aa r X i v : . [ m a t h . D S ] M a r ON THE POLLICOTT–RUELLE RESONANCES
JOEL ANTONIO–V ´ASQUEZ
Abstract.
The purpose of this survey is to present the recent advances about thePollicott–Ruelle resonances. introduction Suppose that M is a smooth compact manifold and let ϕ : M −→ M be a smoothflow such that M is considered a φ t –invariant set for some t >
0. The flow ϕ t iscalled Anosov flow if rather is a hyperbolic set for ϕ t in the sense of [KaHa, Definition6.4.18] and is generated by some smooth vector field V such that ϕ t := e tV . Let f, g ∈ C ∞ ( M ) be smooth functions and µ a φ –invariant probability measure, then thecorrelation function is defined as ρ f,g ( t ) = Z M f ( ϕ − t ( x )) g ( x ) dµ, for any x ∈ M . (1.1)The power spectrum of (1.1) is the Fourier transform such that b ρ f,g ( λ ) = Z ∞ ρ f,g ( t ) e iλt dt, (1.2)is meromorphic for | Im λ | >
0. Thus, the asymtoptic behaviour of ρ f,g is controlledby the poles of the extension b ρ f,g , such poles are known as the Pollicott–Ruelle res-onances . In other words, they are complex numbers, which describe fine of decay ofcorrelations for an Anosov Flow on a smooth compact manifold, and were initiallystudied by M. Pollicott [Po85, Po86] and D. Ruelle [Ru86, Ru87]. From another pointof view, the Pollicott–Ruelle resonances are also the singularities of the meromorphicextension of the Ruelle zeta function, which was conjectured by S. Smale in 1967[Sm]. Such conjecture, has been proved by Giulietti–Liverani-Pollicot [GiLiPo] forcompact manifolds. Later, Arnoldi-Faure-Weich [AFW] defined resonances on openhyperbolic surfaces and Faure–Tsujii [FaTsb] defined resonances for the Grassmanianbundle of an Anosov flow. Recently, Dyatlov–Guillarmou [DyGu14] were able to de-fine Pollicott–Ruelle resonances for open hyperbolic systems on a more general waycompared to [AFW, FaTsb] via a microlocal approach of Faure–Sj¨ostrand [FaSj] andDyatlov–Zworski [DyZw13], holding the results of [Po86, §
7] and as a consequence,they were able to show that the Ruelle zeta function extends meromorphically to theentire complex plane.
Structure of the survey.
Section § § On the compact case
On the functional analysis proof.
In 2012, Giulietti–Liverani–Pollicott [GiLiPo]showed the existence of Pollicott–Ruelle resonances for compact manifolds, such proofwas given thought the Ruelle zeta function for C r Anosov flows for r > C ∞ flows the zeta functionis meromorphic on the entire complex plane. Based on the statement that from C r flows, we can obtain a strip in which ζ Ruelle ( z ) is meromorphic of width unboundedlyincreasing with r [Fr], such work was an expansion of [GoLi, BuLi, LiTs, Lia, Lib, BaLi].2.1.1. Definitions.
In this subsection, we work with the following assumptions: (B1) M is a d –dimensional connected, compact and orientable C ∞ Riemannian man-ifold for some d ∈ Z + , V is a C ∞ nonvanishing vector field on M , and ϕ t = e tV is the corresponding flow; (B2) for each x ∈ M , there is a splitting T x M = E s ( x ) ⊕ E ( x ) ⊕ E u ( x ) , (2.1)where E is the one–dimensional subspace tangent to the flow, such that forsome constants C, γ > | dϕ t ( x ) · v | ≤ Ce − γ | t | | v | if t ≥ , v ∈ E s ; | dϕ t ( x ) · v | ≤ Ce − γ | t | | v | if t ≤ , v ∈ E u ; C − | v | ≤ | dϕ t ( v ) | ≤ C | v | if t ∈ R , v ∈ E . (2.2)We denote d s = dim( E s ) and d u = dim( E u ) to distinguish the dimension of thestable and unstable subspaces, respectively.In the context of the assumptions ( B B ζ ( z ) = Q p (1 − p − z ) − ), replacing p byprimitive closed orbits. Thus, ζ Ruelle ( z ) = Y τ ∈T p (1 − e − zλ ( τ ) ) − , z ∈ C , (2.3)where T p denotes the set of prime orbits and λ ( τ ) denotes the periodic of the closedorbit τ . According to [PaPo, § ζ Ruelle ( z ) isanalytic and nonzero for Re( z ) ≥ h top ( ϕ ) apart for a single pole at z = h top ( ϕ ).In order to understand the whole case, some relevant definitions will be given below,however, the assertions of some concepts will be cited and won’t be part of the mainproofs. N THE POLLICOTT–RUELLE RESONANCES 3
As part of the main definitions, Ruelle [Ru76] related the transfer operator with thedynamical Fredholm determinant and defined D ℓ as the dynamical determinants, whichare functions defined from weighted periodic orbit data of a differentiable dynamicalsystem . Definition 2.1.
