Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space

. Given an Anosov vector ﬁeld X 0 , all suﬃciently close vector ﬁelds are also of Anosov type. In this note, we check that the anisotropic spaces described by Faure and Sj¨ostrand and by Dyatlov and Zworski can be chosen adapted to any smooth vector ﬁeld suﬃciently close to X 0 in C 1 norm.


Introduction
In this note, M will denote a compact manifold of dimension n, and X 0 a smooth Anosov vector field on M . That is, there exists a splitting T M = RX 0 ⊕ E u 0 ⊕ E s 0 . The vector field X 0 never vanishes. The splitting is invariant under the flow of X 0 , which is denoted ϕ 0 t . We also have constants C, β > 0 such that for all t ≥ 0, (here the norm is a fixed norm a priori, and one can check that although the constants depend on that choice of norm, their existence does not). Starting with [2], several authors have built some anisotropic spaces of distributions to study the spectral properties of hyperbolic dynamics, of which Anosov flows are a prime example. This enables the study of so-called Ruelle-Pollicott resonances, originally defined using the (quite different) techniques of Markov partitions [14,13]. They appeared as the poles of some zeta functions, popularized by Smale [15].
Since there is no canonical way to build those spaces, various constructions have been developed. In [1], one can find a thorough review of the literature, weighting the different advantages that each construction has to offer. In 2008, Faure and Sjöstrand [7] introduced such a construction. In its functional analytic aspects, it relied on microlocal analysis tools. Provided one understands standard tools in that field, one can present these spaces in the following fashion: (here Op denotes a classical quantization, and G ∈ S log (T * M )). Using these technique led to new developments. For example the description of the correspondence between classical and quantum spectrum in constant curvature at the level of eigenfunctions. Several results already obtained have also been reproved using these techniques. For example the meromorphic continuation of the dynamical zeta function, proved in general by [9] and in the smooth case with microlocal methods by [6].
So far, one particular aspect of the theory that was not reproduced by the microlocal techniques are perturbations. In this note, we will explain how this can be done. The first step will be to prove: There exists η > 0 such that the following holds: For any R > 0, we can find G ∈ S log (T * M ) such that for X − X 0 C 1 < η and X C ∞ , the spectrum of X acting on H G is discrete in This discrete spectrum corresponds to the poles of the Schwartz kernel of the resolvent of the flow, so it does not really depend on the choice of G; it is called the Ruelle-Pollicott spectrum of resonances of X. Our next result is Considering only the dominating resonance (with maximal real part), this was proved in several contexts [3,10], and the proof for several resonances is the same as for only one; however we could not find a reference for this general statement. A similar statement should also hold in finite regularity (if X 0 is C r with r > 1), but we are not using the best tools to tackle this type of question, so we ignore it altogether.
While this paper was being elaborated, other authors were considering a similar question, to study Fried's conjecture in [12]; discussing with them helped improve the present note. I also thank Y. Chaubet for the discussion that led to Lemma 4.3.

Microlocal proof of the main theorem
In what follows, we consider X to be another vector field, assumed to be close to X 0 in C 1 norm -we will be clear when we use that assumption. We will denote by ϕ t the corresponding flow. The notation * (0) means that the statement is valid both for the object related to X 0 and X.
While the fundamental ideas are always similar, there are several ways to present the proof of Theorem 1.1. We will use the version that was presented in [6]. They use spaces of the form H rG . They assume that as |ξ| → +∞, m being a smooth 0-homogeneous function on T * M . The action of X can be lifted to T * M by considering the Hamiltonian flow Φ t (x, ξ) generated by p = ξ(X): We denote by X * the generator of that flow. Since p is 1-homogeneous, Φ t also is, and it acts on S * M = T * M/R + ∂T * M . The proof of [6] is based on the use of the notion of sinks and sources. A sink for Φ t is a Φ t -invariant conical closed set L ⊂ T * M such that there exists a conical neighborhood U of L ∩ ∂T * M and constants C , β such that for ξ ∈ U, t > 0, One can define subbundles of T * M of particular interest by setting

