Sang Quang

Phan

Spectral monodromy of small non-selfadjoint quantum perturbations of completely integrable Hamiltonians (December 5, 2016, provisional version) 1

Keywords: AMS 2010 Mathematics Subject Classification: 35P20, 81Q12 integrable system, non-selfadjoint, spectral asymptotic, pseudo-differential operators

We define a monodromy, directly from the spectrum of small non-selfadjoint perturbations of a selfadjoint semiclassical operator with two degrees of freedom, which is classically integrable. It is a combinatorial invariant that obstructs globally the existence of lattice structure of the spectrum, in the semiclassical limit. Moreover this spectral monodromy allows to recover a topological invariant (the classical monodromy) of the corresponding integrable system.

Introduction 1.Motivation

We propose in this article a way of detecting the monodromy of a quantum Hamiltonian which is classically integrable, by looking at non-selfadjoint perturbations.

In the classical theory, the classical monodromy is defined for a completely integrable system on symplectic manifolds as a topological invariant that obstructs the existence of global action-angle coordinates on the phase space, see [START_REF] Duistermaat | On global action-angle coordinates[END_REF].

Quantum monodromy was detected a long time ago in [START_REF] Cushman | The quantum mechanical spherical pendulum[END_REF] and completely defined in [START_REF] Ngo | Quantum monodromy in integrable systems[END_REF], in the joint spectrum of system of selfadjoint operators that commute, in the sense of the semiclassical limit, as the classical monodromy of the underlying classical system.

However, a mysterious question is whether a monodromy can be defined for only one semiclassical operator? That is how to detect the modification of action-angle variables from only one spectrum?

We are interested in the globally structure of the spectrum of non-selfadjoint h-Weylpseudodifferential operators with two degrees of freedom, which are small non-selfadjoint perturbations of a selfadjoint operator, in the semiclassical limit. Such an operator is of the form P ε = P (x, hD x , ε; h),

where the unperturbed operator P := P ε=0 is formally selfadjoint, and ε is a small parameter. In this work ε is assumed to depend on the classical parameter h and in the regime h ε = O(h δ ), with 0 < δ < 1. The first answer for the above problem, which was given in Ref. [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF], is a particular case of the present work. In that work, the operators had the simple form P + iεQ, such that the corresponding principal symbols p of P and q of Q commute for the Poisson bracket, {p, q} = 0. Here we develop this result in assuming only that the principal symbol of P ε in (1.1) is of the form

p ε = p + iεq + O(ε 2 ),
with p is a completely integrable Hamiltonian.

It is known from the spectral asymptotic theory (see [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]) that, under some suitable global assumption, the spectrum of the perturbed operators has locally the form of a deformed discrete lattice. The eigenvalues admit asymptotic expansions in h and ε. Moreover, we shaw prove the mail result (Theorem 4.4) that the spectrum is an asymptotic pseudo-lattice (see Definition 4.1). Therefore, as an application from [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF], a combinatorial invariant of the spectral lattice-the spectral monodromy is well defined, directly from the spectrum.

Moreover, this quantum result is strictly related to the classical results. The spectral monodromy can be identified to the classical monodromy of the completely integrable system p.

Brief description for the spectral monodromy

It knows from the spectral asymptotic theory (see [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]) that, under an ellipticity condition at infinity (see (3.10)), the spectrum of the perturbed operators (1.1) is discrete, and included in a horizontal band of size O(ε). Moreover, that work allows give asymptotic expansions of the spectrum located in some small domains of size O(h δ ) × O(εh δ ) of the spectral band, called good rectangles.

As we shaw show in this work that there is a correspondence (in fact a local diffeomorphism, see Proposition 4.6 and a proof for this point in [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF]), denoted by f , from the spectrum contained in a good rectangle

R (a) (ε, h) to a part of hZ 2 , modulo O(h ∞ ), R (a) (ε, h) µ → f (µ, ε; h) ∈ hZ 2 + O(h ∞ ). (1.2)
Here a ∈ R 2 is introduced to fix the center of the good rectangle. This map is called a h-local chart of the spectrum. The spectrum therefore has the structure of a deformed lattice, with horizontal spacing h and vertical spacing εh. See Figure 1 below.

f h σ(P ε ) hZ 2 O(ε) R(χ a ) E(ε, h) Figure 1. h-local chart of the spectrum
Such good rectangles correspond to the Diophantine invariant tori in the phase space, on which the Hamiltonian flow of the unperturbed part, that is p, is quasi-periodic of constant frequency, see (3.29). However, there are many such rectangles in the spectral band due to the existence of Diophantine invariant tori in the phase space. The set of the center of the good rectangles is nowhere dense in the complex plane, but of large measure. It is well known from the classical theory.

We shall call a family of close h-local charts on a small domain U * (ε) a local pseudochart of the spectrum. We notice that each h-local chart is normally valid for one good rectangle. However, an important fact we show in this paper that we may build a local pseudo-chart such that the leading term in asymptotic expansions of their h-local charts in the small parameters h, ε is locally well defined on U * (ε). The construction of local pseudo-charts will be done in detail in Section 4.

With regard to the global problem, we consider the spectrum as a discrete subset of the complex plane. We can apply a result of [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF] about a discrete set, called an asymptotic pseudo-lattice (see also Definition 4.1), for the spectrum. It shows that the differential of the transition maps between two overlapping local pseudo-charts (f i , U i (ε)) and (f j , U j (ε)) is in the group GL(2, Z), modulo O(ε, h ε ):

d( f i ) = M ij d( f j ) + O(ε, h ε ),
where

f i = f i • χ, f j = f j • χ, with χ is the function (u 1 , u 2 ) → (u 1 , εu 2 ), and M ij ∈ GL(2, Z) is an integer constant matrix.
Let U (ε) be a bounded open domain in the spectral band and cover it by an arbitrary (small enough) locally finite covering of local pseudo-charts {(f j , U j (ε))} j∈J , here J is a finite index set. Then the spectral monodromy on U (ε) is defined as the unique 1-cocycle {M ij }, modulo coboundary in the first Čech cohomology group. It is clear that if the spectral monodromy is not trivial, then the transition maps either, and the spectrum hasn't a smooth global lattice structure.

2 Some basics

Weyl-quantization

We will work throughout this article with pseudodifferential operators obtained by the h-Weyl-quantization of a standard space of symbols on T * M = R 2n (x,ξ) , here M = R n or a compact manifold of n dimensions, and in particular n = 2. We denote σ the standard symplectic 2-form on T * M .