The dynamical (also known as Fredholm–Ruelle) determinants aredefined as D ℓ ( z ) = exp − X τ ∈T tr( ∧ ℓ ( D hyp ϕ − λ ( τ ) )) e − zλ ( τ ) µ ( τ ) ǫ ( τ ) | det( − D hyp ϕ − λ ( τ ) ) | ! , (2.4) where ǫ ( τ ) is 1 if the flow preserves the orientation of E s along τ and -1 otherwise.More precisely, ǫ ( τ ) = sign(det( D ϕ − λ ( τ ) | E s )) . The symbol D hyp ϕ − t in Definition 2.1, indicates the derivative of the map induced bythe local transverse sections to the orbit (one at x , the other at ϕ t ( x )) and can berepresented as a ( d − × ( d − ∧ ℓ A we mean the matrixassociated to the standard ℓ –th exterior product of A – see more about dynamicaldeterminants in [Ba16, BaTs08]. Given any φ –invariant probability measure µ on M with h µ ( φ ) being the measure theoric entropy of φ . The topological entropy h top ( φ )can be defined by h top ( φ ) ≡ sup { h µ ( φ ) : µ is a φ –invariant probability measure } . (2.5)Thus, given 0 ≤ ℓ ≤ d − , τ ∈ T , we let χ ℓ ( τ ) = tr( ∧ ℓ ( D hyp ϕ − λ ( τ ) )) ǫ ( τ ) | det( − D hyp ϕ − λ ( τ ) ) | , in order to write Equation (2.4) in a shorter way as D ℓ ( z ) = exp − X τ ∈T χ ℓ ( τ ) µ ( τ ) e − zλ ( τ ) ! . (2.6)Now let us define C r sections of ∧ ℓ ( T ∗ M ) as the space Ω ℓv ( M ) of ℓ –forms on M for all v, ℓ ∈ N . Definition 2.2.
Let Ω ℓ ,v ⊂ Ω ℓv ( M ) be the subspace of forms null in the flow direction,such that Ω ℓ ,v ( M ) = { h ∈ Ω ℓr ( M ) : h ( V, . . . ) = 0 } . For a detailed construction and proof of Definition 2.2 – see [GiLiPo, § Definition 2.3.
For all p ∈ N , q ∈ R + , ℓ ∈ { , . . . , d − } we define the spaces B p,q,ℓ to be the closures of Ω ℓ ,r ( M ) with respect to the norm k·k − ,p,q,ℓ and the spaces B p,q,ℓ + tobe the closures of Ω ℓ ,r ( M ) with respect to the norm k·k + ,p,q,ℓ . JOEL ANTONIO–V ´ASQUEZ
The construction of the B space in Definition 2.3 is detailed in [GiLiPo, § z ) is large enough and follows aproduct analogous [GiLiPo, Equation (2.5)] such that d − Y ℓ =0 D ℓ ( z ) ( − ℓ + ds +1 = ζ Ruelle ( z ) . (2.7)Now, to take care of the t ≤ t , we introduce the dynamical norm k·k p,q,ℓ – see [GiLiPo, § h ∈ Ω ℓτ ( M ), we set k h k p,q,ℓ = sup s ≤ t (cid:13)(cid:13) L ( ℓ ) s h (cid:13)(cid:13) p,q,ℓ , (2.8)where L ( ℓ ) t is a linear operator such that L ( ℓ ) t : Ω ℓ ,r − ( M ) −→ Ω ℓ ,r − ( M ), for some t ∈ R + . Furthermore, L ( ℓ ) t h := ϕ ∗− t h, (2.9)for some h ∈ Ω ℓ ,r − ( M ) – see more [GiLiPo, § e B p,q,ℓ = Ω ℓ ,r k·k p,q,ℓ ⊂ B p,q,ℓ . Let λ i,ℓ be the eigenvalues of X ( ℓ ) . Then for each z ∈ B ( ξ, ρ p,q,ℓ ), we let e B ( ξ − z, ξ ) = Y λ i,ℓ ∈ B ( ξ,ρ p,q,ℓ ) z − λ i,ℓ ξ − λ i,ℓ ψ ( ξ, z ) , (2.10)where ψ ( ξ, z ) is analytic and nonzero for z ∈ B ( ξ, ρ p,q,ℓ ). Thus, the Equation (2.10)shows that the poles of ζ Ruelle are a subset of the eigenvalues of the X ( ℓ ) . Definition 2.4.
Given an operator A ∈ L ( B p,q,ℓ , B p,q,ℓ ) , we define the flat trace astr ♭ ( A ) = lim ǫ → Z M X α,i h ω α,i , A ( j ǫ,α,i,x ) i x ω M ( x ) , (2.11) where ω α,i is the dual of 1–forms such that ω α,i (ˆ e α,j ) = δ i,j and j ǫ,α,i,x ( y ) is defined in [GiLiPo, Equation (5.2)] , then provided the limit exists. For a detailed proof of Definition 2.4 – see [GiLiPo, § On the proof.
We start stablishing in what region ζ Ruelle ( z ) is meromorphic. Lemma 2.5.
