How Lagrangian states evolve into random waves

In this paper, we consider a compact manifold $(X,d)$ of negative curvature, and a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\frac{i}{h} \phi(x)}$ on $X$. For a wide family of phases $\phi$, we show that $f_h$, when evolved by the semiclassical Schr\"odinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.


Introduction
Berry's conjecture.-In his influential paper [4], M. V. Berry, gave a heuristic description of the behavior of high-energy wave-functions of quantum chaotic systems.He suggested that these should, in some sense, at the wavelength scale, behave like stationary Gaussian fields whose spectral measure is uniformly distributed on the unit sphere.The ambiguous comparison between a deterministic system and a stochastic field has given rise to many different interpretations.In the present paper, we are interested in a formulation given by one of the authors in [13] (see also [1] for a similar approach).In this interpretation, we consider a compact connected Riemannian manifold (X d , g) with negative sectional curvature.We will denote by dx the volume measure on X and we will denote by ∆ the Laplace-Beltrami operator on X.The conjecture can be roughly stated as follows: Let (ψ h ) h be a family of functions on X such that h 2 ∆ψ h + ψ h = 0 and normalized so that ψ h 2 2 = 1.Let U ⊂ X be an open subset on which there exists a family of vector fields (V 1 , . . ., V d ) forming an orthonormal frame of the tangent bundle.Given x ∈ U , we write exp x (y) := exp x ( d j=1 y j V j (x)).Let x be a random point in U chosen uniformly with respect to the volume measure dx.For each h > 0 in the index set of (ψ h ) h , let ϕ h x ∈ C ∞ (R d ) be the random field defined by ϕ h x (y) = ψ h ( exp x (hy)).Then, the conjecture can be stated as follows.
Conjecture. -As h → 0 in the index set of (ψ h ) h , the family ϕ h x (y) converges in law as a random field towards a stationary Gaussian field on R d whose spectral measure is the uniform measure on the unit sphere S d−1 .
This conjecture has many consequences in terms of nodal domains and semiclassical limits of (ψ h ), as explained in [13].However, as stated, it seems quite out of reach.
Lagrangian states.-In this paper, instead, we study a much simpler question, in which eigenfunctions are replaced by a well-behaved family of quasi-modes, namely Lagrangian states: Definition 1.1 (Lagrangian states).-A Lagrangian state is a family of functions (u(•; h)) h on X indexed by h ∈ ]0, 1[, defined as follows: (1.1) f h (x) = a(x)e iφ(x)/h , where φ ∈ C ∞ (U ) for some open subset U ⊂ X and a ∈ C ∞ c (U ).The energy measure of f h is the measure on (0, ∞), denoted by µ a,φ , which is the push-forward of the measure |a(x)| 2 dx on X by the map X x → |∂φ(x)| ∈ (0, ∞).We say that the Lagrangian state is monochromatic if it furthermore satisfies |∂φ(x)| = 1 for all x ∈ U .In particular, this implies that µ a,φ is a multiple of δ {1} .
Monochromatic Lagrangian states are quasimodes in the sense that they satisfy (1)   h 2 ∆f h (x) + f h (x) = O C 0 (h).
However, the conjecture above will clearly not hold for them since they vanish on some non-empty open subset of X. (1) Here and in all the sequel, O C k (h α ) denotes a family of functions (g h ) such that g h C k (X) is bounded by a constant times h α .
Hence, instead of studying Lagrangian states of the form (1.1), we will study their evolution by the Schrödinger equation.It can be explicitly described using the WKB method, and is closely related to the dynamics of the geodesic flow.
Such a strategy was already followed in [20], where it was shown that a wide family of monochromatic Lagrangian states evolved during a long time have the Liouville measure as their semi-classical measure.Hence, they satisfy an analogue of quantum unique ergodicity, which is a central conjecture in quantum chaos concerning the genuine eigenfunctions of the Laplacian.In [20], the semi-classical measure associated to the long time evolution of non-monochromatic Lagrangian states is also described explicitly, as a linear combination of Liouville measures at different energies.
A precise description of the long time propagation of Lagrangian states was also used, for instance in [2], [3] and [17], to prove properties about the eigenfunctions and resonances of quantum chaotic systems.
It is thus natural to conjecture that (generic) Lagrangian states evolved during a long time satisfy the same quantum chaotic conjectures as genuine eigenfunctions of the Laplacian.In particular, we can wonder if they satisfy an analogue of Berry's conjecture stated above.
Informal presentation of our results.-The present paper gives a (partial) positive answer to this question.Namely, we consider a "generic" Lagrangian state, and propagate it to a time t by the Schrödinger equation, which gives us a function f t h .To be more precise, recall first that a subset of a topological space is called residual if it contains a countable intersection of dense open subsets.We will first equip the space of phases φ defined on the support of a fixed amplitude a with a natural topology.We then construct a residual subset of the space of phases such that our result will hold under the condition that f h (x) = a(x)e iφ(x)/h where φ belongs to this subset.Similarly to the construction in the previous paragraph, we write f t h,x (y) := f t h ( exp x (hy)), with x chosen uniformly at random in some open set of X.We can then show that f t h,x admits a weak limit for all t large enough, and that, as t → +∞, this limit converges to an isotropic Gaussian field.In the special case where the initial state is monochromatic, we thus obtain the same limit as in Berry's conjecture.There are two major differences between our results and those of [20].
-In [20], the condition on Lagrangian states is completely explicit: one has to assume that the associated Lagrangian manifold is transverse to the stable directions of the classical dynamics (see section 2.3 for more details).Here, we also need transversality to the stable directions, but also some much more subtle conditions.Namely, we will use the WKB method to express the evolved Lagrangian state, locally, as a sum of plane waves.We will need the fact that, generically, these plane waves have directions of propagation which are rationally independent, so that, when observing this sum of waves at a random point, it will behave like a sum of independent complex numbers with uniform argument.Gaussianity will then emerge from the central limit theorem.
-In [20], the Lagrangian states are propagated up to the Ehrenfest time, that is, c| log h| for some c > 0 related to the classical dynamics.Here, we first take h to zero to define our limits, and then let t go to infinity, which is somehow much weaker.We believe that an adaptation of our method could allow us to show Berry's conjecture for generic Lagrangian states propagated up to some time c log | log h| for some c > 0. However, to do so, we would have to change our definition of genericity, from φ belongs to a residual set here to φ belongs to a space of full measure for some suitable measure.This should be pursued elsewhere.Despite these weaknesses, our result can be considered as the first example of a family of functions satisfying Berry's conjecture because of an underlying chaotic classical dynamics.Before that, [6] and [7] (see also [13] and [19]) proved Berry's conjecture for generic families of Laplace eigenfunctions on the two dimensional torus, using some arithmetic arguments.Some examples of families of eigenfunctions in R d satisfying Berry's conjecture are also given in [18].
Organization of the paper.-In section 2, we will present our main result, recalling all the definitions we need regarding local weak limits and Gaussian fields.In section 3, we will show that our main result holds, provided our initial state is a Lagrangian state whose phase φ belongs to a special set.We show in section 5 that this set is in some sense generic.A key step in the proof of the main results presented in section 3 is to give an explicit description of the action of the Schrödinger operator on Lagrangian states.This is Proposition 3.3.The proof of this proposition is the object of section 4, where we recall some properties of the geodesic flow in negative curvature.Finally, in Appendix A, we will recall the facts we need from semi-classical analysis, while in Appendix B, we give a description of the monochromatic phases we consider.
Acknowledgements.-We would like to thank the anonymous referees for their multiple remarks, which greatly helped improve the general presentation of the paper.