One gets the decomposition
The bundles E * u,s,0 are invariant under the corresponding flow Φ t . One can check that E * u0 is a sink, and E * s0 is a source for Φ 0 t . The proofs of Propositions 3.1 and 3.2 in [6] require the following input.
(1) The flow Φ t admits a source E * u and a sink E * s . Both are contained in {p = 0} = X ⊥ .
(2) Given neighborhoods U u and U s as in the definition of sinks and sources, there is a Provided conditions (1)-(3) are satisfied, the proof in [6] applies, and we obtain the following. For any R, one can find an r > 0 such that the spectrum of X on H rG is discrete in { s > −R}. For our purposes this is not sufficient because we cannot let the r depend on X as long as X is close enough to X 0 . However, one can give a rough estimate on the value of r.
Following the proof of Proposition 2.6 in [6], one finds that it applies (and this is the condition for the proof of Propositions 3.1 and 3.2 to also apply) if r satisfies Let us explain how one obtains the constants c and C 0 . The c comes from Lemma C.1 in [6]. For some T 1 > 0, one has |Φ T1 (ξ)| > 2|ξ| for all ξ ∈ U u , and c is defined by We deduce that where |Φ t (x, ξ)| ≤ e Λ|t| |ξ| for ξ ∈ T * M , and C , β are the constants in the definition of being a sink for E * u . The constant Λ can be estimated directly as Λ ≤ dX L ∞ by usual estimates.
The constant C 0 in the proof of Proposition 2.6 is chosen at the end of the proof to absorb some other terms. To be more precise, C 0 has to satisfy We call r X (s) the minimal strength, and this is called the threshold condition. The proof of Theorem 1.1 will thus be done if we can prove the following lemma: There are conical open sets U u and U s , m ∈ C ∞ (S * M ) and η > 0 such that whenever X − X 0 C 1 < η, X satisfies (1), (2) and (3), with U u (resp. U s ) an admissible neighborhood for E * u (resp. E * s ). Additionally, the constants C and β satisfy where C, β are the constants defined in equation (1.1). The minimal strength r X (s) is thus bounded uniformly.
Except for the construction of the weight function m, our Lemma can probably be seen as a corollary of structural stability for Anosov flows [4]. However, we can give a full proof directly, which is quite elementary.
A last remark is that to work with the spaces Op(e −rG )H k (M ) = H rG+k log ξ , the same proof applies. The threshold conditions become r + k > r X (s) and r − k > r X (s), i.e. r > r X (s) + |k|.

Building the weight function
This section is devoted to proving Lemma 2.1. We are now working in S * M . We start by building a weight adapted to X 0 , following the strategy for [8]. We denote by Φ This is a very usual lemma, contained in [11,Theorem 3.2].
In the proof, the constant 0 < C < ∞ may change at every line, but it is always controlled by the lower bound on the angles between the bundles.
Let us now come back to our perturbation problem. Consider X = X 0 + λV with V a smooth vector field with V C 1 ≤ 1, and λ > 0 small. The vector fields generating Φ (0) t are the Hamiltonian vector fields of the principal symbols of −iX (0) , which are ξ(X (0) ). In particular, they involve the first derivative of the vector fields X (0) , so that they are O(λ)-C 0 -close, with a constant depending on V only through V C 1 .
Then the corresponding vector fields on S * M , X ∞ 0 and X ∞ = X ∞ 0 + λV ∞ that generate the boundary flows Φ  We get directly that X ∞ 0 m ≥ 0.
But we also have On the support of χ (m − T ), we have X ∞ 0 m ≥ δ. In particular, with λ smaller than η 0 = δ/CT with C > 0 large enough, we get that X ∞ m ≥ 0.

