.3, associated to Γ reg there exists an escape function G for A 0 := A and an open conic set Γ E * 0 ⊂ Γ 0 so that G is compatible with c X and Γ E * 0 in the sense of Definition 4.1. If in addition Γ E * 0 ∩Φ X A t (WF(P )) = ∅ for all 0 ≤ t ≤ T , then for all 0 ≤ t ≤ T the operator e −tX A P : H N G → H N G is bounded and can be decomposed under the form As a consequence the essential spectral radius of e −tX A P : Proof. Let m ∈ C ∞ (T * M; R) be the order function of the escape function G (see Definition 4.1(1)). Instead of studying e −tX A P on H N G we consider the operatorÂ N G e −tX A PÂ −1 N G on L 2 (M; E) which is a Fourier integral operator. We write this operator aŝ For the newly introduced operator B t we apply Egorov's Lemma (Lemma A.8) and deduce that it is a pseudodifferential operator B t ∈ Ψ is bounded on L 2 , and we can apply Proposition A.9 to this operator. We calculate its principal symbol . Now, using Definition 4.1(3), our assumption that Γ E * 0 ∩ Φ X A t (WF(P )) = ∅ for 0 ≤ t ≤ T ensures that, for any (x, ξ) ∈ WF(P ) and |ξ| sufficiently large, By closedness of Γ E * 0 and WF(P ) this estimate can also be extended to a small conical neighborhood of WF(P ). On the complement of this neighborhood, by the definition of the wavefront set, we deduce lim sup |ξ|→∞ σ 0 p (P )(x, ξ) = 0. We have seen above that e N (G•Φ X A t −G) ∈ S 0 1− . In particular this factor is uniformly bounded. Putting everything together we get lim sup Using Proposition A.9 we can write B t PÂ −1 Now, by (4.7), our operator of interest can be written aŝ , and we get the desired property by setting R N,G (t) := e −tX A R N (t) and K N,G (t) := e −tX A K N (t).
Recall that C L 2 (A) was defined in Formula (4.1). We can now come to the construction of our full parametrix for the Taylor complex: Proposition 4.7. For any A 0 ∈ W, any open cone Γ 0 ⊂ T * M containing E * 0 and satisfy- Furthermore, for any escape function G for A 0 compatible with c X > 0, and Γ E * 0 ⊂ Γ 0 , one has for any N > 0 and δ > 0 that: (1) Q(λ) : H N G Λ j → H N G Λ j−1 is bounded for any λ ∈ a * C and 0 ≤ j ≤ κ, (1) If there is a smooth volume density µ preserved by the Anosov action (e.g. the Haar measure for Weyl chamber flows), and if we consider the scalar case X A = X A , then e tX A is unitary on L 2 (M, µ) and the constant C L 2 (A) vanishes.
(2) For proving that the Ruelle-Taylor spectrum is independent of the choice of H N G it will be essential that the operators Q(λ), F (λ) will only depend on the choice of A 0 and Γ E * 0 but not on the choice of the anisotropic Sobolev space H N G as long as the escape function G satisfies the required compatibility conditions.
Proof of Proposition 4.7. From the definition of C L 2 (A), we deduce that there exists T 0 > 0 so that for T ≥ T 0 , e −T X A 0 ≤ e T (C L 2 (A 0 )+δ/2) ; we choose T so that both T > T 0 and T ≥ 2 ln(3)/δ. For λ ∈ a * C we define the operators X A 0 (λ) := X A 0 + λ(A 0 ) and let We have the relations (4.8) Now we extend Q T (λ) to an operator on C ∞ (M; E) ⊗ Λ a * → C ∞ (M; E) ⊗ Λ −1 a * for each as follows: define the linear map Q T (λ) : a → L(C ∞ (M; E)) by Q T (λ)A 0 = Q T (λ) and Q T (λ)A = 0 if A, A 0 = 0 (recall ·, · is a fixed scalar product on a), and let for u ∈ C ∞ (M; E) and ω ∈ Λ a * . Using the relations of (4.8) and Lemma 3.7 we get Next, we use the microlocal parametrix in the elliptic region from Lemma 4.5 with a carefully chosen microlocal cutoff function: By our assumption that Let us choose a second, smaller conical neighborhood E * u ⊕ E * s ⊂ Γ 2 Γ 1 . Now we fix a microlocal cutoff P = Op(p) ∈ Ψ 0 (M, C) which is microsupported in Γ 1 (i.e. WF(P ) ⊂ Γ 1 ) and microlocally equal to one on Γ 2 (i.e. WF(Id − P ) ∩ Γ 2 = ∅) and which furthermore has globally bounded symbol sup (x,ξ) |p(x, ξ)| ≤ 1. We apply Lemma 4.5 with this choice of P and multiply (4.6) from the left with e −T X A 0 (λ) . Using (4.10), we get We define Q(λ) := Q T (λ) + e −T X A 0 (λ) Q ell (λ) and obtain by adding up (4.9) and (4.11) Let us now show that Q(λ) and F (λ) have the required properties. By precisely the same argument as in Lemma 4.6 (using that X H A 0 m(x, ξ) ≤ 0 for |ξ| large enough) we deduce that e −tX A 0 is bounded on H N G uniformly for t ∈ [0, T ] for any escape function G associated to A 0 compatible with c X > 0 and Γ E * 0 ⊂ Γ 0 . Consequently Q T (λ) and e −T X A 0 (λ) are bounded operators on H N G Λ. AsÂ N G Q ell (λ)Â −1 N G ∈ Ψ −2 (M; E ⊗ Λ), this operator is a bounded operator on L 2 , thus Q ell (λ) is bounded on H N G Λ as well. Putting everything together we deduce that Q(λ) is bounded on H N G Λ for any λ ∈ a * C . As Q T (λ) and Q ell (λ) decrease the order in Λa * by one, Q(λ) has this property as well.
As a consequence we get: Proposition 4.9. For A 0 ∈ W there exists an escape function G such that for any N > 0 the operator d X+λ on H N G Λ defines a Fredholm complex for λ ∈ F N G,A 0 ,0 , i.e. we have σ ess T,H N G (−X) ∩ F N G,A 0 ,0 = ∅. Proof. By Proposition 4.3, there is an escape function G that allows to obtain Proposition 4.7. Then we can use Lemma 3.9 applied to X + λ to deduce Fredholmness.