In the following we represent the quantization in the case M = R n . In the manifold case, the quantization is suitably introduced. We refer to Refs. [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]- [START_REF] Shubin | Pseudodifferential Operators and Spectral Theory[END_REF] for the theory of pseudodifferential operators. Definition 2.1. A function m : R 2n → (0, +∞) is called an order function if there are constants C, N > 0 such that

m(X) ≤ C X -Y N m(Y ), ∀X, Y ∈ R 2n , with notation Z = (1 + |Z| 2 ) 1/2 for Z ∈ R 2n .
Definition 2.2. Let m be an order function and k ∈ R, we define classes of symbols of h-order k, denoted by S k (m) (families of functions), of (a(•; h)) h∈(0,1] on R 2n (x,ξ) by

S k (m) = {a ∈ C ∞ (R 2n ) | ∀α ∈ N 2n , |∂ α a| ≤ C α h k m}, (2.3)
for some constant

C α > 0, uniformly in h ∈ (0, 1]. A symbol is called O(h ∞ ) if it's in ∩ k∈R S k (m) := S ∞ (m).
Then Ψ k (m)(M ) denotes the set of all (in general unbounded) linear operators A h on L 2 (R n ), obtained from the h-Weyl-quantization of symbols a(•; h) ∈ S k (m) by the integral:

(A h u)(x) = (Op w h (a)u)(x) = 1 (2πh) n R 2n e i h (x-y)ξ a( x + y 2 , ξ; h)u(y)dydξ. (2.4)
In this work, we always assume that symbols admit classical asymptotic expansions in integer powers of h. The leading term in this expansion is called the principal symbol of operators.

Classical theory

We recall here some notions and results from the classical theory. Definition 2.3. An integrable system on a symplectic manifold (W, σ) of dimension 2n (n ≥ 1) is given n smooth real-valued functions f 1 , . . . , f n in involution with respect to the Poisson bracket generated from the symplectic form σ, whose differentials are almost everywhere linearly independent. In this case, the map

F = (f 1 , . . . , f n ) : M → R n
is also called an integrable system, or a momentum map.

A smooth function f 1 is called completely integrable if there exists n -1 functions f 2 , . . . , f n such that F = (f 1 , . . . , f n ) is an integrable system.

Let U be an open subset of regular values of F . Then we have, Theorem 2.4 (Angle-action theorem). (Refs. [START_REF] Arnold | On a characteristic class entering into conditions of quantization[END_REF], [START_REF] Arnold | Proof of a theorem of A.N.Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian[END_REF], and [START_REF] Duistermaat | On global action-angle coordinates[END_REF]) Let c ∈ U , and Λ c be a compact regular leaf of the fiber F -1 (c). Then there exists an open neighborhood V = V c of Λ c in W such that F | V defines a smooth locally trivial fibre bundle onto an open neighborhood U c ⊂ U of c, whose fibres are invariant Lagrangian n-tori. Moreover, there exists a symplectic diffeomorphism κ = κ c , Notice that one chooses usually the local chart such that the torus Λ c is sent by κ to the zero section T n × {0}. By this theorem, for every a ∈ U c , then Λ a := F -1 (a) ∩ V c is an invariant Lagrangian n-torus, called a Liouville torus, and we write

κ = (x, ξ) : V → T n × A, with A = A c ⊂ R n is an open subset, such that F • κ -1 (x, ξ) = ϕ(ξ) for all x ∈ T n ,
Λ a κ(Λ a ) = T n × {ξ a } := Λ ξa , (2.5)
with some ξ a ∈ A.

Spectral asymptotics

In this section, we shall apply the spectral asymptotic theory from [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF] for small nonselfadjoint perturbations of selfadjoint operators in two dimensions to give the asymptotics of eigenvalues located in some suitable small windows of the complex plane.

General assumptions

We give here some global geometric assumption on the dynamics of the principal symbol of unperturbed selfadjoint operators, as in Refs. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF], and [START_REF] Hitrik | Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions. I[END_REF]. M denotes R 2 or a connected compact analytic real (Riemannian) manifold of dimension 2 and we denote by M the canonical complexification of M , which is either C 2 in the Euclidean case or a Grauert tube in the case of manifold (see Ref. [START_REF] Burns | Symplectic geometry and the uniqueness of Grauert tubes[END_REF]).

We consider a non-selfadjoint h-pseudodifferential operator P ε on M and suppose that P ε=0 := P is formally selfadjoint.

(3.6)

Note that if M = R 2 , the volume form µ(dx) is naturally induced by the Lebesgue measure on R 2 . If M is a compact Riemannian manifold, then the volume form µ(dx) is induced by the given Riemannian structure of M . Therefore in both cases the volume form is well defined and the operator P ε may be seen as an (unbounded) operator on L 2 (M, µ(dx)). We always denote the principal symbol of P ε by p ε which is defined on T * M .

We will assume the ellipticity condition at infinity for P ε at some energy level E ∈ R as follows:

When M = R 2 , let

P ε = P (x, hD x , ε; h) (3.7)
be the Weyl quantification of a total symbol P (x, ξ, ε; h) depending smoothly on ε in a neighborhood of (0, R) and taking values in the space of holomorphic functions of (x, ξ) in a tubular neighborhood of R 4 in C 4 on which we assume that:

|P (x, ξ, ε; h)| ≤ O(1)m(Re(x, ξ)). (3.8)
Here m is an order function in the sense of Def. 2.1. We assume moreover that m > 1 and P ε is classical of order 0,

P (x, ξ, ε; h) ∼ ∞ j=0 p j,ε (x, ξ)h j , h → 0, (3.9)
in the selected space of symbols as in Section 2.1. In this case, the principal symbol p ε = p 0,ε is the first term of the above expansion, and the ellipticity condition at infinity is

|p ε (x, ξ) -E| ≥ 1 C m(Re(x, ξ)), | (x, ξ) |≥ C, (3.10)
for some C > 0 large enough. When M is a compact manifold, we consider P ε a differential operator on M such that in local coordinates x of M , it is of the form:

P ε = |α|≤m a α,ε (x; h)(hD x ) α , (3.11)
where

D x = 1 i ∂
∂x and a α,ε are smooth functions of ε in a neighborhood of 0 with values in the space of holomorphic functions on a complex neighborhood of x = 0. We assume that these a α,ε are classical of order 0,

a α,ε (x; h) ∼ ∞ j=0 a α,ε,j (x)h j , h → 0, (3.12)
in the selected space of symbols. In this case, the principal symbol p ε in the local canonical coordinates associated (x, ξ) on T * M is

p ε (x, ξ) = |α|≤m a α,ε,0 (x)ξ α , (3.13)
and the elipticity condition at infinity is

|p ε (x, ξ) -E| ≥ 1 C ξ m , (x, ξ) ∈ T * M, | ξ |≥ C, (3.14)
for some C > 0 large enough. Notice here that M has a Riemannian metric, then | ξ | and ξ = (1+ | ξ | 2 ) 1/2 are well defined. It is known from Refs. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF], and [START_REF] Hitrik | Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions. I[END_REF] that with the above conditions, the spectrum of P ε in a small but fixed neighborhood of E in C is discrete, when h > 0, ε ≥ 0 are small enough. Moreover, this spectrum is contained in a horizontal band of size ε:

|Im(z)| ≤ O(ε). (3.15)
Let p = p ε=0 , it is principal symbol of the selfadjoint unperturbed operator P and therefore real. And let q = 1 i ( ∂pε ∂ε ) ε=0 and assume that q is a bounded analytic function on T * M . We can write the principal symbol

p ε = p + iεq + O(ε 2 ). (3.16)
We assume that p is completely integrable, i.e., there exists an integrable system

F = (p, f ) : T * M → R 2 (3.17)
Then the space of regular leaves of F is foliated by Liouville Lagrangian invariant tori by Theorem 2.4.