For any C r Anosov flow ϕ t with r > , then ζ Ruelle ( z ) is meromorphicin the region Re( z ) > h top ( ϕ ) − λ (cid:22) r − (cid:23) (2.12) where λ is determined by the Anosov splitting. N THE POLLICOTT–RUELLE RESONANCES 5
Lemma 2.5 follows by the study of dynamical determinants (Definition 2.1). In fact,to study in what region ζ Ruelle is meromorphic, we must study in what region thedynamical determinants are so. Moreover, for ξ, z ∈ C we let e D ( ξ, z ) = exp − ∞ X n =1 ξ n n ! X τ ∈T χ ℓ ( τ ) µ ( τ ) λ ( τ ) n e − zλ ( τ ) ! . (2.13)Using (2.13), for Re( z ) sufficiently large and | ξ − z | sufficiently small, we can write e D ( ξ − z, ξ ) = exp − ∞ X n =1 ( ξ − z ) n n ! X τ ∈T χ ℓ ( τ ) µ ( τ ) λ ( τ ) n e − ξλ ( τ ) ! = exp − X τ ∈T χ ℓ ( τ ) µ ( τ ) ( e − zλ ( τ ) − e − ξλ ( τ ) ) ! = D ℓ ( z ) D ℓ ( ξ ) . (2.14) Theorem 1. ζ Ruelle ( z ) is analytic for Re( z ) > h top ( ϕ ) and nonzero for Re( z ) > max { h top ( ϕ ) − λ (cid:4) r − (cid:5) , h top ( ϕ ) − λ } . Furthermore, if the flow is topologically mixingthen ζ Ruelle ( z ) has no poles on the line { h top ( ϕ ) + ib } b ∈ R apart from the single simplepole at z = h top ( ϕ ) . By Theorem 1, ζ Ruelle ( z ) is meromorphic in the entire complex plane for smooth geodesicflows on any manifold that asserts the assumptions ( B B ζ Ruelle ( z )has no zeroes or poles on the line { h top ( ϕ + ib ) } b ∈ R , except at z = h top ( ϕ ) where ζ Ruelle ( z ) − has a simple zero. From here, [GiLiPo] specializes to contact Anosov flows.Let λ + ≥ k Dϕ − t k ∞ ≤ C e λ + t for all t ≥ Theorem 2.
For a contact Anosov flow ϕ ∈ C r where r > , with λλ + > there exists τ ∗ > such that the Ruelle zeta function is analytic in { z ∈ C : Re( z ) ≥ h top ( ϕ ) − τ ∗ } apart from a simple pole at z = h top ( ϕ ) .Proof. Equation (2.23) and Equation (2.14) show that the poles of ζ Ruelle ( z ) are asubset of the eigenvalues of X ( ℓ ) . Lemma 2.6.
For any C r Anosov flow ϕ t with r > , ξ ∈ C and z ∈ D ℓ ( ξ ) , then it isanalytic and nonzero in the region Re( ξ ) > h top ( ϕ ) − λ | d s − ℓ | .Proof. Let Ω ℓ ,v ( M ) ⊂ Ω ℓv ( M ) as Definition 2.2, let B p,q,ℓ such that p ∈ N and q ∈ R + as Definition 2.3 and let L ( ℓ ) t ( h ) as Equation (2.8) for some h ∈ Ω ℓ ,v ( M ). By restrictingthe transfer operator L ( ℓ ) t to the space Ω ℓ ,r ( M ) we mimic the action of the standard JOEL ANTONIO–V ´ASQUEZ transfer operators on sections transverse to the flow. The operators (2.9) generalizethe action of the transfer operator L t on the spaces B p,q . Thus, (cid:13)(cid:13) L ℓt h (cid:13)(cid:13) p,q,ℓ ≤ C p,q e σ ℓ t k h k p,q,ℓ , (2.15) k h k p,q,ℓ = sup s ≤ t (cid:13)(cid:13) L ( ℓ ) s h (cid:13)(cid:13) p,q,ℓ . (2.16)By Equations (2.15) and (2.16) imply that for some t < t , then (cid:13)(cid:13)(cid:13) L ( ℓ ) t h (cid:13)(cid:13)(cid:13) p,q,ℓ ≤ max {k h k p,q,ℓ , C p,q e | σ ℓ | t k h k p,q,ℓ } ≤ C p,q k h k p,q,ℓ , while for t ≥ t the required inequality holds trivially. The boundedness of L ( ℓ ) t follows.The second inequality follows directly from the above, for small times and Equation(2.15) for larger times. On e B p,q,ℓ the operators L ℓt form a strongly continuous semigroupwith generators X ( ℓ ) by the above. We consider the resolvent R ( ℓ ) ( z ) = ( z − X ( ℓ ) ) − ,then we have the following Lemma. Lemma 2.7. R ( ℓ ) ( z ) is a quasi–compact operator on e B p,q,ℓ .Proof. This follows by [GiLiPo, Lemma 3.8]. (cid:3)
Although the operator X ( ℓ ) is an unbounded closed operator on e B p,q,ℓ , we can ac-cess to its spectrum thanks to Lemma 2.7. Now, let us make ℓ = d s and let ˜ ω s be avolume. A form on E s normalized so that k ˜ ω s k = 1 and it is globally continuous. Let π s ( x ) = T x M −→ E s ( x ) be the projections on E s ( x ) along E u ( x ) ⊕ E ( x ) such that ω s ( v , . . . , v d s ) = ˜ ω s ( π s v , . . . , π s v d s )by construction ω s ∈ Ω d s , ˜ ω . Note that ϕ ∗− t ω s = J s ϕ − t where J s ϕ − t is the Jacobianrestricted to the stable manifold. Note that, ω s,ε = M ε ω s , for ε small enough, we have h ω s,ε , ω s i ≥ . Hence, Z W α ,G h ω s,ε , L t ω s i ≥ Z W α ,G J s ϕ − t ≥ C Z W α,G J W ϕ − t ≥ C vol( ϕ − t W α,G ) . (2.17)Then the Equation (2.17) implies that the spectral radius of R ( d s ) ( a ) on e B q, ˜ ω,d s isexactly ( a − σ d s ) − . Thuslim n →∞ n n − X k =0 ( a − σ d s ) k R ( d s ) ( a ) k = (cid:8) Q if( a − σ d s ) − ∈ σ e B p,q,ℓ ( R ( d s ) ( a )) , , (2.18) N THE POLLICOTT–RUELLE RESONANCES 7 where Q is the eigenprojector on the associated eigenspace and the convergence takesplaces in the strong operator topology of L ( e B p,q,d s , e B p,q,d s ). Thus, by Equation (2.17), Z W α,G h W s,ε , Y W s i > , we have that Q = 0 and ( a − σ d s ) − belongs to the spectrum. This implies, that if ℓ = d s , then h top ( ϕ ) is an eigenvalue of X and if the flow is topologically transitive h top ( ϕ ) is a simple eigenvalue. Moreover, if the flow is topologically mixing, then h top ( ϕ ) is the only eigenvalue on the line { h top ( ϕ ) + ib } b ∈ R . Thus, this proves Lemma2.6. (cid:3) In the same time, Theorem 2 follows by Lemma 2.6. (cid:3)
Theorem 3 ([Po85, Theorem 2]) . Let ϕ t : Λ −→ Λ be a weak–mixing Axiom A flow,then the Fourier transform b ρ f,g ( z ) has a meromorphic extension to a strip | J ( z ) | ≤ ε ,which is analytic on the real line. Furthermore, b ρ f,g ( t ) tends to zero exponentially fast(for all H¨older continuous functions f, g : Λ −→ R ) only if ζ ( s, F ) has an analyticextension to some strip R ( s ) > P ( F ) − ε , except for the simple pole at s = P ( F ) . Theorem 4 ([ReSi, Paley-Wiener Theorem]) . Let ρ be in S ′ ( R ) . Suposse that b ρ isa function with an analytic continuation to the set { ζ | Im ζ < a } for some a > .Suppose further that for each η ∈ R n with | η | < a , b ρ ( · + iη ) ∈ L ( R n ) and for any b < a, sup | η |
Let ξ as in Lemma 2.6, then the function e D ℓ ( ξ − z, ξ ) is analytic andnonzero for z in the region | ξ − z | < Re( ξ ) − h top ( ϕ ) + | d s − ℓ | λ (2.20) and analytic in z , in the region | ξ − z | < Re( ξ ) − h top ( ϕ ) + | d s − ℓ | λ + λ (cid:22) r − (cid:23) (2.21) Proof.
We can write the spectral decomposition R ( ℓ ) ( z ) = P ( ℓ ) ( z ) + U ( ℓ ) ( z ) where P ( ℓ ) ( z ) is a finite rank operator and U ( ℓ ) ( z ) has spectral radius arbitraly close to ρ ess ( R ( ℓ ) ( z )). Let tr ♭ ( R ( ℓ ) ( z ) n ) < ∞ as in Definition 2.4, then it can be written astr ♭ ( R ( ℓ ) ( z ) n ) = 1( n − X τ ∈T χ ℓ ( τ ) µ ( τ ) λ ( τ ) n e − zλ ( τ ) . (2.22) JOEL ANTONIO–V ´ASQUEZ
Then, we substitute Equation (2.22) in Equation (2.4) and we get that e D ℓ ( ξ, z ) can beinterpreted as the “determinant” of ( − ξR ( ℓ ) ( z )) − , while D ℓ ( z ) can be interpretedas the “determinant” of z − X ( ℓ ) . Thus, e D ( ξ − z, z ) = exp − ∞ X i =0 ( ξ − z ) n n tr ♭ ( R ( ℓ ) ( ξ ) n ) ! = Y λ i,ℓ ∈ B ( ξ,ρ p,q,ℓ ) z − λ i,ℓ ξ − λ i,ℓ ψ ( ξ, z )(2.23)where ψ ( ξ, z ) is analytic and nonzero for z ∈ B ( ξ, ρ p,q,ℓ ).Furthermore, Theorem 5 implies Theorem 1. (cid:3) Theorem 6 ([GiLiPo, Corollary 2.7]) . The geodesic flow ϕ t : M −→ M for a compactmanifold M with better than –pinched negative section curvatures is exponentiallymixing with respect to the Bowen–Margulis measure µ ; that is; there exists α such thatfor f, g ∈ C ∞ ( T M ) there exists a C > for which the correlation function ρ ( t ) = Z f ◦ ϕ t gdµ − Z f dµ Z gdµ, satisfies | ρ ( t ) | ≤ C e − α | t | , for all t ∈ R .Proof. Consider the Fourier transform b ρ ( s ) = R ∞−∞ e ist ρ ( t ) dt of the correlation function ρ ( t ). By Theorem 3 and [Ru87, Theorem 4.1], the analytic extension of ζ Ruelle ( z ) inTheorem 2 implies that there exists 0 < η ≤ τ ∗ such that b ρ ( s ) has an analytic exten-sion to a strip | Im( s ) | < η . Now, without the loss the generality, we fixed each value − η < t < η , such we have that the function σ b ρ ( σ + it ) is in L ( R ). Finally weapply the Paley–Wiener Theorem 4 and result follows. (cid:3) As related to the Equation (1.2), the poles in b ρ ( s ) of Theorem 6 are the Pollicott–Ruelle resonances on a compact manifold which asserts the assumptions ( B B A short microlocal proof.