4.3.
Ruelle-Taylor resonances are intrinsic. So far we have shown that the admissible lift of an Anosov action X acting as differential operators on H N G has a Fredholm Taylor spectrum on F N G,A := F N G,A,0 ⊂ a * C , where A ∈ W and G is an escape function associated to A. Further, we have seen that F N G,A can be made arbitrarily large by letting N → ∞. Proposition 4.10. Let A 0 ∈ W, N ≥ 0 and G be an escape function for A 0 . Then for any λ ∈ F N G,A 0 one has vector space isomorphisms Using this result, we see that the Ruelle-Taylor spectrum is independent of A 0 and of the anisotropic space H N G Λ in the region F N G,A 0 of λ ∈ a * C where the Taylor complex d X+λ is Fredholm on H N G Λ. We can then define the notion of a Ruelle-Taylor resonance as follows: Definition 4.11. We define the Ruelle-Taylor resonances of X to be the set and the Ruelle-Taylor resonant cohomology space of degree j of λ ∈ Res X to be Another consequence of Proposition 4.10 is:

Corollary 4.12 (Location of Ruelle-Taylor resonances). One has
Proof. Assume that there exists an A ∈ W such that Re(λ(A)) > C L 2 (A). Then for some δ > 0, λ ∈ F 0G,A,δ and consequently λ ∈ Res X iff ker L 2 Λ d X+λ / ran L 2 Λ d X+λ = 0. However, by (4.9) there is a bounded operator Q T (λ) : Since Re(λ(A)) > C L 2 (A), the right hand side is invertible on L 2 (M; E ⊗ Λ) provided T > 0 is large enough. As furthermore Id + e −T X A e −T λ(A) and its inverse commute with d X+λ we conclude that ker L 2 Λ d X+λ / ran L 2 Λ d X+λ = 0.
The strategy to prove Proposition 4.10 is to show that in each cohomology class in ker H N G Λ d X+λ / ran H N G Λ d X+λ one can find a representative that lies already in ker To this end we will construct for fixed λ a projector Π 0 (λ) of finite rank such that we can find in each cohomology class a representative in the range of Π 0 (λ). The fact that the range of Π 0 (λ) is independent of the anisotropic Sobolev spaces and contained in C −∞ (4.12) We can thus apply Lemma 3.9, and deduce that if Π 0 (λ) is the spectral projector of F (λ) on its kernel, is an isomorphism. Here, ran(Π 0 (λ)) = Π 0 (λ)H N G . But since C ∞ is dense in H N G , and Π 0 (λ) has finite rank, we deduce that it is equal to Π 0 (λ)C ∞ (M; E ⊗ Λ). We now need the following lemma:

Additionally, it has a continuous extension to
Proof. Recall that Π 0 (λ) : H N G Λ → H N G Λ has been defined as the spectral projector at z = 0 of F (λ) : H N G Λ → H N G Λ, it has finite rank. Since F (λ) and its Fredholmness do not depend on the choice of N , G, as long as λ ∈ F N G,A 0 , neither does its spectral projector at 0. The image of Π 0 (λ) is thus contained in the intersection of the H N G Λ such that λ ∈ F N G ,A 0 .
Let us show that this intersection is contained in C −∞ E * u (M; E ⊗ Λ). We thus take u in all the H N G such that λ ∈ F N G ,A 0 . By Proposition 4.3 for an arbitrary cone Γ reg disjoint from E * u , there exists an escape function G for A 0 compatible with c X and Γ E * 0 ⊂ Γ 0 such that microlocally on Γ reg , H N G is contained in the standard Sobolev space H N /2 (M; E). In particular, taking N arbitrarily large, λ ∈ F N G ,A 0 and WF(u) ∩ Γ reg = ∅. Since Γ reg was arbitrary, WF(u) ⊂ E * u . To prove that Π 0 (λ) has a continuous extension to is also contained in the union of the all the H N G such that λ ∈ F N G ,A 0 . This follows from Definition 4.1,(1), since we know that in a conic neighborhood around E * u we have m(x, ξ) ≤ −1/4. As a consequence, Π 0 (λ) is a linear operator from C −∞ . It is continuous as it has finite rank.
To finish the proof of Proposition 4.10, it suffices to apply a variation of the sandwiching trick presented in the proof of Lemma 3.12. Indeed, since Π 0 (λ) is a bounded projector on We need to show the injectivity of this map. This will follow from the fact that C −∞ Since u belongs to some H N G Λ, we then writeF (λ) = F (λ) + Π 0 (λ), and observe, just as in (3.13), that . But, since Q(λ) andF −1 (λ) are bounded on each H N G Λ such that λ ∈ F N G ,A 0 , this is an element of each such H N G Λ, so it is contained in the intersection thereof. We have seen in the proof of Lemma 4.13 that this intersection is contained in Finally, note that the operator F (λ) : H N G Λ → H N G Λ preserves the order in the Koszul complex, i.e. F (λ) : H N G Λ j → H N G Λ j , and all the subsequent constructions such as Π 0 (λ) do as well. The isomorphism Π 0 (λ) can thus be restricted to the individual cohomology In this section we show that the Ruelle-Taylor resonance spectrum of the admissible lift X : a → Diff 1 (M; E) of the Anosov action, for E a Riemannian vector bundle, is discrete in a * C . Our goal is to use Lemma 3.12. In contrast to just obtaining the Fredholm property of the Taylor complex, this section requires to use a parametrix Q(λ) in Proposition 4.7 that is more intrinsically related to the X action, in particular we shall construct Q(λ) as a function of (X 1 , . . . , X κ ) = (X A 1 , . . . , X Aκ ) if A j ∈ W is an orthonormal basis for some scalar product ·, · on a. This requires to use the slightly better escape function of Lemma 4.4 that provides decay not only in a fixed direction A 1 ∈ W, but also for all other elements in a small neighborhood U of A 1 . Let us now fix an orthonormal basis A 1 , . . . , A κ ∈ U ⊂ W of a in the positive Weyl chamber, and we denote the associated scalar product in a by ·, · . In order to be able to use Lemma 3.12, we will prove the following: Lemma 4.14. For each fixed λ ∈ F N G,A 0 ,δ there is a Lie algebra morphism