We assume also that

p -1 (E) ∩ T * M is connected, (3.18)
and the energy level E is regular for p, i.e., dp = 0 on p -1 (E) ∩ T * M . We would like to notice that the level set p -1 (E) is compact, due to the ellipticity condition at infinity (3.10) or (3.14) Then the energy space p -1 (E) is decomposed into a singular foliation:

p -1 (E) ∩ T * M = a∈J Λ a , (3.19)
where J is assumed to be a compact interval, or, more generally, a connected graph with a finite number of vertices and of edges, see pp. 21-22 and 55 of Ref. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]. We denote by S the set of vertices. For each a ∈ J, Λ a is a connected compact subset invariant with respect to H p . Moreover, if a ∈ J\S, Λ a are the Liouville tori depending analytically on a. These tori are regular leaves corresponding to regular values of F . Each edge of J can be identified with a bounded interval of R and we have therefore a distance on J in the natural way.

We denote H p the Hamiltonian vector field of p, defined by σ(H p , •) = -dp(•). For each a ∈ J, we define a compact interval in R:

Q ∞ (a) = lim T →∞ inf Λa Re q T , lim T →∞ sup Λa Re q T , (3.20)
where q T , for T > 0, is the symmetric average time T of q along the H p -flow, defined by

q T = 1 T T /2 -T /2 q • exp(tH p )dt. (3.21)
Then we can improve (3.15) that the spectrum the of P ε in the neighborhood of E in C is located in the band

Im (σ(P ε ) ∩ {z ∈ C : |Rez -E| ≤ δ}) ⊆ ε inf a∈J Q ∞ (a) -o(1), sup a∈J Q ∞ (a) + o(1) , (3.22
) when ε, h, δ → 0 (see Ref. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]).

From now, for simplicity, we will assume that q is real valued (in the general case, simply replace q by its real part Re(q) in regard to (3.16)).

Each torus Λ a , with a ∈ J\S, locally can be embedded in a Lagrangian foliation of H p -invariant tori. By Theorem 2.4, there are analytic local angle-action coordinates on an open neighborhood

V of Λ a κ = (x, ξ) : V → T 2 × A, (3.23)
such that Λ a Λ ξa , ξ a ∈ A, and p becomes only a function of ξ,

p • κ -1 = p(ξ) = p(ξ 1 , ξ 2 ), ξ ∈ A. (3.24)
Let Λ ⊂ V be an arbitrary Liouville torus (close to Λ a ). We have

Λ Λ ξ , ξ ∈ A. (3.25)
We define q -the average of q on the torus Λ, with respect to the natural Liouville measure on Λ, denoted by q Λ , as following

q Λ = Λ q.
(3.26)

Remark 3.1. In the action-angle coordinates (x, ξ) given by (3.23), we have

q Λ = q Λ ξ = q (ξ) = 1 (2π) 2 T 2 q(x, ξ)dx, ξ ∈ A. (3.27)
In particular, q Λa = q (ξ a ).

It is true that q Λa depends analytically on a ∈ J\S, and we assume that it can be extended continuously on J. Furthermore, we assume that the function a → q Λa = q (ξ a ) is not identically constant on any connected component of J\S, and that dp(ξ) and d q (ξ) are R-linearly independent at ξ a .

(3.28)

Let Λ ⊂ V be a Liouville torus as in (3.25). Then we define the frequency of Λ (also of Λ ξ ) by

ω(ξ) = ∂p ∂ξ (ξ) = ∂p ∂ξ 1 (ξ), ∂p ∂ξ 2 (ξ) , ξ ∈ A, (3.29)
and the rotation number of Λ by

ρ(ξ) = ∂p ∂ξ 1 (ξ) : ∂p ∂ξ 2 (ξ) , ξ ∈ A, (3.30)
viewed as an element of the real projective line. It is clear that ρ depends analytically on ξ ∈ A and we shall assume that the restricted function a → ρ(a) := ρ(ξ a ) is not identically constant on any connected component of J\S.

(3.31) Remark 3.2 (see pp. 56-57 of Ref. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]). For a ∈ J\S, if ρ(a) / ∈ Q, that means the frequency ω(ξ a ) is non resonant, then along the torus Λ a , the Hamiltonian flow of p is ergodic. Hence the limit of q T , when T → ∞ exists, and is equals to q Λa . Therefore we have Q ∞ (a) = { q Λa }.

Asymptotics of eigenvalues

The spectral asymptotic theory (see Refs. [START_REF] Hitrik | Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions. I[END_REF], and [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]) allows us to give the asymptotic description of all the eigenvalues of P ε in some adapted small complex windows of the spectral band, which are associated with Diophantine tori in the phase space. The force of the perturbation ε is small and can be dependent or independent of the classical parameter h. However in this work we consider the case when ε is sufficiently small, dependent on h, and in the following regime

h ε = O(h δ ), (3.32)
where δ > 0 is some number small enough but fixed. In this case, spectral results are related to (h, ε)-dependent small windows.

Definition 3.3. Let α > 0, d > 0, and Λ Λ ξ be a H p -invariant Lagrangian torus, as in (3.25). We say that Λ is (α, d)-Diophantine if its frequency ω(ξ), defined in (3.29), satisfies

ω(ξ) ∈ D α,d = ω ∈ R 2 | ω, k | ≥ α |k| 1+d , ∀ k ∈ Z 2 \{0} . (3.33)
Notice also that when d > 0 is fixed, the Diophantine property (for some α > 0) of Λ is independent of selected angle-action coordinates, see [START_REF] Bost | Tores invariants des systèmes dynamiques hamiltoniens[END_REF]. If Λ is (α, d)-Diophantine, then its frequency must be irrational and we may have the result in Remark 3.2.

It is known that the set D α,d is a closed set with closed half-line structure. When we take α to be sufficiently small, it is a nowhere dense set but with no isolated points. Moreover, its measure tends to large measure as α tends to 0: the measure of its complement is of order O(α). See Refs. [START_REF] Broer | Unfoldings and Bifurcations of Quasi-periodic Tori[END_REF], and [START_REF] Pöschel | A lecture on the classical KAM-theorem[END_REF]. Definition 3.4. For some α > 0 and some d > 0, we define the set of good values associated with an energy level E, denoted by G(α, d, E), obtained from ∪ a∈J Q ∞ (a) by removing the following set of bad values B(α, d, E):

B(α, d, E) = dist(a,S)<α Q ∞ (a) a∈J\S: ω(ξa) is not (α,d)-Diophantine Q ∞ (a) a∈J\S: |d q (ξa)|<α Q ∞ (a) a∈J\S: |ω (ξa)|<α Q ∞ (a) .
Remark 3.5. We note that (i) When d > 0 is kept fixed, the measure of the set of bad values B(α, d, E) in ∪ a∈J Q ∞ (a) (and in q Λa (J)) is O(α), when α > 0 is small enough, provided that the measure of

a∈J\S: ω(ξa)∈Q Q ∞ (a) a∈S Q ∞ (a) (3.34)
is sufficiently small, depending on α (see Ref. [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]).