Unlike [GiLiPo]; whose proofs only work for contactflows; Dyatlov-Zworski [DyZw13] proved in 2013, the meromorphic continuation of theRuelle zeta function for C ∞ Anosov flows under the perspective of microlocal analysis,using semiclassical and scattering tools, and based on the study of the generator of theflow as a semiclassical differential operator. The proofs applies to any Anosov flow forwhich linearized Poincar´e maps P γ , where γ is a closed orbit such that | det( I − P γ ) | = ( − q det( I − P γ ) , with q independent of γ. Furthermore, the assumptions ( B B
2) still hold in this subsection. Let us first listsome important definitions, for a major literature – see more on microlocal analysis[H¨oI-II, H¨oIII-IV, Ve, Mea, Iv], semiclassical analysis [Zwa, GuSt, EvZw, Be] andscattering theory [Meb].
N THE POLLICOTT–RUELLE RESONANCES 9
Definitions.
Victor Guillermin [Gu], defined a Trace formula using distributionaloperations of pullback by some ι ( t, x ) = ( t, x, x ) and some pushforward π : ( t, x ) → t such that tr ♭ e − itP := π ∗ ι ∗ K e − itP , where K • denotes the distributional kernel of operator [Gu, Theorem 6]. As LarsH¨ormander claimed in [H¨oI-II, Theorem 8.2.4], the pullback is well–defined in thesense of distributions sinceWF( K e − itP ) ∩ N ∗ ( R t × ∆( X )) = ∅ , t > , (2.24)where ∆( X ) ⊂ X × X is the diagonal and N ∗ ( R t × ∆( X )) ⊂ T ∗ ( R t × X × X ) is theconormal bundle. Thus, we define Definition 2.8 (Guillemin’s Trace Formula) . tr ♭ e − itP = X γ T ♯γ δ ( t − T γ ) | det( I − P γ ) | , for some t > , (2.25) where T γ is the period of the orbit γ , T ♯γ is the primitive period, P γ is the linearizedPoincar´e map and δ ( • ) is the Dirac delta function. For a detailed proof of Equation (2.25) – see [DyZw13, § Appendix B] and [Gu, § II].Let WF( u ) be the wavefront set for some u ∈ D ′ ( M ) distribution. Since we do need amore robust measure of semiclassical regularity of functions, we define the semiclassicalwavefront set WF h in the sense of [Zwa, § h such h -temperedfamilies of distributions { u ( h ) } The semiclassical wavefront set WF h ⊂ T ∗ M is a subset from thefiber-radially compactified contangent–bundle (i.e., a manifold with interior T ∗ M andboundary ∂T ∗ M = S ∗ M = ( T ∗ M \ / R + , the cosphere bundle – see [DyZw, § E.1] ).Furthermore, WF h measures oscillations on the h –scale and if u is an h –independentdistribution, then WF( u ) = WF h ( u ) ∩ ( T ∗ M \ . (2.26)Now, we consider the semiclassical operator P ∈ Ψ kh ( M, Hom( E )) where E is avector bundle over M such that it is acting on h –tempered families of distributions u ( h ) ∈ D ′ ( M, E ). From Definition 2.9, we denote the natural projection κ : T ∗ M −→ S ∗ M = ∂T ∗ M. (2.27)Let L ⊂ T ∗ M be a closed conic invariant set under the flow e tHp such there is an openneighbourhood U of L – see more [H¨oI-II, § that d ( κ ( e − tH p ( U )) , κ ( L )) → t → + ∞ ;( x, ξ ) ∈ U = ⇒ | e − tH p ( x, ξ ) | ≥ C − e θt | ξ | , for any norm on the fibers and some θ > . (2.28) Definition 2.10. Let κ as in Equation (2.27) and L be a closed conic invariant thatasserts Equation (2.28). Then, we say that L is called a radial source and if we reversethe direction of the flow, then L is called a radial sink. By Definition 2.10 and letting E ∗ s , E ∗ , E ∗ u be the duals of E s , E and E u , respectively,then by Equation (2.28), we say that E ∗ s and E ∗ u are a radial source and a radial sink,respectively. Let P as before such that P : C ∞ ( M ; E ) −→ C ∞ ( M ; E ), besides P ( u ) = 1 i L v u , E = n M j =0 Λ j ( T ∗ M ) , where V is the generator of the flow ϕ t , L denotes the Lie derivate and u is a differentialform on M . Definition 2.11 (Anisotropic Sobolev Spaces) . The Anisotropic Sobolev spaces aredefined using the exponential weight – see [Zwa, Lemma 7.6] and [Zwa, Theorem 7.7] H sG := exp( − sG )( L ( M )) , k u k H sG := k exp( sG ) u k L , (2.29) where G ∈ Ψ ( M ) satisfying σ ( G )( x, ξ ) = m G log | ξ | , where m G = 1 near E ∗ s and m G = − near E ∗ u . For more about Anisotropic Sobolev spaces – see Duistermaat [Du], Unterberger[Un], Zworski [Zwa, § On the proof. Now, let us use some essentials theorems from [DyZw13] in orderto prove the existence of Pollicott–Ruelle resonances on the compact case via microlocalanalysis. Theorem 7 ([DyZw13]) . Supposse M is a compact manifold