Set-up and main results
Our main results state that Lagrangian states converge to some Gaussian field.Hence, we first have to explain our notion of convergence, and then, to describe the Gaussian fields towards which they converge.We will also need our Lagrangian states to be associated with Lagrangian manifolds that are transverse to the stable directions of the geodesic flow, as we will explain in section 2.3.
Recall that (X, g) is a compact connected Riemannian manifold.For each x ∈ X, we will denote by exp x : T x X → X the exponential map at x induced by the metric g on X (as in [14,Def. 1.4.3]).Moreover, given x, y ∈ X, we will denote by d(x, y) the Riemannian distance between the two points x and y.Unless otherwise stated, the spaces C ∞ (X) and C ∞ (R d ) will be equipped with the topology of uniform convergence of derivatives on compact sets.Moreover, when we speak of probability measures on these spaces, we will assume that they are equipped with the Borel σ-algebra.

2.1.
Local limits.-Let us now describe the form of convergence we establish here.To avoid any topological difficulties, we define this convergence locally, though all of our results will hold regardless of the choice of localization.To the point, let U ⊂ X be J.É.P. -M., 2022, tome 9 a small enough open set so that we can define an orthonormal frame V = (V 1 , . . ., V d ) on it, that is to say a family of smooth sections If x ∈ U and y ∈ R d , we will write yV (x (2.1) exp x (y) := exp x (yV (x)).
All the constructions in this section will depend on the choice of this local frame, and will hence not be intrinsic.For the rest of the section, let us fix x a random point in U chosen uniformly with respect to the Riemannian volume measure.
Definition 2.1.-Let (f h ) h>0 be a family of functions in C ∞ (X), and let P be a probability measure on C ∞ (R d ).Then, for each h > 0, we define the h-local measure associated to this family as the law of the random element of C ∞ (R d ) defined by f x,h (y) := f h ( exp x (hy)).We say that P is the local weak limit of (f h ) h in the frame V if, as h → 0, the law of f x,h converges weakly to P.
We insist that, in the definition of f x,h , x is a point chosen uniformly at random in U , so that f x,h is a random element of C ∞ (R d ).Here, C ∞ (R d ) is equipped with its usual topology, given by uniform convergence of derivatives over compact sets.
Hence, saying that P is the local weak limit of (f h ) h in the frame V means that, for any continuous bounded functional Definition 2.2.-Let (f h ) h>0 be a family of functions in C ∞ (X), let (r h ) h>0 be a family of positive real numbers converging to 0, let x 0 ∈ U and let P x0 be a probability measure on C ∞ (R d ).We say that P x0 is the (r h ) h -local limit of (f h ) h>0 at x 0 (in the frame V ) if, as h → 0, the law of the random function f x,h , conditioned on the event that x ∈ B(x 0 , r h ), converges weakly to P x0 .
In other words, P x0 is the (r h ) h -local limit of (f h ) h>0 at x 0 (in the frame V ) if for any continuous bounded functional F : Remark 2.3.-By construction, if (f h ) h has an (r h )-local limit P x0 at almost every x 0 ∈ U , then it has an h-local limit P which satisfies 2.2.Gaussian fields.-As previously, we equip C ∞ (R d ) with its usual topology, given by uniform convergence of derivatives over compact sets.An almost surely (or a.s.) C ∞ (centered) Gaussian field on R d will be a random variable f taking values in C ∞ (R d ) such that for any finite collection of points x 1 , . . ., x k ∈ R d , the random vector (f (x 1 ), . . ., f (x k )) ∈ C d is (centered) Gaussian.We say that two fields f 1 and f 2 are equivalent if they have the same law.In the sequel, unless otherwise stated, we will always identify fields which are equivalent.That is to say that we will speak indifferently of the field and of its law.Let f be an a.s.C ∞ , centered Gaussian field on R d .Then, the covariance function , the matrix K(x i , x j ) i,j is Hermitian.As explained for instance in [16,App. A.11], the function K belongs to C ∞ (R d × R d ) and there is actually a bijection between such functions and a.s.C ∞ centered Gaussian fields on R d (up to equivalence).
Next, recall that, by Bochner's theorem (see for instance [8, §2.1.11]),given a finite Borel complex measure µ on R d , the Fourier transform µ of µ gives rise a continuous positive definite function K : (x, y) → µ(x − y) on R d × R d .If, in addition, µ is compactly supported, its Fourier transform is smooth and gives rise to a unique Gaussian field f on R d (up to equivalence).In this case, we say that µ is the spectral measure of f .Note that K is invariant by the diagonal action of translations on each of its variables.Consequently, the law of f is invariant by translations.We say in this case that f is stationary.
Let us now apply this recipe to define a family of Gaussian fields on R d .Fix λ 1 , λ 2 such that0 < λ 1 < λ 2 , and let µ be a Borel measure on [λ 1 , λ 2 ].Consider the measure λ µ on R d which is given by (2.2) where ω d−1 is the uniform measure on S d−1 .If µ = µ a,φ with a, φ as in Definition 1.1, we simply write λ a,φ instead of λ µ a,φ .
Definition 2.4.-The isotropic Gaussian field with energy decomposition µ is the unique law for an a.s.continuous stationary Gaussian field f on R d whose spectral measure is λ µ .In other words, for each x, y ∈ R d , We will denote by P µ the law of f , which is a probability measure on C ∞ (R d ).
Note that, when |∂φ(x)| = 1 for all x in the domain of φ, µ a,φ is a 2 L 2 (X) δ {1} .In particular, if f is a Gaussian field with law P µ a,φ , then a −1 L 2 (X) f is (equivalent to) the random monochromatic wave.