(ii) Let G ∈ G(α, d, E) be a good value, then by Definition of B(α, d, E) and Remark 3.2, there are a finite number of corresponding (α, d)-Diophantine tori Λ a 1 , . . . , Λ a L , with L ∈ N * and {a 1 , . . . , a L } ⊂ J \ S, in the energy space p -1 (E) ∩ T * M , such that the pre-image

q -1 (G) = {Λ a 1 , . . . , Λ a L }.
In this way, when G varies in G(α, d, E), we obtain a family of large measure of (α, d)-Diophantine invariant tori in the phase space satisfying {p = E, q = G}.

For G ∈ G(α, d, E) is a good value, we define in the horizontal band of size ε of complex plan, given in (3.22), a suitable window of size O(h δ ) × O(εh δ ), around the good center E + iεG, called a good rectangle,

R (E,G) (ε, h) = (E + iε G) + - h δ O(1) , h δ O(1) + iε - h δ O(1) , + h δ O(1) . ( 3

.35)

Now let G ∈ G(α, d, E) be a good value. As in Remark 3.5, there exists L elements in pre-image of G by q . We shall assume that L = 1 and we write

q -1 (G) = Λ a ⊂ p -1 (E) ∩ T * M, a ∈ J \ S, (3.36)
Note that this hypothesis can be achieved if we assume that the function (p, q ) is proper with connected fibres. Let Λ a , a ∈ J\S be an invariant Lagrangian torus and let κ be the action-angle local coordinates in (3.23). The fundamental cycles (γ 1 , γ 2 ) of Λ a are defined by

γ j = κ -1 ({(x, ξ) ∈ T * T 2 : x j = 0, ξ = ξ a }), j = 1, 2.
Then we note η ∈ Z 2 the Maslov indices and S ∈ R 2 the action integrals of these fundamental cycles,

S = (S 1 , S 2 ) = γ 1 θ, γ 2 θ , (3.37)
where θ is the Liouville 1-form on T * M .

Definition 3.6 (Refs. [START_REF] De Verdière | Méthodes semi-classiques et théorie spectrale[END_REF], and [START_REF] Robbin | The Maslov index for paths[END_REF]). Let E be a symplectic space and let Λ(E) be his Lagrangian Grassmannian (which is set of all Lagrangian subspaces of E). We consider a bundle B in E over the circle or a compact interval provided with a Lagrangian subbundle called vertical. Let λ(t) be a section of Λ(B) which is transverse to the vertical edges of the interval in the case where the base is an interval. The Maslov index of λ(t) is the intersection number of this curve with the singular cycle of Lagrangians which do not cut transversely the vertical subbundle.

The Maslov index appears in the statement of spectral asymptotic results. In our work it is treated as a standard constant.

We recall here spectral asymptotic results for the standard case at the energy level E = 0, cited from [START_REF] Hitrik | Diophantine tori and spectral asymptotics for nonselfadjoint operators[END_REF]. However these results can be immediately generalized for any energy lever E by a translation. Theorem 3.7 ([4]). For E = 0. We suppose that P ε is an operator satisfying Assumptions 3.1 (from (3.6) to (3.31)), and in the regime h ε = O(h δ ) for 0 < δ < 1. Let G ∈ G(α, d, 0) be a good value as Def. 3.4, and assume that (3.36) is true. Suppose that the action-angle coordinates κ in (3.23) send Λ a to the zero section T 2 ×{ξ a = 0} ∈ T * T 2 . Assume that dp(ξ) and d q (ξ) are R-linearly independent at ξ a = 0, where p = p(ξ) as in (3.24) and q (ξ) is the average of q on tori, given in (3.27).

Then the eigenvalues µ of P ε with multiplicity in the good rectangle R (0,G) (ε, h) of the form (3.35) have the following expression,

µ = P (∞) h(k - η 4 ) - S 2π , ε; h + O(h ∞ ), k ∈ Z 2 , (3.38)
where P (∞) (ξ, ε; h) is a smooth function of ξ defined in a neighborhood of (0, R 2 ) and ε, h in neighborhoods of (0, R). Moreover P (∞) (ξ, ε; h) is real valued for ε = 0, admits the following polynomial asymptotic expansion in (ξ, ε, h) for the C ∞topology:

P (∞) (ξ, ε; h) ∼ α,j,k C αjk ξ α ε j h k . (3.39)
In particular P (∞) is classical in the space of symbols with h-leading term:

p (∞) 0 (ξ, ε) = p(ξ) + iε q (ξ) + O(ε 2 ). ( 3

.40)

The fact that the horizontal spectral band is of size O(ε) suggests us introducing the function

χ : R 2 u = (u 1 , u 2 ) → χ u = (u 1 , εu 2 ) ∈ R 2 (3.41) ∼ = u 1 + iεu 2 ∈ C, in which we identify C with R 2 .
Noticing that σ(P ε ) = σ(P εχ a ) + χ a and applying Theorem 3.7 (in the standard case) for the operator (P ε -χ a ) with respect to the good rectangle R (0,G) (ε, h), we obtain easily the asymptotic eigenvalues of P ε in any good rectangle R (E,G) (ε, h), as following.

Theorem 3.8. For E ∈ R. We suppose that P ε is an operator satisfying Assumptions 3.1 (from (3.6) to (3.31)), and in the regime ,d,E) be a good value as Def. 3.4, and assume that (3.36) is true. Suppose that by κ, given in (3.23), Λ a Λ ξa , with ξ a ∈ A. Assume that dp(ξ) and d q (ξ) are R-linearly independent at ξ a , where p = p(ξ) as in (3.24) and q (ξ) as in (3.27).

h ε = O(h δ ) for 0 < δ < 1. Let G ∈ G(α
Then the eigenvalues µ of P ε with multiplicity in the good rectangle R (E,G) (ε, h) of the form (3.35), modulo O(h ∞ ), are h-locally given by

µ = P (∞) ξ a + h(k - η 4 ) - S 2π , ε; h + O(h ∞ ), k ∈ Z 2 , (3.42)
where P (∞) (ξ; ε, h) is a smooth function defined in a neighborhood of (ξ a , R 2 ) and of ε, h in neighborhoods of (0, R). Moreover P (∞) (ξ; ε, h) admits asymptotic expansions of the form (3.39), with h-leading term of the form (3.40).

Remark 3.9. We can write the total symbol P (∞) in the reduce form:

P (∞) (ξ, ε; h) = p(ξ) + iε q (ξ) + O(ε 2 ) + O(h), (3.43)
uniformly for ε and h small.

Remark 3.10. We can show that the function P (∞) in the above theorem is a local diffeomorphism from a neighborhood of (ξ a , R 2 ) into its image, that is in a O(ε)horizontal band (see Ref. [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF]). Therefore the eigenvalues of P ε in a good rectangle form a deformed h-lattice. It's the image of a square lattice of hZ 2 by a local diffeomorphism. Moreover, we can show that the lattice has a horizontal spacing O(h) and a vertical spacing O(εh).

4 Spectral asymptotic pseudo-lattice and its monodromy

In the previous section we have seen that the spectrum locally is a deformed h-lattice.

In this section we are studying globally the spectrum in showing that the spectrum is an asymptotic pseudo-lattice. Then we can define a monodromy for the spectrum.

Monodromy of an asymptotic pseudo-lattice

We recall here the definition of an asymptotic pseudo-lattice and its monodromy given in [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF]. This is a discrete subset of R 2 admitting a particular property.