and ϕ t : M −→ M isa C ∞ Anosov flow with orientable and unstable bundles. Let { γ } denote the set ofprimitive orbits of ϕ t , with T ♯γ their periodics. Then the Ruelle zeta function, ζ Ruelle ( λ ) = Y γ (1 − e iλT ♯γ ) , (2.30) which converges for Im λ ≫ , has a meromorphic continuation to C . One of the main Propositions in [DyZw13], is: N THE POLLICOTT–RUELLE RESONANCES 11 Theorem 8 ([DyZw13, Proposition 3.4]) . Fix a constant C > and ε > . Then for s > large enough depending on C and h small enough, the operator P δ ( z ) : D sG ( h ) → H sG ( h ) , − C h ≤ Im z ≤ , | Re z | ≤ h ε , is invertible, and the inverse, R δ ( z ) , satisfies k R δ ( z ) k H sG ( h ) → H sG ( h ) ≤ Ch − , WF ′ h ( R δ ( z )) ∩ T ∗ ( M × M ) ⊂ ∆( T ∗ M ) ∪ Ω + , with ∆( T ∗ M ) , Ω + defined in [DyZw13, Propostion 3.3] , and WF ′ h ( • ) ⊂ T ∗ ( M × M ) isdefined for an h –tempered family of operators • ( h ) : C ∞ c ( M ) −→ D ′ ( M ) .Proof. The proof of Theorem 8 is assumed by k u k H sG ( h ) ≤ k u k H sG ( h ) ≤ Ch − k f k H sG ( h ) , u ∈ D sG ( h ) , f = P δ ( z ) u . (2.31)Then for some A ∈ Ψ h ( M ), we can get bounds on A u as are detailed in [DyZw13,Proposition 3.4] which arrive to the Equation (2.31). (cid:3) From Theorem 8, we can deduce that:(1) H sG and D sG are topologically isomorphic to H sG ( h ) and D sG ( h ) , respectively. And Q δ : D sG −→ H sG is smoothing and thus compact – see [DyZw13, Proposition3.1]).(2) If Im λ > C , u ∈ H sG ⊂ H − s and ( P − λ ) u = f ∈ H sG , then u = − Z ∞ ∂ t ( e iλt ϕ ∗− t u ) dt = i Z ∞ e iλt ϕ ∗− t f dt, where the integrals converge in H − s . This also implies that ( P − λ ) is injectiveand invertible D sG −→ H sG . Then( P − λ ) − = i Z ∞ e iλt ϕ ∗− t dt, (2.32)where ϕ ∗− t : C ∞ ( X ; E ) −→ C ∞ ( X ; E ) is the pullback operator by ϕ − t on differentialforms and the integral on the right–hand side converges in operator norm H s → H s and H − s → H − s – see [DyZw13, Proposition 3.2].(3) By [Zwa, § D.3], R ( λ ) = R H ( λ ) + P J ( λ ) j =1 A j / ( λ − λ ) j where λ is a near pole and A j are operators of finite rank such thatΠ := − A = 12 πi I λ ( λ − P ) − dλ, where [Π , P ] = 0. Thus A j = − ( P − λ ) j − Π and ( P − λ ) J ( λ ) Π = 0 – see [DyZw13,Proposition 3.3]. (4) Since Q δ is pseudodifferential and supposing the fact that R ( λ ) = h ( R δ ( z ) − i R δ ( z ) Q δ R δ ( z )) − R δ ( z ) Q δ R ( λ ) Q δ R δ ( z ) , we get thatWF ′ h ( R δ ( z ) − i R δ ( z ) Q δ R δ ( z )) ∩ T ∗ ( M × M ) ⊂ ∆( T ∗ M ) ∪ Ω + – see [DyZw13, Proposition 3.3].The Pollicott–Ruelle resonances are the poles of Re( λ ) in the region Im λ > − C of themeromorphic continuation of the Schwartz Kernel of the operator given by the right–hand side of (2.32), and thus are independent of the choice of s and the weight G . Forthe microlocal proof of the meromorphic continuation of Theorem 7 – see [DyZw13, § Further developments. Many applications had been development since the proofof those methods: • Dyatlov–Zworski [DyZw15], showed that Pollicott–Ruelle resonances are thelimits of eigenvalues of V /i + iεδ g , as ε → − δ g is any Laplace–Beltrani operator on X . • Jin–Zworski [JiZw], proved that for any Anosov flows there exists a strip withinfinitely many resources and a counting function which cannot be sublinear. • Colin Guillarmou [G1], studied regularity properties of cohomological equationsand provides applications. Guillarmou [G2] also established a deformation lensrigidity for a class of manifolds including manifolds with negative curvatureand strictly convex boundary. • Dyatlov–Guillarmou [DyGu14], proved meromorphic continuation for ( P − λ ) − and zeta functions for non–compact manifolds with compact hyperbolic trappedsets. • Dyatlov–Faure–Guillarmou [DyFaGu], described the complex poles of the powerspectrum of correlations for the geodesic flow on compact hyperbolic manifoldsin terms of eigenvalues of the Laplacian on certain natural tensor bundles. • Dyatlov used [DyZw13, Proposition 2.4] and [DyZw13, Proposition 2.5] in [Dya]as part to establish a resonance free strip for condimension 2 symplectic nor-mally hyperbolic trapped sets. To see a major literature about resonances forinfinite–area hyperbolic surfaces – see [Bo16]. • Dyatlov–Zworski [DyZw15], used microlocal methods similar to [DyZw13] in or-der to show stochastic stability of Pollicott–Ruelle