Q(λ) : a → L(H N G ) ∩ L(C ∞ (M; E)) and commuting with X(λ) := X + λ in the sense that [X A j (λ), Q A k (λ)] = 0 for all j, k, such that Proof. Let T j > 0 for j = 1, . . . , κ, and consider χ j ∈ C ∞ c ([0, ∞[; [0, 1]) non-increasing with χ j = 1 in [0, T j ] and supp χ j ∈ [0, T j + 1]. Then we set and we make it act on C ∞ (M; E)⊗Λa * by Q j (λ) : u⊗w → (Q j (λ)u)⊗ι A j ω. As in Proposition 4.7, we compute and note that R j (λ) = R j (λ) ⊗ Id is scalar. We thus have . First we observe that Q(λ) = δ Q(λ) is the divergence associated to the Lie algebra morphism Q(λ) : a → L(C ∞ (M; E)) defined by We notice that Q A j (λ) commutes with X A i (λ) for each i, j. As in the proof of Proposition 4.7, we have that Q(λ) maps to L(H N G ) and Q(λ) is bounded on H N G Λ: notice that here we use Lemma 4.4 as it is important that the order function m satisfies X H A j m ≤ 0 for |ξ| large enough and for all j = 1, . . . , κ. We take P microsupported in a neighbourhood of E * u ⊕E * s and WF(P ) in a sufficiently close conical neighbourhood of E * u ⊕ E * s , as in the proof of Proposition 4.7, and follow the arguments given there, which were based on Lemma 4.6: if T j := T is chosen large enough (as in proof of Proposition 4.7) where R(t, λ) L(H N G ) j χ j L ∞ ≤ 1/2 and K(t, λ) is compact on H N G for all t ∈ [T, T + 1] κ (both depend on N, G). This shows that the operator (F (λ) − Id)P decomposes as (F (λ) − Id)P = R(λ) + K 1 (λ) with R(λ) L(H N G Λ) < 1/2 and K 1 (λ) compact on H N G Λ. Next, we claim that using that P ∈ Ψ 0 (M) is scalar with WF(Id − P ) not intersecting a conic neighborhood of E * u ⊕ E * s , we see that K 2 (λ) := (F (λ) − Id)(Id − P ) is a compact operator on H N G Λ. Indeed, let us first take a microlocal partition of (Id − P ) so that (Id − P ) − κ k=1 P k ∈ Ψ −∞ (M) with P k ∈ Ψ 0 (M) and WF(P k ) not intersecting a conic neighborhood of the characteristic set {(x, ξ) ∈ T * M | ξ(X A k ) = 0}. Let us show that R k (λ)P k is compact on H N G : first, but Z k (λ) being compact on H N G , we get that R k (λ)P k is compact on H N G using (4.16). Next, we write The former operator is compact since all the R k (λ) are bounded on H N G and commute with each other and R k (λ)P k is compact. Putting everything together we deduce that F (λ) has the desired properties by setting K(λ) := K 1 (λ) + K 2 (λ).
Remark: we notice that in the proof above, it is sufficient to take only one of the T j to be large while the others can be small, as this is sufficient to get the norm estimate R(λ) L(H N G ) < 1/2.
As a corollary, using Lemma 3.12 and Lemma 3.10, we deduce the following: Proposition 4.15. For an admissible lift of an Anosov action X, the Ruelle-Taylor resonance spectrum is a discrete subset of a * C . Moreover, This completes the proof of Theorem 4. In the scalar case (i.e. E is the trivial bundle) we will show in Corollary 4.16 below that part 3) of Theorem 4 can be sharpened using the dynamical parametrix Q(λ) in Lemma 4.14 (the same argument also works for admissible lifts under the condition e −tX A f L(L ∞ ) ≤ C for all t ∈ R): Corollary 4.16. Let X be an Anosov action. Then one has Proof. Let A ∈ W and assume that λ ∈ a * C satisfies Re(λ(A)) > 0, then we will show that λ can not be a Ruelle-Taylor resonance. We use the parametrix Q(λ) of Lemma 4.14 with A 1 := A and (A j ) j ∈ W κ forming a basis of a with A j in an arbitrarily small neighborhood of A 1 so that Re(λ(A j )) > 0 for all j. We get that (4.15) holds with F (λ) having discrete spectrum near z = 0. Let Π 0 (λ) be the spectral projector of the Fredholm operator F (λ) at z = 0, which can be written for some small enough > 0. We notice that for f ∈ L ∞ (M), we have This shows that by choosing the T j > 0 (that had been introduced in Lemma 4.14) large enough, (Id − F (λ)) L(L ∞ ) < 1/2. In particular F (λ) is invertible on L ∞ and therefore Π 0 (λ) = 0 since the expression (4.17) holds also as a map C ∞ (M) → C −∞ (M). This ends the proof.
Let us end the section with a statement about joint Jordan blocks for an admissible lifts X: Therefore given α ∈ N κ we define X α (λ) := κ j=1 (X A j + λ j ) α j .
Proposition 4.17. For any Ruelle-Taylor resonance λ ∈ Res X there is J ∈ N * which is the minimal integer such that, whenever for some u ∈ C −∞ E * u (M) and k ∈ N * one has X β (λ)u = 0 for all |β| = k then X α (λ)u = 0 for all |α| = J. Moreover the space of generalized joint resonant states is the finite dimensional space given by where Π 0 (λ) is the spectral projector of F (λ) at z = 0, defined in (4.17).
We also notice that the non-triviality of the space (4.18) (with J minimal) implies that λ is a Ruelle-Taylor resonance, since for u in this space, there is an α with |α| = J − 1 such that v := X α (λ)u = 0 satisfies v ∈ Res X,Λ 0 (λ). We also note that equality in (4.18) does in general not hold. One rather has: Proposition 4.18. If Π 0 (λ) is the spectral projector of R(λ) from (4.19) then where J ∈ N is the integer from Proposition 4.17.
Proof. First note that ran Π 0 (λ) is finite dimensional and X j invariant, thus we can decompose the space into joint generalized eigenstates. If η is such a joint eigenvalue then Proposition 4.15 implies that η is also a Ruelle-Taylor resonance. Now let u ∈ ran Π 0 (λ) be a joint eigenstate of X with eigenvalue η then by (4.19) Thus u is an eigenstate of R(λ) but as u is also required to be in the generalized eigenspace of eigenvalue 1, we deduce that jψ j (−i(λ j − η j )) = 1. This shows, that the left hand side is contained in the right hand side.