Lemma 3.4.
There is an 0 < η ≤ η 0 such that the following holds. Let X be a C 1 vector field such that Proof. Following the arguments in the proof of Lemma 3.2, we can find open cones V 1 ⊂ V 2 containing E * u0 as small as desired, such that V 1 ⊂ V 2 , and for some t > 0, Since we can write this as a decreasing intersection of compact sets (compact in T * x M ∪S * x M ), it is non empty, closed, and it is a cone by linearity of Φ t . We deduce that E * u is a sink for Φ t . Likewise, we can find similar cones around E * s0 for negative times, and obtain that the corresponding E * s is a source. For the points that are neither in the source nor in the sink, we can directly use Lemma 3.2. Finally, since we could choose the neighborhood V 2 as small as desired, since E * u ⊂ V 2 , and since m = +1 in a neighborhood of E * u0 , the proof is complete.

Perturbation of resonances
The main step in the proof of [6] is to prove the following (their Proposition 3.4). They use semi-classical spaces Op h being now a semi-classical quantization with small parameter h > 0. This space is the same as H rG , albeit with a different, equivalent norm depending on h > 0. We fix Q a positive self-adjoint pseudor which is microsupported and elliptic around the zero section.
One can check that the constants h 0 and C can be estimated as where the number of derivatives N (r) depends only on r and the dimension. Now, we consider a smooth family X of vector fields perturbating X 0 . From Lemma 2.1, we can find 0 > 0, and r(s) a non increasing function of s, so that r(s) ≥ r X (s) for all ∈ ]− 0 , 0 [. Using Lemma 4.1, we get a uniform control lemma: Consider X a smooth family of vector fields perturbating X 0 . Then there is 0 > 0 such that the following holds. Given any s 0 > 0, k and r > r We observe that vanishes. For fixed X C 1 -close to X 0 , F (X, ·) is a holomorphic function in Ω h,s0 . Now, we consider a smooth family X . We observe that Since in the Tr in the formula for ∂ F the operators are smoothing, this trace does not depend on the Sobolev space with respect to which we are taking the trace. We will denote by · Tr the norm on the space L 1 (H h,rG , H h,rG ). Additionally, is uniformly bounded (see equation B.5.15 in [5]). From the formula, we deduce that ∂ F (X , s) defines a holomorphic function in the s parameter. In particular, to obtain estimates on its derivatives in s, it suffices to estimate (with a constant C changing at every line). Here, we needed (X − h −1 Q − s) −1 to be bounded on H h,rG−log ξ , so that the computation is only valid for h < h 1 .
By an induction argument, we obtain that for h < h k , Since the resonances do not depend on the choice of space, it does not matter if we were using H rG or H h,rG+k log ξ .
By the Rouché theorem, we deduce that the zeroes of F (X , s) in Ω δ can be parametrized by continuous functions, which are C ∞ when the resonances are simple. This proves Corollary 1.2.
As a remark, let us consider spectral projectors. We come back to classical operators (h = 1). Retaking the notations of Corollary 1.2, we consider an open set Ω, and λ 1 ( ), . . . , λ N ( ) continuous functions such that the spectrum of X intersected with Ω is exactly {λ 1 ( ), . . . , λ N ( )} counted with multiplicity. Next, we pick a closed curve γ contained in Ω, assuming that it does not contain any λ i (0). Then this remains true on some interval | | < . As a consequence, for each such , the following operator is well defined: By usual arguments, one can show that for r > r(s 0 ) + |k|, where s 0 = inf γ(t), it is a bounded projector in L(H rG+k log ξ ). Using the resolvent formula, we get for | | < , which is a bounded operator from H rG to H rG−log ξ and also from H rG+log ξ to H rG . On the other hand, since Π γ has finite rank and is a projector, its derivatives also have finite rank. This comes from the relation ∂ Π = Π∂ Π + (∂ Π)Π.
Since it has finite rank, the range of ∂ Π is contained in H rG , and it is bounded on that space (provided r > r 0 + 1). By induction, we deduce Lemma 4.3. Consider a closed curve γ such that for | | < 0 no resonance crosses γ. Then, for r > r(s 0 ) + k + 1, → Π γ ( ) is a C k family of bounded operators on H rG .
Since the generator is elliptic in the direction of the flow, one could probably refine this statement to show that Π γ gains regularity in that direction, however we will not investigate this here.