resonances, more precisely,let P E = i V + i E ∆ g and let { λ j ( E ) } ∞ be the set of its L –eigenvalues. Fur-thermore, let { λ j } ∞ j =0 be the set of the Pollicott–Ruelle resonances of the flow ϕ t , then λ j ( E ) → λ j as E → 0+ with convergence uniform for λ j in a compactset – see the proof in [DyZw15, § N THE POLLICOTT–RUELLE RESONANCES 13 • Similar to [DyZw15], Zworski [Zwc] showed scattering resonances of − ∆ + V where V ∈ L ∞ c ( R n ), are the limits eigenvalues of − ∆ + V − iεx as ε → 0+ viacomplex scaling method [Zwc, § 2] – to see more about scattering resonances[DyZw, Zwb]. • Alexis Drouot [Dr], showed that for a compact manifold and negatively curved M , the L –spectrum of the infinitesimal generator of the Kinetic Brownianmotion on the cosphere bundle as a stochastic process modeled by the geodesicequation perturbed with a random force of size ε , converges to the Pollicott–Ruelle resonances as ε goes to 0.3. On the open systems case In 2014, Dyatlov–Guillearmou [DyGu14] defined Pollicott–Ruelle resonances for opensystems, more precisely, geodesic flows on noncompact asymptotically hyperbolic neg-atively curved manifolds, as well as for more general open hyperbolic systems relatedto Axiom A flows. They used many generalized microlocal tools from [DyZw13, FaSj]and functional analysis tools from [GiLiPo], and used anisotropic Sobolev spaces tocontrol the singularities at fiber infinity, and using complex absorbing potentials on theboundary and complex absorbing pseudodifferential operators beyond the boundaryto obtain a global Fredholm problem for the extension of X to a compact manifoldwithout boundary M – see more [DyGu14, § Definitions. We use the same notation as in [DyGu14], given a n –dimensionalcompact manifold U with interior U and boundary ∂ U , then X is a smooth C ∞ –nonvanishing vector field on U such that for some t , the corresponding flow is definedas ϕ t = e tX . Furthermore, ∂ U is strictly convex (i.e., for some x ∈ U then Xρ ( x ) =0 = ⇒ X ρ ( x ) < ρ ∈ C ∞ ( U )). Definition 3.1. The incoming ( Γ + ) and outgoing ( Γ − ) tails are subsets from U suchthat Γ ± = \ ± t ≥ ϕ t ( U ) . (3.1)From Definition 3.1, let K = Γ + ∩ Γ − be the trapped set such that for some x ∈ K there is a splitting in T x M in the sense of Equation (2.1) and Equation (2.2), where M is a compact manifold without boundary such that U is embedded in M – seemore about dynamical assumptions of the trapped set in [Dyb, § E be the smooth complex vector bundle over U and the first order differential operator X : C ∞ ( U ; E ) −→ C ∞ ( U ; E ) such that X ( f u ) = ( Xf ) u + f ( Xu ) , f ∈ C ∞ ( U ) , u ∈ C ∞ ( U ; E ) . (3.2) Fixing a smooth measure µ on M and the norm L ( M ; E ), we define the transferoperator e − t X : L ( M ; E ) −→ L ( M ; E ) and by Equation (3.2) we have that thesupport of e − t X is: e − t X ( f u ) = ( f ◦ ϕ − t ) e − t X u , f ∈ C ∞ ( M ) , u ∈ C ∞ ( M ; E ) . (3.3) Definition 3.2. Let R be the restricted resolvent defined as R ( λ ) = U ( X + λ ) − U : C ∞ ( U ; E ) −→ D ′ ( U ; E ) , Re λ > C , (3.4) where λ ∈ C . Then, for each j ≥ the space of generalized resonant states Res ( j ) X ( λ ) = { u ∈ D ′ ( U ; E ) | supp u ⊂ Γ + , WF( u ) ⊂ E ∗ + , ( X + λ ) j ( u ) = 0 } , (3.5) where E ∗ + ⊃ E ∗ u is the extended unstable bundle over Γ + . For a detailed construction of E ∗ + in Definition 3.2, see [DyGu14, Lemma 2.10]. Fur-thermore, the subbundle E ∗ + is a generalized radial sink and E ∗− ⊃ E ∗ s is a generalizedradial source; which is a modification from Equation (2.28). As related to Defini-tion 2.11, [DyGu14] defined the anisotropic Sobolev space H rh ; in order to control thesingularities at fiber infinity; as H rh = exp( − rG ( h ))( L ( M ; E )) , k u k H rh = k exp( rG ( h )) u k L ( M ; E ) , (3.6)where G is the operator defined as G ( h ) ∈ T λ> Ψ kh ( M ) – see more [DyGu14, § V ∈ C ∞ ( U ; C ),then γ ♯ : [0 , T γ ♯ ] −→ K of ϕ t of period T γ ♯ , thus V γ ♯ = 1 T γ ♯ Z T γ♯ V ( γ ♯ ( t )) dt (3.7)be the average of V over γ ♯ . Thus, we define the Ruelle zeta function as the productover all primitive closed trajectories of ϕ t on K : ζ Ruelle V ( λ ) = Y γ ♯ (1 − exp( − T ♯γ ( λ + V γ ♯ ))) , Re λ ≫ . (3.8)For express Pollicott–Ruelle resonances of X as poles, let E be the vector bundle over U by E ( x ) = { η ∈ T ∗ x M | h X ( x ) , η i = 