For the converse inclusion we note that any joint resonant state (X j + η j )u = 0 whose joint resonance fulfills jψ j (−i(λ j − η j )) = 1 is an eigenstate of R(λ) with eigenvalue 1 and thus contained in ran(Π 0 (λ)). For the generalized eigenstates of higher order we argue as above in Proposition 4.18 by induction.

The leading resonance spectrum
In this section we study the leading resonance spectrum, i.e. those resonances with vanishing real part and show that they give rise to particular measures and are related to mixing properties of the Anosov action. In this section the bundle E will be trivial. 5.1. Imaginary Ruelle-Taylor resonances in the non-volume preserving case. In this section, we investigate the purely imaginary Ruelle-Taylor resonances and in particular the resonance at 0 for the action on functions. We assume that the Anosov action X does not necessarily preserve a smooth invariant measure. We choose a basis A 1 , . . . , A κ of a, with dual basis (e j ) j in a * , and set X j := X A j , and we use dv g the smooth Riemannian probability measure on M. Let us choose iλ ∈ ia * purely imaginary and fix non-negative functions χ j ∈ C ∞ c (R + ), equal to 1 on a large interval [0, T j ], with χ j ≤ 0 and use the parametrix Q(iλ) in the divergence form from Lemma 4.14 so that by (4.15) and writing ψ j := −χ j ∈ C ∞ c ((0, ∞)) and λ j := λ(A j ) We proved that R(iλ) has essential spectral radius < 1 in the anisotropic space H N G , and the resolvent (R(iλ) − z) −1 is meromorphic outside |z| < 1 − for some , and the poles in |z| > 1 − are the eigenvalues of R(iλ). Moreover, for f ∈ L ∞ , one has Since R(iλ) is bounded, for |z| large enough one has on H N G but the L ∞ estimate (5.2) shows that this series converges in L(L ∞ ) and is analytic for |z| > 1. Therefore, using the density of C ∞ (M) in H N G , we deduce that R(iλ) has no eigenvalues in |z| > 1. We will use the notation u, v for the distributional pairing associated to the Riemannian measure dv g fixed on M that also extends to a complex bilinear pairing , this is simply M uv dv g . Accordingly, we also write u, v L 2 for the pairing M uv dv g and its sesquilinear extension to the pairing The next three lemmas (Lemma 5.1) characterize the spectral projector of R(iλ) onto the possible eigenvalue 1. Keep in mind that by Lemma 3.9 this spectral projector is closely related to the Ruelle Taylor resonant states. Finally in Proposition 5.4 we will use the knowledge about this spectral projector to characterize the leading resonance spectrum and to define physical measures.
Proof. We take u ∈ H N G such that (R(iλ) − z) −1 u has a pole of order > 1 at z = τ . By density of C ∞ in H N G , we can always assume that u is smooth. Denoting by ψ (k) = ψ * · · · * ψ (k-th convolution power), we can write Note that R(0)1 = 1. We take v another smooth function, then We deduce that for |z| > 1 This is in contradiction with the assumption that τ is a pole is of order > 1.
Then we can prove the following Lemma 5.2. For λ = κ j=1 λ j e j ∈ a * , R(iλ) has an eigenvalue of modulus 1 on H N G if and only if iλ is a Ruelle-Taylor resonance. In that case, the only eigenvalue of modulus 1 of R(iλ) in H N G is τ = 1 and the eigenfunctions of R(iλ) at τ = 1 are the joint Ruelle resonant states of X at λ. Moreover, if Π(iλ) is the spectral projector of R(iλ) at τ = 1, one has, as bounded operators in H N G , Proof. First, if iλ is a Ruelle-Taylor resonances, Proposition 4.17 implies that R(iλ) has 1 as an eigenvalue and the resonant states are included in the range of the spectral projector of R(iλ) at 1. Conversely, let Π(iλ) be the spectral projector of R(iλ) at τ ∈ S 1 : it commutes with the X j , so we can use Lemma 3.8 to decompose ran Π(iλ) in terms of joint eigenspaces for X j . Let u be a joint-eigenfunction of X j in ran Π(iλ), with X j u = ζ j u. By Lemma 5.1, R(iλ) has no Jordan block at τ , thus u ∈ H N G is a non-zero eigenfunction of R(iλ) with eigenvalue τ ∈ S 1 . Then For τ to have modulus 1, we need κ j=1 |ψ j (λ j − iζ j )| = 1. But since R ψ j = 1 and the ζ j 's have non-negative real part, so Re(ζ j ) = 0 and |ψ j (λ j − iζ j )| = 1 for all j. But then there is α ∈ R so that 1 = R ψ j (t) = R cos(t(λ j + Im(ζ j )) + α)ψ j (t)dt and thus cos(t(λ j + Im(ζ j )) + α) = 1 on supp(ψ j ) since ψ j ≥ 0, but this implies that ζ j = −iλ j and α ∈ 2πZ. Then we get τ = 1. In particular, with K(iλ)Π(iλ) = Π(iλ)K(iλ) = 0, and K(iλ) having spectral radius r < 1 on H N G , thus satisfying that for all > 0, there is n 0 large so that for all n ≥ n 0 K(iλ) n L(H N G ) ≤ (r + ) n . We can chose r + < 1, which implies that To conclude the proof, we want to prove that (X j + iλ j )Π(iλ) = 0 for all j = 1, . . . , κ. By the discussion above, 0 is the only joint eigenvalue of (X 1 + iλ 1 , . . . , X κ + iλ κ ) on ran Π(iλ), i.e. there is J > 0 such that κ j=1 (X j + iλ j ) α j Π(iλ) = 0 for all multi-index α ∈ N κ with length |α| = J. We already know that R has no Jordan block, and we want to deduce that this is also true for the X j 's. By Proposition 4.17, we get In particular this space does not depend on the choice of χ j (and thus ψ j ). The operator e − j t j (X j +iλ j ) : ran Π(iλ) → ran Π(iλ) is represented by a finite dimension matrix M (t) with t = (t 1 , . . . , t κ ), and R(iλ)| ran Π(iλ) = Id (since R(iλ) has no Jordan block), thus for all choices of χ j (and ψ j = −χ j ). We can thus take, for T = (T 1 , . . . , T κ ), the family ψ j converging to the Dirac mass δ T j and we obtain M (T ) = Id. This shows that M (t) = Id for all t ∈ R κ + large enough such that Lemma 4.14 can be applied and therefore (X j + iλ j )Π(iλ) = 0 for all j. This implies that ran Π(iλ) is exactly the space of Ruelle resonant states for X at iλ.