2.3.
Transversality to the stable directions.-We denote by Φ t : T * X → T * X, t ∈ R the geodesic flow on T * X.For each λ > 0, let us write Since X has negative curvature, (Φ t ) t , restricted to some S * λ X, is an Anosov flow (see [10] for a proof of this fact).We will recall in section 4.1 the definition of an Anosov flow.In particular, J.É.P. -M., 2022, tome 9 we defer to this section for the definition, for each ρ ∈ S * λ X, of the unstable, stable and neutral subspaces of T ρ S * λ X.For any λ 1 , λ 2 such that 0 < λ 1 < λ 2 and any open subset Ω ⊂ X, we write To each φ ∈ E (λ1,λ2) (Ω), we can associate a Lagrangian manifold We then define the set of phases associated to Lagrangian manifolds that are transverse to the stable directions as We may now state our main result.To this end, we introduce the semi-classical Schrödinger propagator U h (t) := e ith∆/2 : L 2 (X) → L 2 (X).Moreover, we recall once more that, a subset of a topological space is called residual if it contains a countable intersection of dense open subsets.
Thanks to Remark 2.3, Theorem 2.6 implies the following result.
Corollary 2.7.-With the notation of Theorem 2.6, for each t T 0 , the family (U h (t)f h ) h>0 has an h-local limit µ t which converges weakly to P µ a,φ as t → +∞.
Remark 2.8.-Note that, although the law f h depends on U and on the choice of frame V , the limiting measure P µ a,φ depends only on a and φ.
Remark 2.9.-Let us finally observe that by Remark 2.5, each point of X admits an open neighbourhood Ω ⊂ X for which E T,irr (λ1,λ2) (Ω) is non empty (and even uncountable).Hence, although we do not have a global "generic" statement, Theorem 2.6 does yield a wide family of Lagrangian states whose pointwise local weak limits converge to that of the isotropic stationary a.s.smooth Gaussian field on R d with spectral measure µ a,φ from Definition 1.1 as t → +∞, under the action of the Schrödinger flow.
2.5.The case of monochromatic phases.-We would now like to state an analogue of Theorem 2.6 for monochromatic phases, i.e., phases satisfying (2) |∂φ| = 1.At first glance, it would seem natural to work with the space of phases which we would equip with the C ∞ (Ω) topology.However, this set appears to be very hard to work with: it is not trivial to perturb a function in E 1 (Ω) while remaining in this set.Hence, the set E 1 (Ω) could contain isolated points, which would make our approach based on genericity irrelevant.We will therefore use another approach to study phases satisfying |∂φ| = 1.
Let Σ ⊂ X be an embedded orientable simply connected hypersurface.Let us denote by ν a vector field defined on Σ such that for each y ∈ Σ, ν(y) has unit norm and is orthogonal to T y Σ.We write We then define ( 2) The case |∂φ| = λ for some λ > 0 can be recovered from the case |∂φ| = 1 by a simple rescaling.
J.É.P. -M., 2022, tome 9 Moreover, any two functions with these properties must coincide on a neighbourhood of Σ.Furthermore, by Lemma B.2, for any x ∈ Ω u , there exists a unique pair (y, t) In particular, we see from (4.1) that u ∈ C T (Σ) if and only if there exists The same argument as in Remark 2.5 shows that C T (Σ) is non-empty when Σ is small enough, and that, if We may now state our analogue of Theorem 2.6 for monochromatic phases.To this end, we equip the set C (Σ) with the C ∞ (Σ) topology (i.e., the topology of uniform convergence of derivatives on compact sets).Note that, unlike in the polychromatic case, the pointwise local weak limits exist here for all x 0 , and not just for almost all of them.Theorem 2.10.-Let X be a compact connected Riemannian manifold with negative sectional curvature, and let Σ ⊂ X be an embedded orientable simply connected hypersurface with a normal vector field ν.There exists a residual subset C T,irr (Σ) of C T (Σ) such that, for any u ∈ C T,irr (Σ), there exists T 0 0 such that the following holds.Let φ u and Ω u be as in (2.7) an open set, and V be an orthonormal frame on U .Let α be such that 1/2 < α < 1.Then for every x 0 ∈ U , and every t T 0 , the family (U h (t)f h ) h>0 has an (h α ) h>0 -pointwise local weak limit at x 0 , which we denote by µ t,x0 .Furthermore, µ t,x0 converges weakly to P µ a,φ as t → +∞.
Remark 2.11.-Note that, in this case, as explained in section 2.5, if f has law P µ a,φ , then, a −1 L 2 f is in fact the monochromatic wave.In particular, although the construction depends on U , on the choice of frame (V (x)) x , on a and on φ, the limit is (up to a multiplicative constant) independent of all of these choices.
Remark 2.12.-As for the case of Theorem 2.6, C T,irr (Σ) is non-empty and we obtain a wide family of Lagrangian states have pointwise local weak limits converging to the monochromatic wave under the action of the Schrödinger flow.

Proof of Theorems 2.6 and 2.10
The aim of this section is to describe explicitly the sets E T,irr (λ1,λ2) (Ω) and C T,irr (Σ) appearing respectively in the statements of Theorem 2.6 and Theorem 2.10, and to prove these theorems, postponing the proof of the fact that E T,irr to the next section.Throughout the present section, we will therefore fix Ω ⊂ X an open subset, as well as constants λ 1 , λ 2 such that 0 < λ 1 < λ 2 , and consider phases in E (λ1,λ2) (Ω).Likewise, for the monochromatic case, we fix Σ ⊂ X a simply connected embedded orientable hypersurfaces of X and ν a section of T X| Σ such that for each y ∈ Σ, ν(y) has unit norm and is orthogonal to T y Σ in T y X.We will also consider monochromatic phases of the form φ u with u ∈ C (Σ) as defined in section 2.5.
Finally, in order to describe local limits, we also fix U ⊂ X equipped an orthonormal frame V as in section 2.1.
The proof will go as follows.In section 3.1 we state a compactness criterion.Thanks to this criterion, proving convergence of finite marginals will yield convergence in C ∞ (R d ) topology.In section 3.2 we will describe the effect of the Schrödinger propagator on a Lagrangian state whose phase belongs to E T (λ1,λ2) (Ω).In section 3.3 we first describe the sets E T,irr (λ1,λ2) (Ω) and C T,irr (Σ).Assuming that φ belongs to one of these sets we let h → 0 for some fixed (large enough) t and describe the local limits associated to the propagated Lagrangian state at time t around some point x 0 (which we assume to be generic in the former case).In section 3.4 we let t → +∞ and describe the asymptotic behavior of the local limit around x 0 .Finally, in section 3.5 we fit the pieces together and complete the proofs of Theorems 2.6 and 2.10.