0 } , x ∈ U , and let P x,t : E ( x ) −→ E ( ϕ t ( x )) be the Poincar´e map such that P x,t = ( dϕ t ( x )) − T | E ( x ) .Now, for each u ∈ C ∞ ( M ; E ), we put α x,t ( u ( x )) = e − t X u ( ϕ t ( x )) where α x,t is the par-allel transport defined as α x,t : E ( x ) −→ E ( ϕ t ( x )), then if u ( x ) = 0 implies that e − t X u ( ϕ t ( x )) = 0 by Equation (3.3). Thus, for the operator α ϕ t ( x ) ,T : E ( ϕ t ( x )) −→E ( ϕ t ( x )) where T > α ϕ t ( x ) = tr α ϕ t ( x ) ,T , det( I − P γ ) = det( I − P ϕ t ( x ) ,T ) = 0 . (3.9) N THE POLLICOTT–RUELLE RESONANCES 15 Definition 3.3. Using the wavefront set WF in the sense of Definition 2.9, of any u ∈ D ′ ( M ) and considering wavefront sets WF ′ ( B ) ⊂ T ∗ ( M × M ) \ , where B : C ∞ ( M ) −→ D ′ ( M ) are operators, we define WF ′ ( B ) = { ( x, ξ, y, − η ) | ( x, ξ, y, η ) ∈ WF( K B ) } , (3.10) where the Schwartz Kernel K B ∈ D ′ ( M × M ) is given by Bf ( x ) = Z M K B ( x, y ) f ( y ) dy, f ∈ C ∞ ( M ) . (3.11)Let V, W ∈ T ∗ M be open sets, such that e − T H p ( x, ξ ) ∈ V and e − tH p ( x, ξ ) ∈ W for t ∈ [0 , T ]. We denote the open subsetCon p ( V ; W ) ⊂ T ∗ M , the set of such points – see [DyGu14, Proposition 2.5]. Let A, B, B ∈ Ψ h ( M ) beoperators such that q ≥ h ( B ), where WF h ⊂ Con p (ell h ( B ); ell h ( B )), thusthe trajectories of e − tH p starting on WF h ( A ) either pass though ell h ( B ) or converge tosome closed set L , while staying on ell h ( B ) – see [DyGu14, Definition 3.3].3.2. On the proof. Dyatlov–Guillarmour used sharp G˚arding inequatlity – see [Zwa, § P is nottrivial, that is, assume that P ∈ Ψ m +1 h ( M ; E ) is principally scalar, A ∈ Ψ h ( M ), andRe σ h ( P ) ≤ U ⊂ T ∗ M of WF h ( A ). Then, there exist a constant C such that for each N and u ∈ H m +1 / h ( M , E ),Re h P A u , A u i L ≤ Ch k A u k H mh + O ( h ∞ ) k u k H − Nh . (3.12)Furthermore, if L and P satisfies that Im( P − iQ ) . − h on H sh near L , for all s , where Q ∈ Ψ h ( M ), Im σ h ( P ) ≤ L and Re σ h ( Q ) > L . Then, for some aditional p := Re σ h ( P ) ∈ Hom ( T ∗ M ; R ) and assuming that L ⊂ ∂T ∗ M , where L is invariantunder e tH p . Fix a metric | · | on the fibers of T ∗ M . Then,(1) Assume that there exist c, γ > | e tH p ( x,ξ ) || ξ | ≥ ce γ | t | for ( x, ξ ) ∈ L, t ≤ . (3.13)Then there exists s such that for all s > s , Im P . − h near L on H sh .(2) Assume that there exist c, γ > | e tH p ( x,ξ ) || ξ | ≥ ce γ | t | for ( x, ξ ) ∈ L, t ≥ . (3.14)Then there exists s such that for all s < s , Im P . − h near L on H sh .For the proofs of Equations (3.12), (3.13) and (3.14) – see [DyGu14, § Theorem 9. The family of { R ( λ ) } , defined in the sense of Equation (3.4), continuesmeromorphically to λ ∈ C , with poles of finite rank. Theorem 10 ([DyGu14, Theorem 4]) . Define for Re λ ≫ F X ( λ ) = X γ e − λT γ T ♯γ tr α γ | det( I − P γ ) | , (3.15) where the sum is over all closed trajectories γ inside K, T γ > is the period of γ , and T ♯γ is the primitive period. Then F ( λ ) extends meromorphically to λ ∈ C . The polesof F ( λ ) are the Pollicott–Ruelle resonances of X and the residue at a pole λ is equalto the rank of Π λ .Proof. We define the flat trace in the sense of the operator A : C ∞ ( M ; U ) −→D ′ ( M ; U ) such that WF ′ ( A ) ∩ ∆( T ∗ M\ 0) = ∅ , thentr ♭ A = Z M tr End( E ) K A ( x, x ) dx. (3.16)Making F X ( λ ) = tr ♭ ( χe − t ( X + λ ) R ( λ ) χ ) (3.17)for λ > C , C > χ ∈ C ∞ ( U ) and t > t < T γ for all γ .Then, by [DyGu14, Theorem 2] and Equation (3.17) we have thattr ♭ J ( λ ) X j =1 ( − j − χe − t ( X + λ ) ( X + λ ) j − Π λ χ ( λ − λ ) j = rank Π λ λ − λ + Hol( λ ) , (3.18)where Hol( λ ) is holomorphic near λ . (cid:3) The proof (and in fact the work of Dyatlov–Guillarmou) is really complex, for a fulldetailed and several particular cases of Theorem 10 – see [DyGu14, § − ih U R ( ihλ ) U gives the mero-morphic continuation of R ( λ ) in the region [ − C , h − ] + i [ − C , C ] for h small enough.Since C and C can be chosen arbitraly and h can be arbitrally small, we obtain thecontinuation to the entire complex plane and Theorem 9 follows. (cid:3) References [AFW] Jean-Franois Arnoldi, Fr´ed´eric Faure and Tobias Weich, Asymptotic spectral gap and Weyllaw for Ruelle resonances of open partially expanding maps, Erg. Thoery Dyn. Syst. 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