From what we have shown in Lemma 5.2, we deduce that we can write the spectral projector as Π(iλ)f = J k=1 v k f, w k L 2 with v k ∈ H N G spanning the space of joint Ruelle resonant states of the resonance iλ and w k ∈ H * N G H −N G . Recall that we have shown that the space of joint Ruelle resonant states (i.e. the range of Π(iλ)) is intrinsic, i.e. does not depend on the precise form of the parametrix. But surely the operator R(iλ) depends on the choice of the cutoff functions ψ j (see (5.1)) and thus also Π(iλ) might depend on that choice. In order to see that this is not the case, let us consider X * j = −X j + div vg (X j ) which are the adjoints with respect to the fixed measures v g . Note that by the commutativity of the X j , the operators X * j also commute and are admissible operators (in the sense of Definition 2.4) for the inverted Anosov action τ − (a) := τ (−a) which is obviously again an Anosov action (with the same positive Weyl chamber after swaping the stable and unstable bundles). Therefore we can apply the results of Section 4 to the admissible operators X * j , in particular they have discrete joint spectrum on the spaces H −N G . Using (X j + iλ j )Π(iλ) = 0 and the fact that [X j , Π(iλ)] = 0 we deduce that (X * j − iλ j )w k = 0 and thus all w k , k = 1, . . . , J are joint resonant states of the X * j . We can even see that they span the space of joint resonant states: one can perform the same parametrix construction Lemma 4.14 to X * and if we choose the same cutoff functions as in the parametrix for X j at the beginning of this section we find is the spectral projector of R X * (iλ) onto the eigenvalue 1 then we obtain Π X * (−iλ)f = Π X (iλ) * f = J k=1 w k f, v k L 2 with adjoint defined as above. By Lemma 3.9 the space of joint resonant states (X * j − iλ)w = 0, w ∈ H −N G is in the range of Π X * (−iλ), consequently the w j span the space of joint resonant states of X * with joint resonance −iλ and the w j span the space of joint resonant states of the resonance iλ. Putting everything together, we have: Lemma 5.3. Let λ ∈ a * such that iλ is a Ruelle-Taylor resonance of X. Then −iλ is also a Ruelle-Taylor resonance of X * and spaces of joint resonant states have the same dimension.
If v 1 , . . . , v J ∈ C −∞ E * u (M) and w 1 , . . . , w J ∈ C −∞ E * s (M) are such that they span the space of joint resonant states of X at iλ and X * at −iλ respectively and fulfill v j , w k L 2 = δ jk , then we can write Π(iλ) = J k=0 v k ·, w k L 2 . In particular Π(iλ) depends only on the X j but not on the choice of R(iλ).
We can now identify resonant states on the imaginary axis with some particular invariant measures.
(1) For each v ∈ C ∞ (M, R + ), the map is a non-negative Radon measure with mass µ v (M) = M v dv g , invariant by X j for all j = 1, . . . , κ in the sense µ v (X j u) = 0 for all u ∈ C ∞ (M).
is (3) Let f ∈ L 1 (W; [0, 1]) with compact support contained in W and W f > 0. Then for any u, v ∈ C ∞ (M) where dA is the Lebesgue-Haar measure on a. (4) Similarly, for λ ∈ a * , v ∈ C ∞ (M) the map } is finite dimensional and coincides with the space of finite complex measures µ with WF(µ) ⊂ E * s which are equivariant in the above sense. (5) Let v 1 , v 2 ∈ C ∞ (M, R + ) with v 1 ≤ Cv 2 for some C > 0 and iλ ∈ ia * a Ruelle-Taylor resonance. Then µ λ v 1 is absolutely continuous with bounded density with respect to µ v 2 = µ 0 v 2 . In particular any µ λ v is absolutely continuous with respect to µ 1 . Proof. First R(0)1 = 1 is clear and X has a Taylor-Ruelle resonance at λ = 0 by Lemma 3.10. If u, v ∈ C ∞ (M) are non-negative, we have a k := R(0) k u, v ≥ 0 and Note also that for each k, and each u ∈ C ∞ (M) non-negative This implies that for each v ∈ C ∞ with v ≥ 0, µ k v : u → R(0) k u, v is a Radon measure with finite mass µ k v (M) = M v dv g and thus µ v is as well. The invariance of µ v is a direct consequence of Lemma 5.3. The same holds for property (2). The invariance of the space spanned by these measures with respect to X j follows from Π(0)X j = X j Π(0) = 0, obtained by Lemma 5.2.
Let us next show that for an arbitrary Ruelle-Taylor resonance iλ ∈ ia * we get complex measures µ λ v and in the same turn prove the absolute continuity statement (5). We consider u ∈ C ∞ (M), v 1 , v 2 ∈ C ∞ (M, R + ) with v 1 ≤ v 2 and get for all k . This proves that µ λ v 1 is a complex measure. A priori, µ λ v 1 is absolutely continuous with respect to µ v 2 , so it has a L 1 density f with respect to µ v 2 . This density is actually bounded by 1 or equivalently, for every Borel set A. If A is a closed set, we can find a sequence of smooth functions g n , valued in [0, 1], which converges simply to the characteristic function of A. By dominated convergence µ λ v 1 (g n ) → µ λ v 1 (A), and likewise µ v 2 (g n ) → µ v 2 (A). Taking products of sequences of functions, or sequences 1 − g n , and a diagonal argument, we see that the set of Borel sets A for which (5.8) holds contains closed sets, and is stable by countable intersection, and complement. It is thus equal to the whole tribe of Borel sets, and the proof of f L ∞ ≤ 1 is complete.