For any subset U of R 2 we denote

U (ε) = χ(U ), (4.44)
where χ is the map in (3.41).

Definition 4.1. Let U be an open subset of R 2 with compact closure and let Σ(ε, h) (which depends on small h and ε) be a discrete subset of U (ε). For h, ε small enough and in the regime h ε, we say that (Σ(ε, h), U (ε)) is an asymptotic pseudo-lattice if:

for any small parameter α > 0, there exists a set of good values in R 2 , denoted by G(α), whose complement is of size O(α) in the sense:

| C G(α) ∩ I |≤ Cα | I |,
with a constant C > 0 for any domain I ⊂ R 2 ; for all c ∈ U , there exists a small open subset U c ⊂ U around c such that for every good value a ∈ U c ∩G(α), there is an adapted good rectangle R (a) (ε, h) ⊂ U c (ε) of the form (3.35) (with a = (E, G)), and a smooth local diffeomorphism f = f (•, ε; h) which sends R (a) (ε, h) on its image, satisfying

Σ(ε, h) ∩ R (a) (ε, h) µ → f (µ, ε; h) ∈ hZ 2 + O(h ∞ ). (4.45)
Moreover, the function f := f • χ, with χ defined by (3.41), admits an asymptotic expansion in (ε, h ε ) for the C ∞ -topology in a neighborhood of a, uniformly with respect to the parameters h and ε, such that its leading term f 0 is a diffeomorphism, independent of α, locally defined on whole U c and independent of the selected good values a ∈ U c .

We also say that the couple

(f (•, ε; h), R (a) (ε, h)) is a h-chart of Σ(ε, h), and the family of h-charts (f (•, ε; h), R (a) (ε, h)), with all a ∈ U c ∩ G(α), is a local pseudo-chart on U c (ε) of (Σ(ε, h), U (ε)).
Remark 4.2. A standard lattice is obviously an asymptotic pseudo-lattice. An another example is a known lattice with some similar but lighter properties, named asymptotic lattice, given in [START_REF] Ngo | Quantum monodromy in integrable systems[END_REF]. That lattice is modeled on the joint spectrum of system of commuting operators. It is locally defined, while the asymptotic pseudo-lattice is very delicate, it is h-locally defined.

The introduction of this discrete lattice aims to show that the combinatorial invariant that we will define is directly built from the spectrum of operators. If different operators have the same spectrum, then they have the same monodromy.

Let {U j } j∈J , here J is a finite index set, be an arbitrary (small enough) locally finite covering of U . Then the asymptotic pseudo-lattice (Σ(ε, h), U (ε)) can be covered by associated local pseudo-charts {(f j (•, ε; h), U j (ε))} j∈J . Note from Definition 4.1 that the leading terms f j,0 (•, ε; h) are well defined on whole U j and we can see them as the charts of U . Analyzing transition maps, we had the following result. Proposition 4.3 ([17]). On each nonempty intersection U i ∩ U j = ∅, i, j ∈ J , there exists an unique integer linear map M ij ∈ GL(2, Z) (independent of h, ε) such that: (4.47)

d f i,0 • ( f j,0 ) -1 = M ij . ( 4
It does n't depend on the selected finite covering {U j } j∈J and the small parameters h, ε.

We can also associate the class [M] with its holonomy, that is a group morphism from the fundamental group π 1 (U ) to the group GL(2, Z), modulo conjugation. The triviality of [M] is equivalent to the one of its holonomy.

Spectral monodromy of P ε

Now let F = (p, f ) be an integrable system as in (3.17) and denote U r the set of all regular values of F . We recall that the space of regular leaves of F is foliated by Liouville invariant tori by Theorem 2.4.

Theorem 4.4. Let U be an open subset of U r with compact closure. In the regime h ε = O(h δ ) for 0 < δ < 1, we suppose that P ε is an operator satisfying the assumptions of Sec. 3.1 (from (3.6) to (3.31)) for any energy level E in the projection of U on horizontal axis. For any Liouville torus Λ ⊂ F -1 (U ) such that Λ Λ ξ in a local angle-action coordinate κ = (x, ξ) as in (3.23), we assume that dp(ξ) and d q (ξ) are R-linearly independent, where p = p(ξ) as in (3.24) and q (ξ) as in (3.27) are expressions of p and q in the local angle-action variables. Moreover we assume that the map (p, q ) on F -1 (U ) is proper with connected fibers. Then (σ(P ε ), U (ε)) is an asymptotic pseudo-lattice.

Proof of Theorem 4.4. Let an any point c ∈ U . There exists uniquely a Liouville torus Λ c = (p, q ) -1 (c) in T * M . We note that p and q commute in neighborhood of each Liouville torus, due to the fact that q is invariant under the flow of p. Therefore we can use a local action-angle coordinate in a neighborhood V c ⊂ T * M of Λ c as in Theorem 2.4 with F = (p, q ) and ϕ(ξ) = (p(ξ), q (ξ)), ξ ∈ A c . Now, for any point a = (E, G) in a small enough neighborhood U c such that G is a good value, i.e., G ∈ G(α, d, E), see Definition 3.4. We notice also that with assumptions of the theorem, the condition (3.36) is valid (L = 1). So the corresponding Liouville torus Λ a = F -1 (a) is (α, d)-Diophantine, as discussed in Remark 3.5. Suppose that Λ a Λ ξa , with ξ a = ϕ -1 (a) ∈ A c . Then with the help of Theorem 3.8 the eigenvalues µ of P ε in the good rectangle R (E,G) (ε, h) of the form (3.35), are h-locally given by (3.42).

On the other hand, we have a classical result that for any a

∈ U c , S 2π -ξ a := τ c ∈ R 2 , ( 4.48)
is locally constant in c ∈ U r (see Ref. [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF]). Then from Remark 3.10, the Eq. (3.42) provides a smooth local diffeomorphism at

E + iεG ∈ C, denoted by f = f (µ, ε; h), that sends µ to hk ∈ hZ 2 , modulo O(h ∞ ), of the form f = f (µ, ε; h) = S 2π -ξ a + h η 4 + P -1 (µ) (4.49) = τ c + h η 4 + P -1 (µ) σ(P ε ) ∩ R (E,G) (ε, h) µ → f (µ, ε; h) ∈ hZ 2 + O(h ∞ ),
here P is the map P (∞) in Theorem 3.42, for ease of notation. We shall show that this map should be a h-chart of σ(P ε ) on the good rectangle

R (E,G) (ε, h). Let f = f • χ, then we have f = τ c + h η 4 + P -1 • χ. (4.50)
To analysis P -1 • χ, we first consider its inverse, P := χ -1 • P . It is obtained from P by dividing the imaginary part of P by ε. As P admits an asymptotic expansion in (ξ, ε, h), so it is true that P admits an asymptotic expansion in (ξ, ε, h ε ) (here h ε). Moreover, from Remark 3.9 we can write P in the reduce form:

P (ξ, ε; h) = P 0 (ξ) + O(ε) + O( h ε ) = P 0 (ξ) + O(ε, h ε ), (4.51)
uniformly for h, ε small and h ε, with P 0 (ξ) = p(ξ) + i q (ξ) ∼ = ϕ(ξ). The Proposition 4.6 (as below) ensures that the map P -1 = P -1 • χ admits an asymptotic expansion in (ε, h ε ) whose first term is ( P 0 ) -1 = ϕ -1 . Hence the map f in (4.50) admits an asymptotic expansion in (ε, h ε ) with the leading term

f 0 = τ c + ϕ -1 . (4.52)
It is clear that the leading term f 0 is a local diffeomorphism, defined on U c . It does not depend on the selected good rectangle in

U c (ε). So (f, R (E,G) (ε, h)) is a h-chart of σ(P ε ) on U c (ε).
On the other hand, in the domain U c (ε), there are many good rectangles, due to the fact that the set of good values is of large measure, see Remark 3.5. A family of such as these h-charts with common leading term satisfying (4.52) forms a local pseudo-chart for σ(P ε ) on U c (ε).