3.1.
A criterion for convergence of local measures.-Here we record a compactness criterion for the convergence of probability measures on C ∞ (R d ).Let a = (a k, ) k, ∈N 2 be a sequence of positive real numbers depending on two parameters.We define It follows from the Arzelà-Ascoli theorem that K (a) is a compact subset of C ∞ (R d ) for the topology of convergence of all derivatives over all compact sets.
Let us write F for the set of functionals Then F forms an algebra which separates points.Hence, by the Prokhorov theorem, we obtain the following result, which we will use several times in the sequel.See [13, §3] for more details.
Lemma 3.1.-Let a = (a k, ) k, ∈N 2 be a sequence of positive real numbers depending on two parameters.Let (P n ) be a sequence of Borel probability measures on C ∞ (R d ), which is supported in K (a), and let µ be a Borel probability measure on C ∞ (R d ).
Suppose that, for any F ∈ F , we have Then (P n ) converges weakly to P. Remark 3.2.-More generally, using Markov inequality, the condition that (P n ) is supported in K (a) can be replaced by the following: For every k, ∈ N, there exists a k, > 0 such that for all n ∈ N, we have The dynamics of a Lagrangian state by the Schrödinger flow is easy to describe in terms of the evolution of Λ φ under the geodesic flow on X.The main point of this section is to describe the effect of the Schrödinger propagator acting on a Lagrangian state on a manifold X of negative sectional curvature.We do so in Proposition 3.3.The proof of this proposition, which is essentially an application of the WKB method, relies on the techniques developed in [2], [3], [17], and we will recall it in section 4.5 below for the reader's convenience.Recall that U h (t) = e ith∆/2 is the Schrödinger propagator and that Φ t : T * X → T * X is the geodesic flow.
(4) There exists a constant C 2 > 0 such that for all t T 0 and all j ∈ {1, . . ., M (t)}, we have For the rest of the section, we fix a ∈ C ∞ c (Ω) and φ ∈ E T (λ1,λ2) .For each h > 0 and t ∈ R, we set ). Proposition 3.3 applies to f t h .For each x 0 ∈ X, t T 0 , j, j ∈ {1, . . ., M (t)}, we will write j ∼ x0,t j if x 0 ∈ U j,t ∩U j ,t and ∂ φ j,t • exp x0 (0) = ∂ φ j ,t • exp x0 (0).Up to reordering the terms {1, . . ., M (t)}, we may suppose that there exists N (t; x 0 ) ∈ N such that the set {1, . . ., N (t; x 0 )} contains exactly one representative of each of the different equivalence classes.In the sequel, since x 0 will be fixed most of the time, we will just write N (t) instead of N (t, x 0 ).
We then write, for every j ∈ {1, . . ., N (t)} , . . ., β t,x0 N (t) ).3.3.Convergence to pointwise local limits at fixed times.-In this section, we first define the residual sets of phases (3.6) and (3.7) which appear in the statements of Theorems 2.6 and 2.10 respectively.Then, assuming that the phase belongs to (3.6) we describe the pointwise local limits at fixed time t large enough (see Proposition 3.5 below).
Recall the definitions of for almost every x 0 ∈ X, the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent for all t T 0 (φ) , where the (ξ t,x0 j ) j are obtained from φ by the construction (3.5) which follows from Proposition 3.3.The set E T,irr (λ1,λ2) (Ω) is precisely the set appearing in the statement of Theorem 2.6.We will show in section 5 that the space is a residual subset of E T (λ1,λ2) (Ω) equipped with the convergence of all derivatives on all compact sets.For the monochromatic case, we will consider the following analogous set.Recall that, in section 2.5, given an oriented hypersurface Σ, we saw how to associate to each function u ∈ C (Σ) an open neighbourhood Ω f of Σ and a map φ u ∈ E 1 (Ω u ).If u ∈ C T , we thus denote by (ξ t,x0 j ) j=1,...,Nx 0 (t) the vectors obtained by applying Proposition 3.3 to φ u (see (3.5)).We then define the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent for all t T 0 (φ) .
J.É.P. -M., 2022, tome 9 We will see in section 5.2 that this set is a residual subset of C T (Σ) equipped with the topology of uniform convergence of derivatives on compact sets.
From now on, we will always suppose that the phase φ introduced in section 3.2 belongs to E T,irr (λ1,λ2) (Ω), and take x 0 such that the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent for all t T 0 (φ).
Let us now describe the measures P t,x0 appearing in Theorem 2.6 associated to the family (f t h ) introduced in section 3.3.To do this, recall that at the beginning of section 3 we fixed U an open subset of X equipped with an orthonormal frame V .We will always implicitly consider h-local limits in this frame.The local limits of (f t h ) for various fixed t will belong to a family of probability laws on C ∞ (R d ) which we now define.-Let α be such that 1/2 < α < 1, and let t T 0 (φ).Let x 0 ∈ U be such that the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent.Then (f t h ) h has an h α -pointwise local weak limit at x 0 , which is given by P β t,x 0 ,ξ t,x 0 .
Proof of Proposition 3.5 First step: a criterion for convergence.-Let t T 0 (φ) and x 0 ∈ U be such that the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent.Equation (3.2) implies that, for any R > 0 and any k ∈ N, we have This quantity is thus bounded independently of h, t being fixed.This implies that we may find a sequence a such that for all h small enough and all x in B(x 0 , h α ), the function f t x,h belongs to K (a), with K (a) as in (3.1).Hence, thanks to Lemma 3.1, it suffices to show that for any k ∈ N, any y 1 , . . ., y n ∈ R d and any where and where the first expectation is taken with respect to x ∈ B(x 0 , h α ).
Second step: local expressions.-Next, we are going to use Taylor expansions to obtain a simpler asymptotic expression for F (f t x,h ).If x ∈ B(x 0 , h α ), the fact that b j,t is C 1 implies that, for every fixed y ∈ R d , we have b j,t ( exp x (hy)) = b j,t (x 0 ) + O(h α ).
Since α > 1/2, the error terms vanish as h → 0. Therefore, if we define the continuous function .
Third step: computing the expectation.-To compute the expectation of this quantity, we note that x is a random variable on B eucl (0, 1), whose density we denote by (1/Vol(B euc' (0, 1)))ρ h (z) dz.Since d 0 exp x0 is an isometry, we have that for all z ∈ B eucl (0, 1) for some C < +∞ which depends only on (X, g) and on the choice of frame (V (x)) x∈X .Therefore, if z denotes a uniform random variable on B eucl (0, 1), we have To compute this expectation, we want to use a multidimensional Kronecker theorem, whose proof we recall.Suppose first of all that Γ is of the form Γ(θ 1 , . . ., θ N (t) ) = e 2iπ(n1θ1+•••+n N (t) θ N (t) ) , where n := (n 1 , . . ., n N (t) ) ∈ Z N (t) {0}.Let us write ξ x0,t n := N (t) j n j ξ x0,t , which is non-zero since the ξ x0,t j are rationally independent.Therefore, we have But B eucl (0,1) e 2iπh α−1 ξ x 0 ,t n •z dz is the Fourier transform of the indicator of the unit ball evaluated at h α−1 ξ x0,t n .Since α < 1 and ξ x0,t n = 0, this goes to zero as h → 0. For a general Γ, we may approach it uniformly by a trigonometric polynomial having the same mean (this is a consequence of Fejér's theorem), and we see from what precedes that only the constant term will give a non-vanishing contribution to the expectation as h → 0. Therefore, we have This quantity is exactly E P β,ξ [F ], and the result follows.
Remark 3.6.-We used the fact that the ξ x0,t j are rationally independent only in the last step of the proof.If they are not rationally independent, then the phases get equidistributed along an affine sub-torus of T N (t) .The linear part of this torus depends only on the ξ x0,t j , and not on h.However, the affine torus depends on the ϑ x0,t 1 (h), . . ., ϑ x0,t N (t) (h) , so we do not have convergence to a measure independent of h (and hence, existence of a pointwise local weak limit).However, we may extract subsequences h n such that ϑ x0,t 1 (h N ), . . ., ϑ x0,t N (t) (h N ) converges.Doing so, we ensure the existence of pointwise local weak limits, even when the ξ x0,t j are not rationally independent.We will not use this construction in the sequel, since we don't want to extract subsequences.