Let us finally show (5.7). For each f ∈ L 1 (a; [0, 1]) with supp(f ) ⊂ W being compact and f > 0, we want to prove that Assume that (5.9) is satisfied for a dense set in L 1 (W) of compactly supported functions F := (f i ) i∈I ∈ C c (W). Then, for all δ > 0 small, one can find a sequence f i(n) ∈ F so that which implies (5.9) for f by our assumption on f i and since δ is arbitrarily small. We shall then show (5.9) for functions of the form ω(t 1 )q(t/t 1 ) if (t 1 ,t) ∈ R + × R κ−1 are coordinates associated to bases of vectors in small cones C with closure contained in W ∪ {0}, and ω ∈ C ∞ c (0, 1), q ∈ C ∞ c (R κ−1 ) such that supp(ω(t 1 )q(t/t 1 )) ⊂ C. We fix a small open cone C ⊂ W with arbitrarily small conic section with closure contained in W and choose a basis (A j ) κ j=1 of a so that A j ∈ C. Up to rescaling A j by some fixed large T > 0, we can assume that Lemma 4.14 applies with T j = 1/2 in the construction of Q(λ) and R(λ). We then identify a R κ by identifying the canonical basis (e j ) j of R κ with (A j ) j , and we define a scalar product on a by deciding that A j are orthonormal. We let Σ = C ∩ {A 1 + κ j=2 t j A j | t j ∈ R} be a hyperplane section of the cone C. Choose ψ ∈ C ∞ c ((−1/2, 1/2)) non-negative even with R ψ = 1, and for each σ ∈ R κ , define ψ σ (t) := κ j=1 ψ(t j − σ j ). The operators Q(0), R(0) constructed in Lemma 4.14 can be defined, for σ close to e 1 , with the cutoff function χ j so that −χ j (t j ) = ψ(t j − σ j ), we then denote Q σ , R σ the corresponding operators, which in turn are locally uniform in σ. Then µ v is given by µ v (u) = lim k→∞ R σ (0) k u, v locally uniformly in σ. This means that viewing Σ as an open subset of e 1 + R κ−1 containing e 1 , taking any q ∈ C ∞ c (Σ) with R κ−1 q(t)dt = 1 and any ω ∈ C ∞ c ((0, 1)) with 1 0 ω = 1, we have for σ(t) : Indeed, one can write, if supp(ω) ⊂ ( , (1 − )) for some > 0 and we use 1 N N k=1 ω k N → 1 0 ω = 1 as N → ∞, and R k σ u, v → µ v (u) uniformly in σ as k → ∞, so that we get the result by dominated convergence.
As noted in the introduction, we will call physical measures the measures µ v , and µ 1 will be called the full physical measure.

Imaginary
Ruelle-Taylor resonances for volume preserving actions. In this section, we are going to study the dimensions of the Ruelle-Taylor resonance at λ = 0 in the case where there is a smooth measure preserved by the action. First, we want to prove Proposition 5.6. Assume that there is a smooth invariant measure µ for the action, i.e. L X A µ = 0 for each A ∈ W. Then, for each λ ∈ ia * imaginary, there is an injective map Fix a basis A 1 , . . . , A κ ∈ W close to A 1 and write λ j := λ(A j ) and X A j (λ) := X A j +λ j . Let T j > 0 for j = 1, . . . , κ, let ε > 0 be small and consider χ j ∈ C ∞ c ([0, ∞[; [0, 1]) non-increasing with χ j = 1 in [0, T j ] and supp χ j ∈ [0, T j + ε]. We use the parametrix Q(λ) = δ Q(λ) of the proof of Lemma 4.14 and get (4.15) with those χ j . As in the proof of Lemma 4.14, F (λ) − Id = R(λ) + K(λ) with K(λ) compact on H N G and R(λ) < 1/2, and by the Remark following Lemma 4.14, we can choose T 1 > 0 large and T j > 0 small for j = 2, . . . κ so that this still holds. Using Lemma 4.13, we deduce ran(Π 0 (λ)) ⊂ C −∞ E * u (M; Λa * ) if Π 0 (λ) is the spectral projector of F (λ) at z = 0. We will show that the range of the spectral projector Π 0 (λ) at z = 0 of F (λ) actually satisfies ran Π 0 (λ) ⊂ C ∞ (M; Λa * ). (5.18) Since F (λ) is a scalar operator, we can work on scalar valued distributions, and we shall then identify F (λ) with an operator H N G → H N G for some N > 0 large enough, and fixed. Using Lemma 5.1, z = 1 is at most a pole of order 1 of (Id − F (λ) − z) −1 , so that each u ∈ ran(Π 0 (λ)) satisfies F (λ)u = 0. Then let u ∈ H N G such that F (λ)u = 0.
For T 1 > 0 large enough but fixed and T 2 , . . . , T κ small enough, one can find a closed neighborhood W u of E * u ∩ ∂T * M in the fiber radial compactification of T * M, which is conic for |ξ| large, 0 < c 1 < c 2 such that for all t 1 ∈ [T 1 /2, T 1 + ε] and all t j ∈ [0, T j + ε] when j ≥ 2 we have First we choose b 0 ∈ S 0 (M) with the following properties: first, then, for each t = (t 1 , . . . , t κ ) with t j ∈ [T j , T j + ε] the symbol is equal to 1 on {(x, ξ) ∈ E * u | |ξ| ∈ [c 1 , c 2 ]}, and third, by a partition of unity we can insure that there is c 0 ∈ S 0 (M) such that b 2 0 + c 2 0 = 1. If B 0 = Op h (b 0 ) and C 0 = Op h (c 0 ) are the corresponding operators then B 0 B * 0 + C 0 C * 0 = Id + hR with R ∈ Ψ −1 h (M). As in the construction of a parametrix, one can modify the symbols by lower order terms and get symbols b, c such that for B = Op h (b) and C = Op h (c) we have Id−B * B −C * C ∈ h ∞ Ψ 0 h (M). Indeed, for S ∈ Ψ −1 h (M) with real principal symbol (using Op h (q) − Op h (q) * ∈ hΨ 0 h (M) if q ∈ S 0 (M) is real-valued), we have M) and, as B 0 + C 0 is semiclassically elliptic, we can invert it microlocally and find s ∈ S 0 (M) so that S = Op h (s) satisfies R + 2S(B 0 + C 0 ) ∈ hΨ 0 (M) and we gain a power of h if we correct B 0 , C 0 by hS. This argument then can be iterated. Note furthermore that the regions where b 0 = 0 respectively b 0 = 1 are still valid for b.