The above construction ensures that (σ(P ε ), U (ε)) is an asymptotic pseudo-lattice.

With the help of Theorem 4.4 and the construction in Sec. 4.1 we may now give the definition of the monodromy for operators as following. Definition 4.5. For ε, h > 0 sufficiently small such that h ε = h δ , 0 < δ < 1, we suppose that the domain U and the operator P ε satisfies all assumptions of Theorem 4.4. Then we define the spectral monodromy of P ε on the domain U (ε), denoted by [M sp (P ε )], as the monodromy of the asymptotic pseudo-lattice (σ(P ε ), U (ε)), given in (4.47). Proposition 4.6. (see [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF]) Let P = P (ξ; X) be a complex-valued smooth function of ξ near 0 ∈ R 2 and X near 0 ∈ R n . Assume that P admits an asymptotic expansion in X near 0 of the form P (ξ; X)

∼ α C α (ξ)X α ,
with C α (ξ) are smooth functions and C 0 (ξ) := P 0 (ξ) is a local diffeomorphism near ξ = 0. Then, for | X | small enough, P is also a smooth local diffeomorphism near ξ = 0 and its inverse admits an asymptotic expansion in X near 0 whose the first term is ( P 0 ) -1 .

Spectral monodromy recovers the classical monodromy

Suppose that {U j } j∈J is an arbitrary small enough finite open covering of U . Here J is a finite index set. With notations as in Theorem 2.4, F -1 (U ) can be covered by a finite covering of angle-action charts {(V j , κ j )} j∈J . Then the classical monodromy is defined as the Z nbundle H 1 (Λ c , Z) → c ∈ U , whose transition maps between trivializations are

{ t d((ϕ i ) -1 • ϕ j ) -1 } = { t M cl ij -1 }.
Here H 1 (Λ c , Z) is the first homology group of the Liouville torus Λ c , and M cl ij are a locally constant matrixes of GL(2, Z) defined on the nonempty overlaps V i ∩ V j , i, j ∈ J , see [START_REF] Duistermaat | On global action-angle coordinates[END_REF].

Hence, in regard to the definition of the spectral monodromy by (4.47), (4.46) and the result (4.52), we can state that the spectral monodromy of non-selfadjoint small perturbed operators allows to recover the classical monodromy of the underlying completely integrable system of selfadjoint unperturbed operators. More precisely we have. This result is seminar to an one of paper [START_REF] Phan | Spectral monodromy of non selfadjoint operators[END_REF], operators in both cases are related to completely integrable systems.

In conclusion, the spectral monodromy of small non-selfadjoint perturbations of a selfadjoint operator, admitting a principal symbol that is completely integrable, is well defined directly from its spectrum, independent of small perturbations and the classical parameter.

Moreover, the spectral monodromy allows to recover the classical monodromy of integrable systems. It shows the important relationship between quantum and classical mechanics.

  and ξ ∈ A, and here ϕ = ϕ c : A → ϕ(A) = U c is a local diffeomorphism. We call (x, ξ) local angle-action variables near Λ c and (V, κ) a local angle-action chart.

  .46) Then we define the (linear) monodromy of the asymptotic pseudo-lattice (Σ(ε, h), U (ε)) as the 1-cocycle of {M ij }, modulo coboundary, in the Čech cohomology of U with values in the integer linear group GL(2, Z), denoted by [M] ∈ Ȟ1 (U, GL(2, Z)).

Theorem 4 . 7 .

 47 The spectral monodromy of P ε is the adjoint of the classical monodromy defined by p.

FITA, Vietnam National University of Agriculture, Hanoi, Vietnam & Vietnam Institute for Advanced Study in Mathematics, Hanoi, Vietnam // E-mail: pqsang@vnua.edu.vn

Acknowledgement

This work was completed during my visit to the Vietnam Institute for Advanced Study in Mathematics (VIASM). I would like to thank the Institute for its support and hospitality. I would like also to thank the Vietnam National University of Agriculture, who gave me a good opportunity to develop my research.


Jamal Bouoiyour
email: jamal.bouoiyour@univ-pau.fr

Refk Selmi

Political elections and uncertainty -Are BRICS markets equally exposed to Trump's agenda?

Keywords: US presidential election, Trump's agenda, stock markets, BRICS, event study, social media, search queries, public opinion polls

There certainly is little or no doubt that politicians, sometimes consciously and sometimes not, exert a significant impact on stock markets. The evolving volatility over the Republican Donald Trump's surprise victory in the US presidential election is a perfect example when politicians, through announced policies, send signals to financial markets. The present paper seeks to address whether BRICS (Brazil, Russia, India, China and South Africa) stock markets equally vulnerable to Trump's plans. For this purpose, two methods were adopted.

The first presents an event-study methodology based on regression estimation of abnormal returns. The second is based on vote intentions by integrating data from social media (Twitter), search queries (Google Trends) and public opinion polls.

Our results robustly reveal that although some markets emerged losers, others took the opposite route. China took the biggest hit with Brazil, while the damage was much more limited for India and South Africa. These adverse responses can be explained by the Trump's neo-mercantilist attitude revolving around tearing up trade deals, instituting tariffs, and labeling China a "currency manipulator". However, Russia looks to be benefiting due to Trump's sympathetic attitude towards Vladimir Putin and expectations about the scaling down of sanctions imposed on Russia over its role in the conflict in Ukraine.

Introduction

Major stock indices around the world witnessed huge negative fluctuations during the initials hours after Donald Trump won the US presidential election. The MSCI Emerging Markets index collapsed markedly since 9 November 2016.

Among emerging markets, the BRICS (Brazil, Russia, India, China and South Africa) shares fell significantly. In a weekly report published by the Brazil's Central Bank on Monday 14 November 2016, the GDP retraction for 2016 has been revised from 3.31 percent to 3.37 percent. In fact, the São Paulo's stock market index fluctuated extremely since the announcement of the US election outcome.

After decreasing by about 0.98 percent early in Wednesday, 09 November morning, the index bounced back by 0.11 percent in the afternoon. Since the event day, the MSCI India stock index fell by 7.1 percent compared to 4.9 percent drop in emerging markets more broadly; also, the rupee lost more than 3 percent against the dollar. Likewise, China's stock index dropped significantly (i.e., the Shangai index plunged as much as 3.6 percent on 9 November by mid-afternoon) on fears that President-elect Trump's protectionist proclivities will harm their trade and then exacerbate the current Chinese economic slowdown. For Trump, China's manufacturing hub and low-cost production have threatened the US economy.