3.4.
Long time behaviour of local limits.-The aim of this section is to prove the following proposition, which is the last step in the proof of Theorem 2.6, except for the fact that E T,irr (λ1,λ2) is a residual set.Recall that we fixed a phase φ ∈ E T,irr (λ1,λ2) (Ω), and a function a ∈ C ∞ c (Ω).We now also fix point x 0 such that the vectors (ξ t,x0 j ) j=1,...,Nx 0 (t) are rationally independent for all t T 0 (φ).Recall the definition (2.2) of λ µ associated to some measure µ and those of λ a,φ and λ µ a,φ given just below (2.2).Proposition 3.7 (Pointwise local limits at long time).-The measures P β t,x 0 ,ξ t,x 0 converge weakly, as t → +∞, to P µ a,φ .This proposition follows from the following two lemmas, which we prove below.For any compact set K ⊂ R d and any k ∈ N, we have which is bounded independently of t, by assumption.We may therefore apply Lemma 3.1 and Remark 3.2 to prove the result.
To this end, we fix y 1 , . . ., y k ∈ R d and we study the convergence of the vector (f t (y 1 ), . . ., f t (y k )) as t → +∞.We wish to apply a multivariate Lindeberg Central Limit Theorem to the sum over j of the random vectors η j (t) = (β t j e iξ t j •y1+iϑj , . . ., β t j e iξ t j •y k +iϑj ).By construction, the η j 's are mutually independent.Moreover, for each t 0 and j ∈ {1, . . ., N (t)}, E[η j (t)] = 0 and the covariance of η j (t) has coefficients E[η h j (t)η j (t)] = (β t j ) 2 e iξ t j (y h −y ) .Thus, the sum of their covariance matrices M t = (m t h ) h has coefficients which converge to m h = R d e iξ(y h −y ) dλ µ (ξ) by the first assumption of the lemma.But the matrix (m h ) h thus constructed is the covariance matrix of the random vector (f (y 1 ), . . ., f (y k )), where f is a random function following the law P µ .In particular, the matrix M t is invertible for all large enough t.Lastly, since sup j β t j − −− → t→0 0, we have (deterministically) sup j |η(t)| = o(N (t)), which implies the remaining condition for the multivariate Lindeberg Central Limit Theorem (3) .Thus, as t → +∞, the vector (3)  Before proceeding with the proof of Lemma 3.9, let us introduce some notations.Recall that V = (V 1 , . . ., V d ) is an orthonormal frame defined in a neighbourhood U of x 0 .Using the Riemannian metric, it naturally induces an orthonormal co-frame x X, we write (V * x ) −1 (ξ) for the unique y ∈ R d such that yV * (x) = ξ.We refer the reader to Appendix A for the definition and standard results regarding semiclassical measures, which we use in the proof.Recall also that Φ t : T * X → T * X is the geodesic flow.
Proof of Lemma 3.9.-The sequence (ae iφ/h ) h>0 has a semi-classical measure, which we denote by ν 0 .By Egorov's theorem (Theorem A.1 below), the semi-classical measure of . By the previous remarks, we have with λ a,φ and µ a,φ as in section 2.2.On the other hand, by Proposition 3.3 we know that, as h → 0, e iφj,t(x)/h b j,t (x) + o L 2 (1), so that, by (A.1) and the L 2 -continuity of semi-classical measures (which follows for instance from [21, Th. 5.1]), J.É.P. -M., 2022, tome 9 To obtain the second line, we used the smoothness of the vector fields V * i and of χ 2 .Note that (V * x0 ) −1 (∂φ j,t (x 0 )) = ξ t j and recall that |B j,t (x 0 )| = β t j .For the last line, we use the fact that, since ν t = Φ t * ν 0 as observed at the start of the proof, the total mass of ν t is constant.We deduce that w).In other words, converges weakly to λ a,φ .

Conclusion of the proofs.
-In this section we use the results from sections 3.1, 3.2, 3.3 and 3.4, as well as Propositions 5.1 and 5.2 from the following section, to prove Theorems 2.6 and 2.10.
Proof of Theorem 2.6.
Proof of Theorem 2.10.-The proof is very close to that of Theorem 2.6.The only differences are the following.The set E T,irr (λ1,λ2) (Ω) should be replaced by C T,irr (Σ) and Proposition 5.1 should be replaced by Proposition 5.2.For the rest of the proof, one takes u ∈ C T,irr (Σ), which induces a phase φ u defined on an open subset Ω u .The rest of the proof carries over with φ u (resp.Ω u ) in place of φ (resp.Ω).

Classical and quantum dynamics of Lagrangian submanifolds
The aim of this section is to prove Proposition 3.3.In sections 4.1 and 4.2 we introduce basic definitions and properties related to the hyperbolic dynamics on S * X.In section 4.3, we then apply these to state Lemma 4.8, which is a key estimate needed in the proof (more precisely, we need it to prove (3.3).In section 4.4, we prove Lemma 4.8.Finally, in section 4.5, we prove Proposition 3.3.In all this section, we fix an arbitrary metric g 0 on T * X.