Note that the escape function G can be chosen so that the order function m ≥ 0 in the region T * M \ W u for |ξ| large enough. Since u ∈ H N G , we thus have Bu ∈ L 2 . Let χ ∈ C ∞ (R κ ) be given by χ(t) = (−1) κ κ j=1 χ j (t j ) ≥ 0 for t ∈ R κ . Recalling that F (λ) is the operator introduced in (4.15), we can write, using the semiclassical Egorov Lemma in its simple form of a coordinate change [DZ19, Proposition E19], ). This gives, using e −t j X j (λ) L(L 2 ) = 1 by the fact that µ is invariant, and using Cauchy-Schwarz We can then write, since (R + ) κ χ(t)dt = 1, Next, recall that supp χ(t) ⊂ j≥1 [T j , T j + ]. We claim that for t ∈ suppχ there is e t ∈ S 0 (M; [0, 1]) such that we have and such that e t (x, ξ) = 1 + O(h) in the region {(x, ξ) ∈ E * u | |ξ| ∈ [c 1 , c 2 ]}. Indeed, E t is microlocally equal to C t := e κ j=1 t j X A j (λ) Ce − κ j=1 t j X A j (λ) on WF h (B t ) and to B on the complement of WF h (B t ). This implies, thanks to (5.19), We thus obtain We then conclude that WF(u) ∩ E * u = ∅, which also shows that u ∈ C ∞ and (5.18). Then we define the following map which is well defined since ran Π 0 (λ) ⊂ C ∞ Λ. We claim that this map is injective: let u = d X(λ) v ∈ ran Π 0 (λ) with v ∈ C ∞ Λ j , then we need to show that u = d X(λ) w for some w ∈ ran Π 0 (λ). But is suffices to use [d X(λ) , Π 0 (λ)] = 0 to see that u = Π 0 (λ)u = d X(λ) Π 0 (λ)v. This proves the claim and concludes the proof of the lemma by using also the isomorphism (4.14).
Lemma 5.7. Assume that there is a smooth invariant measure µ for the action, i.e. L X A µ = 0 for each A ∈ a, and that supp(µ) = M. Then the periodic tori are dense in M.
Proof. Since M is compact, the measure is finite, so we can apply Poincaré's recurrence theorem: almost every point x of M is recurrent, i.e. its orbit comes back infinitely close to x infinitely many times (and this for each direction of the action). Katok-Spatzier [KS94, Theorem 2.4] proved a closing lemma for Anosov actions: there is C, δ > 0 such that whenever there is x ∈ M and t ∈ W with d(τ (t)x, x) < δ, t > C, then there is a periodic torus for the action at distance at most 1 δ d(τ (t)x, x) from x.
Proposition 5.8. Assume that there is a smooth invariant measure µ for the action, with supp(µ) = M. Then and the cohomology space is generated by the constant forms e i 1 ∧ · · · ∧ e iκ if (e j ) j is a basis of a * .
Proof. In the proof of Proposition 5.6 with λ = 0, we have defined an operator F (0) that is Fredholm on H N G and Π 0 (0) is its spectral projector at z = 0, with ran(Π 0 (0)) ⊂ C ∞ (M). Recall also that F (0) is scalar and can thus be considered as an operator on functions. Let us show that ran(Π 0 (0)) = R consists only of constants under our assumptions. Pick u ∈ C ∞ (M) such that F (0)u = 0. Let x ∈ M belong to a closed orbit in the Weyl chamber, i.e. ϕ X A t 0 (x) = x for some A ∈ W and t 0 > 0. Then it is a classical result that the orbit T x := {ϕ XÃ s (x) | s ∈ R,Ã ∈ a} is a closed κ-dimensional torus (a proof compatible with the present notation can e.g. be found in [GBGW, Lemma 3.1]). It is isomorphic to R κ /Z κ by the map for some basis A i ∈ a. Note that ψ * x (e s X A u)(t) = ψ * x u(t + s). Let us restrict the identity F (0)u = 0 or equivalently R(0)u = u on T x . We can decompose v := ψ * x u into Fourier series Recall that the basis for which R(0) was constructed in Proposition 5.6 was denoted by A 1 , . . . , A κ ∈ a. We can express this basis in terms of the basis A j of the periodic torus via some base change matrix A j = i M ij A i (using κ =1 s A = ,i s M i A i ) the identity Using that χ ≥ 0 and χ(s) > 0 in some open set and that M is invertible, we see that either v k = 0 or k = 0, i.e. v(t) = v(0) is constant. Therefore u is constant on each periodic torus. Since u is smooth and the periodic tori are dense, this implies that d X u = 0 and u(ϕ X A t (x)) = u(x) for each x ∈ M, t ∈ R and A ∈ a. Taking A ∈ W, there is ν > 0 such that for each t > 0 large enough so that |dϕ X A t v| ≤ e −νt |v| for each v ∈ E s . Thus Letting t → ∞, we conclude that du| Es = 0. The same argument with t < 0 shows that du| Eu = 0 and therefore du = 0. Since F (0)1 = 0, this shows that, when viewed as an operator on Λa * , ran Π 0 (0) is exactly the space of constant forms. We can then use the isomorphism (4.14) to conclude the proof since it is direct to see that constant forms e i 1 ∧ · · · ∧ e i j form a basis of ker d X /Im d X on Im(Π 0 (0)) (as d X | ran Π 0 (0) = 0).
Note that in their paper [KS94] Katok-Spatzier study the first cohomology group of dynamical systems and show that any smooth cocycle is smoothly conjugated to a constant function [KS94, Theorem 2.9 a)]. In our language of Taylor complexes, this result implies that for standard Anosov actions dim(ker C ∞ Λ 1 d X / ran C ∞ Λ 1 d X ) = κ and is spanned by the constant forms. Combining this fact with Proposition 5.8, we obtain: Corollary 5.9. If the Anosov R κ -action is standard in the sense of [KS94], then the map (5.17) is an isomorphism for j = 1.

5.3.
Ruelle-Taylor resonances and mixing properties. In this section we do not assume anymore that a volume measure is preserved, and want to establish the following relation of Ruelle-Taylor resonances and mixing properties.