Besides, South Africa's share index tumbled as much as 4 percent on Wednesday, but it rebounded very slightly in the end of the day given the surge in gold miners as traders and investors search generally for safe havens under uncertain period. In general, emerging markets struggled as the rally in the dollar following the Trump's triumph dampened demand for emerging market assets. If there is one country viewed gaining from political risk from Washington, it's Russia. Unlike the rest of BRICS equities, the Market Vectors Russia (RSX) exchange traded fund rose by 4.5 percent, beating the S&P 500 and the MSCI Emerging Markets Index.

Even though many analysts and one recent research (Bouoiyour and Selmi 2016 b) are talking about how US stock markets might react to Trump's win, there are a number of other countries that saw their markets respond significantly after the polls close. The Trump's America First protectionist plans may hurt heavily the emerging markets including the BRICS. Indeed, Trump claimed the abandon of the tariff-cutting Trans Pacific Partnership trade agreement between the U.S. and particular emerging nations. Although it is still unclear whether or not Trump's promises will translate into actual economic and political policies, market participants appear to be concerned by ongoing volatility because of the Trump's protectionist rhetoric. All BRICS leaders aim, undoubtedly, to promote economic growth and curtail foreign capital flight while controlling for political turmoil and overcoming the harmful protectionism consequences. The International Monetary Fund (IMF2016) anticipated that a rise in global protectionism could decrease the global GDP by more than 1.5 percent over the next years. But it might be relevant for market participants to differentiate between the countries best able to weather the storm, and those unable to avoid the adverse effects of uncertainty surrounding the Trump's economic agenda.

Given these considerations, the present study point out the prominence of answering some critical questions. What Trump's election victory means for BRICS shares? Do BRICS move from markets' strength to vulnerability? Are BRICS stock markets equally exposed to Trump's plans? To address these questions, we use the standard market model event study methodology as originally described by [START_REF] Dodd | On corporate governance: a study of proxy contests[END_REF] and [START_REF] Brown | Using daily stock returns: the case of event studies[END_REF]. The event study aims at investigating the average stock market response to a specific stock market event (here the announcement of Trump's victory on 08 November 2016).

Beyond the analysis of the effect of the day relative to the announcement of Trump winning on the abnormal returns, this study also assesses how respond the BRICS stock returns to vote intentions based social media (Twitter), search queries (Google Trends) and polling data as indicators of public interest-levels in the US presidential election.

In doing so, we unambiguously document that the BRICS stock markets are heterogeneously exposed to Trump's stunning triumph. While some markets emerged losers (China, Brazil, India and South Africa, in this order), others appeared winners (Russia). The victory of Donald Trump is viewed as detrimental for BRICS markets especially because of Trump's protectionist rhetoric. However, a potential factor which makes investors more bullish toward Russian shares is the possible easing of the western sanctions regime against Russian companies.

The outline of the paper is as follows: Section 2 includes a brief discussion of the theory on the effect of political uncertainty on financial markets. Section 3 outlines the methodology adopted and describes the data. Section 4 reports and discusses the main findings. Section 5 concludes.

Political elections, uncertainty and financial markets: Some theoretical considerations

The interest in examining the relationship between stock markets and political uncertainty is among researchers for a long time. In general, the political risk is associated with heavier stock return volatility. Normally, the stock markets have to make important choices based on the expected future economic policy decisions of the new government and the resulting policy circumstances [START_REF] Brogaard | The asset-pricing implications of government economic policy uncertainty[END_REF][START_REF] Schiereck | Brexit: (Not) another Lehman moment for banks?[END_REF]Bouoiyour and Selmi 2016 a). Such policy changes put downward pressure on stock prices, particularly if the uncertainty is extreme [START_REF] Pastor | Uncertainty about government policy and stock prices[END_REF]. Once the political uncertainty become less pronounced, stock prices would bound back [START_REF] Pantzalis | Political elections and the resolution of uncertainty: the international evidence[END_REF]. But some events may have persistent effects. For instance, in the case of Brexit, the uncertainty is likely to still higher until it becomes clearer what the future relations among the UK and the European Union will be, continuing to exert a harmful influence on stock prices (Bouoiyour and Selmi 2016 b;[START_REF] Schiereck | Brexit: (Not) another Lehman moment for banks?[END_REF].

Even though political uncertainty takes various shapes and forms including changes in the government and changes in the domestic and foreign policies, the present research focuses on one kind of political uncertainty, which is associated with elections. The latter constitute a major event for re-distribution of political power, which may have meaningful implications for the future political and economic prospects of a country. There is considerable debate regarding the impact of elections on asset price variation [START_REF] Kim | What makes the stock market jump?An analysis of political risk on Hong Kong market returns[END_REF][START_REF] Akmedov | Opportunistic Political Cycles: Test in the Young Democracy Setting[END_REF][START_REF] Canes-Wrone | Elections, Uncertainty, and Irreversible Investment[END_REF]Bouoiyour and Selmi 2016 a).

Nevertheless, there is a large consensus that political uncertainty makes financial markets extremely volatile, particularly after close elections or in response to election results that may lead to radical policy changes (Canes-Wrone and Jee-Kwang 2012). There are at least three reasons that election may exacerbate the financial market volatility. Firstly, a potent political uncertainty surrounding the election outcome may intensify the asymmetries between informed and uninformed market participants. Secondly, the deeper uncertainty over the US presidential election may amplify the ambiguity across market participants about economic fundamentals influencing the share values. The Trump's storming victory has ramped up uncertainty over the policies he will pursue. Several analysts proclaimed that the only certainty about US President-elect Donald Trump's incoming administration is the uncertainty that will attend it. This is seemingly true with regard to a main sensitive policy area. If Trump administration cut taxes and undertake a massive infrastructure program, America's budget deficits will increase substantially. This accompanied with the Federal Reserve's gradual interest-rate hikes will appreciate the dollar, and deteriorate the so-called emerging-market currencies, and shift money from the rest of the world to the US. This is viewed as a very anxious -if not terrifying-prospect. Thirdly, the political uncertainty in election may disrupt the normal functioning of financial markets since the Trump's proclamations on different topics (the withdrawal from NAFTA, the renegotiation free-trade agreements resulting more isolated and less open US markets) remain conditional to the overall congress opinion and the legal challenges from private firms which may play a pivotal role in deterring Trump's administration from implementing these measures (Bouoiyour and Selmi 2016 a). All of these considerations may be of utmost relevance for "politically sensitive" industries, i.e., the companies whose economic fortunes are more likely to be significantly influenced by political continuity or discontinuity.

The literature has put much effort in refining measures of uncertainty [START_REF] Bloom | The Impact of Uncertainty Shocks[END_REF][START_REF] Bloom | Really Uncertain Business Cycles[END_REF][START_REF] Cesa-Bianchi | Uncertainty and economic activity: A global perspective[END_REF]Jurado et al. 2015, etc…). In general, uncertainty is defined as the conditional volatility of a disturbance that is unforecastable. A challenge in empirically analyzing the uncertainty and its dependence to other macroeconomic and financial phenomena is that no objective measure of uncertainty exists. Throughout the rest of our study, we analyze the uncertainty over US political elections via two dimensions:

(a) the way in which the 2016 US presidential election was communicated by media and social networking and the public opinion polls; and (b) the time leading up to an election or the time of government transition after the election by using a dummy variable for the day relative to the announcement of election result.