4.1.
Hyperbolicity. -For each λ > 0, we denote by Φ t λ : S * λ X → S * λ X, t ∈ R the geodesic flow on S * λ X.Since X has negative curvature, (Φ t λ ) t is an Anosov flow (see [10] for a proof of this fact).It means that for each ρ ∈ S * λ X, there exist E + ρ , E − ρ and E 0 ρ subspaces of T ρ S * λ X, respectively called the unstable, stable and neutral direction at ρ such that: ρ and E 0 ρ depend Hölder continuously on ρ.
-The distribution E 0 ρ is one dimensional and generated by d dt | t=0 Φ t λ (ρ).In particular, dΦ t λ | E 0 is bounded from above and below uniformly in t. -E + ρ and E − ρ are both d − 1 dimensional, and for each t ∈ R, we have (4.1) -There exists C > 0 and A > 1 such that for each ρ ∈ S * λ X, t > 0, In a basis adapted to this decomposition, we have where M ρ,t is a (d−1)×(d−1) matrix such that M ρ,t ξ CA t ξ for any ξ ∈ R d−1 .It follows from (4.1) and ( 4.3) that E − ρ ⊕ E 0 ρ and E + ρ ⊕ E 0 ρ are Lagrangian spaces.If σ denotes the canonical symplectic structure on T * X, we may find a constant C 0 > 0 such that, for all ρ ∈ T * X and all ξ, ζ ∈ T ρ T * X, we have Furthermore, the map Φ t being symplectic, we have σ(ξ, ζ) = σ(d ρ Φ t (ξ), d ρ Φ t (ζ)).Combining this with (4.3) and letting t → ±∞, we see that In particular, forms a vector space of dimension 2, and there is a unique symplectic form on Finally, we define the stable and weak stable manifolds of ρ as ) remains bounded as t → +∞}.W − (ρ) and W −0 (ρ) are then manifolds, whose tangent space at ρ are respectively E − ρ and ).Here, we will focus on a special family of Lagrangian submanifolds, which can be written as graphs.
Definition 4.1 -Let Y be a smooth Riemannian manifold.We say that a Lagrangian submanifold Λ ⊂ T * Y is projectable if there exist an open subset Ω Λ ⊂ Y and a smooth real-valued function φ defined on a neighbourhood of Ω Λ such that -We call φ a phase function and say that it generates Λ.Note that φ is monochromatic if and only if Λ ⊂ S * Y .We call Ω Λ the support of Λ.
-Given also Λ ⊂ T * Y a Lagrangian submanifold, we say that Λ is a Lagrangian extension of Λ if Λ ⊂ Λ .Remark 4.2.-Let Λ be a submanifold of T * Y .Then, Λ is a projectable Lagrangian manifold if and only if Λ is the graph of a smooth section of T * Y defined over an open subset Ω Λ , which can be extended smoothly to some neighbourhood of Ω Λ .Definition 4.3.-Let λ 1 , λ 2 be such that 0 < λ 1 < λ 2 , and let Λ ⊂ S * [λ1,λ2] X be some Lagrangian submanifold.We say that Λ is transverse to the stable directions if it admits a Lagrangian extension Λ such that for any ρ ∈ Λ , we have In the case where Λ is a section of T * X, this is equivalent to the fact that this section is transverse at ρ to the unique stable manifold W − (ρ ) containing ρ.This motivates our use of the term transverse in this context.Definition 4.4.-Let λ 1 , λ 2 be such that 0 < λ 1 < λ 2 , and let Λ ⊂ S * [λ1,λ2] X be some Lagrangian submanifold.We say that Λ is nowhere stable if it admits a simply connected Lagrangian extension Λ such that for any ρ 1 , ρ 2 ∈ Λ , we have ] Y be a precompact Lagrangian submanifold transverse to the stable directions.Then there exists finitely many Lagrangian submanifolds Λ 1 , . . ., Λ n such that Λ = n i=1 Λ i and each Λ i is nowhere stable.
Since Λ is a smooth manifold and the dependence of the unstable directions in ρ is Hölder, we see that ρ → ε ρ is continuous.Hence ε 0 := inf ρ∈Λ ε ρ is > 0.
Let us consider a covering of Λ by finitely many balls of radius ε 0 /2, and check that each element of this covering is nowhere stable.If ρ 1 , ρ 2 belong to the intersection of Λ with a ball of radius ε 0 /2, then we have Therefore, we have (ρ 2 ∈ W − (ρ 1 )) =⇒ (ρ 2 = ρ 1 ), as announced.

Evolution of Lagrangian manifolds on Hadamard manifolds.
-Next we will focus on the evolution of nowhere stable Lagrangian submanifolds on the universal cover of X, which we denote by X.The manifold X is then a Hadamard manifold, i.e., a complete simply connected manifold of negative curvature.In particular, we state the key estimate Lemma 4.8 needed in the proof of Proposition 3.3.Definition 4.6.-Let Y be a Riemannian manifold and let π : T * Y → Y be the canonical projection.Let Λ ⊂ T * Y be a Lagrangian submanifold.Let (Φ t ) t be the geodesic flow, acting on T * Y .We say that Λ is expanding if there is a Lagrangian extension Λ of Λ such that for any ρ, ρ ∈ Λ which do not belong to the same geodesic, the function t -Let Y be a complete simply connected manifold of negative curvature, and let Λ ⊂ S * [λ1,λ2] Y be a Lagrangian submanifold which is nowhere stable.Then there exists T 0 (Λ) > 0 such that for all t T 0 , Φ t (Λ) is an expanding projectable Lagrangian submanifold.
Suppose that this map converges to zero as t → +∞, so that it is decreasing.Then we must also have dist T * Y (Φ t ρ 1 , Φ t ρ 2 ) converging to zero.Indeed, if this were not the case, we could find large times t at which the points Φ t ρ 1 and Φ t ρ 2 are very close when projected on Y , but have directions which are not close to each other.The distance on the base of such points cannot be a decreasing function.Therefore, we must have ρ 2 ∈ W − (ρ 1 ), which contradicts the fact that Λ is nowhere stable.
The following lemma, which we prove in the next section, gives us an estimate on the regularity of the maps g t1,t2 which will be essential to obtain the first point in Proposition 3.3.
] X be a Lagrangian manifold which is transverse to the stable directions, and let T 0 (Λ) be as in Lemma 4.7.Then for all t T 0 (Λ), log(|det(d x g 0,t )|) is continuous in x, uniformly in (x, t).