Proposition 5.10. Let X be an Anosov action on M then the following are equivalent: (1) There is a direction A 0 ∈ a such that ϕ X A 0 t is weakly mixing with respect to the full physical measure µ 1 .
(2) 0 is the only Ruelle-Taylor resonance on ia * and there is a unique normalized physical measure µ 1 . (3) For each A ∈ W, ϕ X A t is strongly mixing with respect to the full physical measure µ 1 .
Proof. Obviously (3) ⇒ (1). So let us prove (1) ⇒ (2): Assume that there is either a non-zero Ruelle-Taylor resonance iλ ∈ ia * or a non-unique normalized physical measure. Then by Proposition 5.4(5) there is a non-constant bounded density f ∈ L ∞ (M, µ 1 ) with X A f = iλ(A)f for all A ∈ a (setting λ = 0 if the density comes from the non-uniqueness of the physical measure). As f is non-constant there exists g ∈ L ∞ (M, µ 1 ) with g dµ 1 = 0 but gf dµ 1 = 0. With these two functions, the correlation function is not weakly mixing. We will now prove (2)⇒(3) using the regularity of a joint spectral measure: Let us first introduce these measures. We consider the space L 2 (M, µ 1 ). Since the measure µ 1 is flowinvariant, the flow acts as unitary operators on L 2 (M, µ 1 ). In particular, for each A ∈ a, X A is skew-adjoint when acting on L 2 (M, µ 1 ) with domain D(X A ) = u ∈ L 2 (M, µ 1 ) lim t 1 t (e tX A u − u) exists = {u ∈ L 2 (M, µ 1 ) | X A u ∈ L 2 (M, µ 1 )}.
Additionally, since the flows commute, the X A are strongly commuting, so that we can apply the joint spectral theorem -see Theorem 5.21 in [Sch12]. There exists a Borel, L 2 (M, µ 1 )projector valued, measure ν on a * such that for u ∈ L 2 (M, µ 1 ), We will prove the following regularity result of these measures below: Lemma 5.11. Let X be an Anosov action. Assume that there is no non-zero purely imaginary Ruelle-Taylor resonance and a unique normalized physical measure. Then for any f, g ∈ C ∞ (M) with M f dµ 1 = M g dµ 1 = 0 we consider ν f,g (θ) := ν(θ)f, g L 2 (M,µ 1 ) which are finite complex valued measures on a * . Then the analytic wavefront set 6 7 WF a (ν f,g ) ⊂ a * × a fulfills WF a (ν f,g ) ∩ (a * × W) = ∅.
Before proving this Lemma let us show that it implies (3). Take A 0 ∈ W, f, g as in the above Lemma, then the spectral theorem yields Given any ε > 0, using the fact that ν f,g is finite, there is a cutoff function χ K ∈ C ∞ c (a * , [0, 1]) equal to 1 on a sufficiently large compact set K ⊂ a * such that | a * e −iϑ(A 0 )t (1−χ K )dν f,g (ϑ)| ≤ ε/2 uniformly in t. Furthermore by the fact that the wavefront set is empty in the direction of the Weyl chamber W we deduce that there is T such that | a * e −iϑ(A 0 )t χ K dν f,g (ϑ)| ≤ ε/2 for any t > T thus lim t→∞ C f,g (t, A 0 ) = 0. The passage to arbitrary L 2 (M, µ 1 ) functions follows by the density of the smooth functions.
Proof of Lemma 5.11. Let us pick any A 0 ∈ W and a basis A 1 , . . . , A κ ∈ W such that these elements span an open cone around A 0 . With this basis we identify the joint spectral measure with a measure on R κ . Recall the definition of R σ (iλ) from the proof of Proposition 5.4 which 6 See [Fol89, §3.3] for the definition and basic properties of the analytic wavefront set. In our proof, we need to use a non-quadratic phase, and only the quadratic case is treated in Folland; however this is just a slight technical hurdle, as mentionned by Folland at the start of p. 160. For completeness, however, we will refer to [Sjo82] (in French).
Recall furthermore that we had identified a * ∼ = R κ by the above choice of the basis A j , thus our result implies that there is no analytic wavefront set in a * × { c j A j |c j ∈ R + }, but as the A j span an arbitrary subcone of W we also get the absence of analytic wavefront set in a * × W and we have completed the proof of Lemma 5.11.

Appendix A. Tools from microlocal analysis
We recall here some essentials of microlocal analysis. In the paper, we are working with pseudodifferential operators acting on C ∞ (M; E) ⊗ Λa * C ∼ = C ∞ (M; E ⊗ Λa * C ). Note that by fixing an arbitrary scalar product on a * the bundle E ⊗ Λ := E ⊗ Λa * C → M is again a Riemannian bundle. We will therefore introduce notations for pseudodifferential operators on general Riemannian bundles E → M over a compact Riemannian manifold M. Only when we want to exploit some specific structures of E ⊗ Λ, will we refer to this particular bundle.
If furthermore A and B are holomorphic families of operators, then Q can be chosen to be holomorphic as well.
Lemma A.8. Let F ∈ Diffeo(M) be a smooth diffeomorphism and let F ∈ Diffeo(E) be a lift of F , i.e. F acts linearly in the fibers and π • F = F • π for π : E → M the fiber projection. Define the transfer operator there exists a decomposition A = K + R, where K ∈ Ψ −∞ (M; E) is a smoothing and hence L 2 -compact operator and R L 2 →L 2 ≤ C. If A t is a smooth family in Ψ 0 ρ (M; E) for t ∈ [t 1 , t 2 ], the decomposition A t = R t + K t can be chosen so that t → R t and t → K t are continuous in t.
Proof. See [FRS08, Lemma 14] for the proof. The dependence in t is straightforward from the proof.
We conclude this appendix by mentioning that one can use a small semiclassical parameter h > 0 in the quantization, in which case we shall write Op h , by using the expression in a local chart Op h (a)f (x) = 1 (2πh) n e i(x−x )ξ h a(x, ξ)f (x )dξdx if a is supported in a chart. We do not use this semiclassical quantization except in the subsection 5.2 and we refer to [DZ19, Appendix E] for the results on semiclassical pseudodifferential operators that we will use. The class One of the advantages is that one can get the estimate Op h (a) L 2 →L 2 ≤ sup x,ξ |a(x, ξ)| + O(h) for small h > 0 and if a ∈ S 0 (M; E).