4.4.
Proof of Lemma 4.8.-In this section, we prove Lemma 4.8 but before doing so, we state and prove a final auxiliary lemma.Recall that we fixed a metric on T * X, which allows us to define angles between vectors of T ρ T * X for any ρ ∈ S * [λ1,λ2] X.
Definition 4.9.-Let η 0 > 0. We say that a Lagrangian submanifold Λ ⊂ S * [λ1,λ2] X is η 0 -transverse to the stable directions if, for any ρ ∈ Λ, the angle between any vector of T ρ Λ and any vector of E − ρ is at least η 0 .
Proof.-First of all, note that there exists c > 0 such that for all ρ ∈ S * [λ1,λ2] X and all ξ = (ξ For a given ρ, this follows from the fact that all norms are equivalent on a finitedimensional space, and the constant c involved depends on the angle between the directions E 0 ρ , E 0 ρ and E + ρ .By compactness, the constant may hence be taken independent of ρ. Let us fix η 0 > 0 and Λ as in the statement.Let ρ ∈ Λ and let t ∈ R. Write ρ t := Φ t (ρ) and ξ t := (d ρ Φ t )(ξ) ∈ T ρt Φ t (Λ).Decomposing ξ t as Thanks to (4.2) and (4.10), we have On the other hand, (4.2) and (4.10) also imply that Thus, we have and the first claim follows.
We may now proceed with the proof of Lemma 4.8.Recall that it says that, if we consider the family of functions D t ∈ C ∞ (X) indexed by t T 0 (Λ), defined by D t (x) = log(|det(d x g 0,t )|), then for all ε > 0, there exists µ > 0 such that for all t T 0 (Λ) and all x, x ∈ Ω T0 at mutual distance at most µ, Proof of Lemma 4.8.
-By compactness, we may find η 0 > 0 such that Λ is η 0 -transverse to the stable directions.Let ε > 0, let δ 0 > 0 which we will choose later, depending on ε, and let T 1 = T 1 (η 0 , δ 0 ) be as in Lemma 4.10, which we may assume to be greater than T 0 .Clearly, it is enough to establish (4.13) for t T 1 .Let t T 1 .By (4.9), for any x ∈ Ω t , defining ρ ∈ Λ t by π(ρ) = x, we have (4.14) To study the right-hand side of this decomposition, we make the three following observations.
-Thanks to the second part of Lemma 4.10, for each t T 1 , we may find a subspace E 0 ρ ⊂ E 0 ρ ⊕ E 0 ρ such that the orthogonal projector P ρ : ).-For all s 0, π| Λs : Λ s → Ω s is a diffeomorphism and uniformly bi-Lipschitz in s.Moreover, for each s > 0, the maps Φ −s : Λ s → Λ and g s,s for s < s are contracting.Consequently, the mappings x → d ρ πP * ρ , x → d Φ T 1 −t (ρ) πP * Φ T 1 −t and x → d g T 1 ,t (x) g 0,T1 (where π(ρ) = x) are Hölder continuous on Ω t , uniformly in t T 1 .

4.5.
The action of the Schrödinger propagator on Lagrangian states.-The aim of this section is to prove Proposition 3.3, which describes the action of the Schrödinger propagator on Lagrangian states that are transverse to the stable directions.We will start with the following proposition, which treats the case of Lagrangian states that are nowhere stable, on a complete simply connected manifold.The discussion is simplified by the fact that, here, we only consider Lagrangian states associated with Lagrangian manifolds that are projectable.We start by establishing the following proposition, which is an adaptation of results from [17, §4.1].
Proposition 4.11.-Let X be the universal cover of a Riemannian manifold of negative sectional curvature, let λ 1 , λ 2 be such that 0 < λ 1 < λ 2 , and let Λ ⊂ S * [λ1,λ2] be a projectable Lagrangian submanifold of T * X with support Ω 0 , generated by a phase function φ 0 .Suppose that Λ is nowhere stable.Then there exists T 0 0 such that, for any t T 0 , Φ t (Λ) is a projectable Lagrangian manifold, whose support and phase we denote by Ω t and φ t .Furthermore, for any symbol a ∈ C ∞ c (Ω 0 ), any t T 0 and any k ∈ N, the application of the operator U h (t) to the Lagrangian state ae iφ0/h associated with Λ 0 can be written as for some β(t) ∈ R. Here, the b t are smooth compactly supported functions on X such that φ t is smooth on a neighbourhood of the support of b t .The two functions are defined as follows: (1) Let π X : T * X → X be the canonical projection.For any y ∈ Ω t , there exists a unique x ∈ Ω 0 such that π X Φ t (x, ∂φ 0 (x)) = y.We then write Φ t (x, ∂φ 0 (x)) = (y, ∂φ t (y)).
(2) The function b t is defined by where g t (x) = g 0,t (x) is the projection on the base manifold of Φ −t (x, ∂ t φ(x)).Furthermore, log |b t (x)| is continuous in x, uniformly in (x, t).
Proof.-This proposition essentially follows from the fact that the Schrödinger propagator is a Fourier Integral Operator, and by using the WKB method.This method has been described in [17,Lem. 4 reduce the proof to this setting.In the coming steps we will use tools from semiclassical analysis, some of which are presented in section A of the appendix.
Step 1: the Schrödinger propagator as a Fourier Integral Operator.-For any t ∈ R, we denote We claim that if t = 0, Λ(t) is a projectable Lagrangian submanifold of T * ( X 2 ).Indeed, since X has negative curvature, [14, Th. 4.8.1]implies that for any x, x ∈ X and any t ∈ R {0}, there exists unique ξ ∈ T x X, ξ ∈ T x X, depending smoothly on x and x , such that Φ t (x, ξ) = (x , ξ ).In other words, Λ(t) is a smooth section of T * ( X 2 ), so it is a projectable Lagrangian manifold thanks to Remark 4.2.Next, recall the standard fact that the frequency-localized Schrödinger propagator is a Fourier Integral Operator associated to the geodesic flow, whose proof is similar to [15, Th. 2.1] (see also [21,Th. 10.4]).This means that, if ψ t is a phase generating Λ(t), and if (4)  A ∈ Ψ comp h ( X), then there exists u ∈ C ∞ c ( X 2 ) such that U (t)A is the sum of an operator whose Schwartz kernel is u(x, x )e iψt(x,x )/h and of an O L 2 →L 2 (h ∞ ) remainder.Here, u can depend on h, but its supports and C k norms are bounded independently of h.
Step 2: using coordinates.-Fourier Integral Operators are easier to describe in some system of coordinates.Since X is a complete simply connected manifold of negative curvature, by the Cartan-Hadamard theorem, there exists a diffeomorphism κ : X → R d , which is simply given by the exponential map at any point.We denote by T * X the co-tangent bundle of X, and by Φ t : T * X → T * X the geodesic flow at time t.We equip T * X with its natural symplectic structure.The diffeomorphism κ can be lifted to a symplectomorphism by (4.16)K : For any t ∈ R, let us write We deduce from the previous step that Λ(t) is a projectable Lagrangian submanifold of T * R 2d .Furthermore, if ψ t is a phase generating Λ(t), and if ) is the sum of an operator whose Schwartz kernel is (4.17) u(y, y )e i ψt(y,y )/h , (4) The space Ψ comp h is the space of (Weyl) pseudo-differential operators with compactly supported symbols.It is defined in section A of the appendix.and of an O L 2 →L 2 (h ∞ ) remainder.Here, again, u can depend on h, but its supports and C k norms are bounded independently of h.
Step 3: using the WKB method.-Let φ 0 ∈ C ∞ (Ω 0 ) and a ∈ C ∞ c (Ω 0 ) be as in the statement of Proposition 4.11.Recall that We thus want to apply the operator (κ −1 ) * U h (t)(κ) * to the Lagrangian state ae i h ϕ0 .Up to a O H k (h ∞ ) (for any k ∈ N), it does therefore amount to applying the operator (κ −1 ) * U h (t)A(κ) * , whose integral kernel is described by (4.17).We are then exactly in the framework of [17,Lem. 4.1], and we can conclude using the following lemma, the last point in Proposition 4.11 coming from Lemma 4.8.Lemma 4.12.-Let Λ 0 be a projectable Lagrangian submanifold of T * R d , generated by a phase function ϕ 0 , with support ω 0 ⊂ R d .Fix t > 0. Suppose that Φ t (Λ 0 ) is a projectable Lagrangian submanifold of T * ω t , for some open subset ω t ⊂ R d , generated by a phase function ϕ t .Then, for any a ∈ C ∞ c (ω 0 ) and any k ∈ N, the application of the operator (κ −1 ) * U h (t)(κ) * to the Lagrangian state ae iϕ0/h associated with Λ 0 can be written as where r h H k = O(h), and where a t ∈ C ∞ c (R d ) is given by (4.18) a t (x) = e iβ(t)/h |det dg 0,t (x)| 1/2 (a • g 0,t )(x), for some β(t) ∈ R. Here, g 0,t : ω t → ω 0 is given by g 0,t = κ −1 • g 0,t • κ, with g 0,t : ω t → ω 0 as in (4.8).
We may now proceed with the proof of Proposition 3.3, after introducing a few notations.Let us write pr : X → X for the covering map of X.It induces a projection pr * : T * X → T * X, such that pr * • Φ t = Φ t •pr * .We shall write pr −1 (x) := {y ∈ X | pr(y) = x}.We also define a map Π : C 0 ( X) → C 0 (X) by (Πf )(x) = y∈pr −1 (x) f (y).Let us denote by U h (t) : L 2 ( X) → L 2 ( X) the semi-classical Schrödinger propagator on X.If f ∈ C 0 ( X), we have -In this section we prove Proposition 5.1.For the proof, we will need the following definition.Let k be an integer no smaller than two.For each finite sequence of relative, non-zero integers, n = (n 1 , . . ., n k ), let T n = ((x, ξ 1 ), . . ., (x, ξ k )) | x ∈ X, ξ 1 , . . ., ξ k ∈ T * x X {0}, the ξ j are not all equal and k j=1 n j ξ j = 0 .Proposition 5.1 will be a consequence of the following result.Proof of Proposition 5.1.-A countable intersection of residual sets is still a residual set.Hence, thanks to the previous lemma, we know that there exists a residual subset E ⊂ C ∞ (Ω) such that for all φ ∈ E, the following holds.For all k 2 and all n ∈ N k , the sets O n are countable unions of one dimensional submanifolds.Let φ ∈ E, and let k 2. We shall write Ψ φ : Ω k × R (x 1 , . . ., x k , t) → π X (Φ t (x 1 , ∂φ(x 1 ))) ∈ X.Then, for all n ∈ N k , Ψ φ (O n ) has measure zero.
We claim that the set Ψ φ (O n ) is exactly the set of x ∈ X such that there exists x 1 , . . ., x k ∈ X, t ∈ R and ξ 1 , . . ., ξ k ∈ T * x X such that Φ t (x j , ∂φ(x j )) = (x, ξ j ) for all j = 1, . . ., k and k j=1 n j ξ j = 0. Indeed, by the discussion after Proposition 3.3, the directions ξ 1 , . . ., ξ k are all different, so the claim follows from definition of T n .
Since Φ t is a diffeomorphism, the map Ψ is a submersion.Moreover, T n is a submanifold of (T * X) k of codimension kd.Therefore, Ψ −1 (T n ) is a submanifold of Proof.-The lemma follows by the method of characteristics.Indeed, for the PDE (B.1), the hypersurface Σ is non-characteristic (in the sense of [11, (36) p. 106]).Therefore, we can apply the method of characteristics and deduce local existence and uniqueness of the solution near each point of Σ (see [11]).Then, piecing together local solutions using the uniqueness, we obtain a global solution near Σ.Moreover, given two solutions (ψ 1 , Ω 1 ) and (ψ 2 , Ω 2 ) of (B.1), by local uniqueness, for each x ∈ Σ, there exists U x ⊂ Ω 1 ∩ Ω 2 on which they coincide.In particular, Ω 3 = x∈Σ U x is a neighbourhood of Σ on which they coincide.
To better understand the solutions to Equation (B.1), we check that solutions to this equation satisfy a property that makes them simple do describe in terms of the initial condition.

(4. 19 ) 5 . 1 .
Π U h (t)f = U h (t)Πf,because both side satisfy the same differential equation with the same initial conditions.Proof of Proposition 3.3.-Thanks to Lemma 4.5, we know that we may find finitely many open sets (O n ) n=1,...,N in X such that T * O n ∩ Λ is nowhere stable.Let two following propositions, which we use in the proofs of Theorems 2.6 and 2.10 respectively.We equip all these sets with the topology of uniform convergence of derivatives on compact sets.Proposition 5.1.-The set E T,irr (λ1,λ2) (Ω) is a residual subset of E T (λ1,λ2) (Ω).Proposition 5.2.-The set C T,irr (Σ) is a residual subset of C T (Σ).The polychromatic case: proof of Proposition 5.1.
J.É.P. -M., 2022, tome 9 j on R d converge weakly as t → +∞ to λ a,φ .Let us start with the proof of Lemma 3.8.Proof of Lemma 3.8.-For each t 0, consider the random function f t (y) := N (t) j=1 β t j e iξ t j •y+iϑ t j , where for each t, the ϑ t j are independent random variables uniformly distributed on [0, 2π].Thus f t has law P β t ,ξ t .