Methodology and data

To quantify the effects of Trump's victory on BRICS stock markets, we conduct two methodological steps. On the one hand, we analyze the impact of the 2016 US election event (a dummy variable that takes a value 1 on 08 November 2016 and 0 otherwise) on the BRICS abnormal returns. On the other hand, we assess the impact of the vote intentions on the BRICS stock returns. Precisely, we offer a new approach to identify the peoples' opinions about Trump's win by using data from social media, search queries and public opinion polls.

The event study methodology

The event study methodology, first proposed byWarner (1983) and [START_REF] Brown | Using daily stock returns: the case of event studies[END_REF], is designed to examine the impact of a specific event on a dependant variable. A commonly used dependent variable in event studies is the stock price. Accurately, an event study is an analysis of the changes in stock price beyond expectations (Abnormal returns) during a precise period of time (event window), such as the abnormal returns are attributed to the onset of such event.

Overall, the purpose is to test if there is an abnormal stock price effect associated with an event. We define the day "0" as the day of the event for a given stocks.

Thereafter, the estimation and event windows can be determined (see Figure 1).

The interval [T1+1, T2] is the event window with length L2=T2-T1-1, whereas the interval [T0+1, T1] is the estimation window with length L1=T1-T0-1. The length of the event window often depends on the ability to accurately date the announcement date. If one is able to date it precisely, the event window will be less lengthy and capturing the abnormal returns will be more adequate.

Figure 1. Event study windows

For our case of study, we use for each BRICS equity a maximum of 120 daily stock returns observations for the period around the ultimate election result, beginning at day -110 and ending at day + 5 relative to the event. The first 110 days (-115 through -5) is denoted as "the estimation period", and the following 11 days (-5 through + 5) is designated as "the event period". The cumulative abnormal return (CAR) for a sector i during the event window [ τ 1 ; τ 2 ] surrounding the event day t = 0, where [

τ 1 ;τ 2 ] = ∈ [ -5 ;+5] , is expressed as follows: ) ( , , ] , [ , 2 1 2 1 t M i t i t i i R R CAR            (1) where ] , [ , 2 1   i CAR
is the cumulative abnormal return of share i during the event window [τ 1; τ 2 ], R i, t is the realized return of stock i on day t 4 , R M, t is the return of the benchmark index of stocks i, i  ˆ and i  ˆare the regression estimates from an ordinary least squares (OLS) regression for 110 trading day estimation period until t = -5. We utilize the MSCI emerging stock market return as the benchmark index.

We set our event day for the Trump's win event to 8 November 2016.

Then, an OLS regression of the observed cumulative abnormal return for each BRICS shares on the announcement day of the Trump winning is estimated.

For this purpose, we use daily data for the stock market indices of Brazil's Ibovespa, China's Shanghai index, Russia's RTS index, the India's NSE and South Africa's FTSE/JSE.

The equation to estimate is denoted as:

i i Event CAR         1 0 ] , [ , 2 1 (2) where ] , [ , 2 1   i CAR
is the cumulative abnormal returns (the dependent variable), Event is a dummy variable which takes the value of 1 on the day of the US election outcome and 0 otherwise, and i  is the error term.

Another objective of this research is to see whether the event-study findings are sensitive to the inclusion of potential control variables. Generally, major global financial and economic factors could be channels through which fluctuations in the world's economic and financial conditions are transmitted to BRICS stock markets.

These factors include the West Texas Intermediate (WTI) oil price, the world gold price (Gold) and the silver price (Silver). The WTI has been widely used in the literature as the benchmark price for global oil markets. The WTI crude oil is among the most traded oil on the world markets, and therefore is significantly affected by macro-financial variables. Due to their surges under uncertain circumstances, the precious metals (gold and silver) have been perceived as a hedge against sudden shocks and also a safe haven over extreme stock market fluctuations [START_REF] Baur | Is Gold a Hedge or a Safe Haven? An Analysis of Stocks, Bonds and Gold[END_REF]Hood and Mallik 2013). According to [START_REF] Baur | Is gold a safe haven? International evidence[END_REF], we characterize safe havens by their negative and significant correlations with asset markets during financial turmoil or troubled times. In addition, the Bitcoin 5 's considerable climb alongside the announcement of Trump's victory has led to affirm its validity as a safe haven investment. As a reaction to the uncertainty surrounding the US election result, the asset markets around the world plunged as investors were concerned about ongoing volatility. This has yielded to a trend towards questioning the effectiveness of standard economic and financial structures which govern the conventional monetary and financial system. Here, the digital currency is leading the charge by providing a completely decentralized secure alternative to fiat currencies during times of economic and geopolitical unrest. The WTI, Gold and Silver prices data are sourced from DataStream of Thomson Reuters, while the Bitcoin price data in US dollars are collected from CoinDesk at www.coindesk.com/price. The variables under study were transformed by taking natural logarithms to correct for heteroskedasticity and dimensional differences.

The function to estimate is expressed as follows:

  i t t t i Bitcoin Silver Gold WTI Event CAR                 5 4 3 2 1 0 , , 2 1 (3) where ] , [ , 2 1   i CAR
is the cumulative abnormal returns and i  is the error term.

A regression-based intention votes

Unlike the event-study methodology based on regression estimation of abnormal returns that helps to answer whether BRICS equities uniformly respond to the announcement of Trump winning, in this section, we introduce the concept of internet concern as quantitative measure to test if extracting public moods related to 5 Bitcoin was created in 2009 by an anonymous programmer under the pseudonym Satoshi Nakamoto and has since achieved a widest level of international recognition. Unlike the fiat currencies, Bitcoins are digital coins which are decentralized, not issued by any government or legal entity and not redeemable for gold or any other commodity. Bitcoins rely on cryptographic protocols and a distributed network of users to mint, store, and transfer. Instead, investors perform their business transactions themselves without any intermediary. The peer-to-peer network eliminates the trade barriers and makes business easier (Bouoiyour et al. 2016).

US election exerts a significant influence on BRICS stock markets. Millions of people daily interact with search engines, creating valuable sources of data regarding the 2016 US election. In brief, the Internet search seems a potential tie allowing analyzing the public opinions towards the election.

Recent literature evaluated how online information predicts Grexit (Mitchell

      1 0 (4) i t t Twitter STR       1 0 (5) i t t polls STR       1 0 (6)
where i  , i  and i  are the error terms.

To avoid possible methodological bias regarding omitted variable, a vector of additional explanatory variables (discussed above) is incorporated in the models (4), ( 5) and ( 6). Precisely, we estimate the following equations: By accounting for WTI, Gold, Siver and Bitcoin (Conditional analysis, Table 2), some changes with respect the strength of the Trump's victory effect (the Event's coefficient become stronger by moving from the unconditional (Table 1) to the conditional analysis (Table2); this holds true over [0; 0] event day and [+1; +5] event window) were noticed. However, we usually find that the announcement of the Trump triumph in 2016 US election has varying effects across BRICS area.

This event divides the BRICS equities into losers (China, Brazil, India, South Africa, in this order) and winners (Russia).The WTI is likely to differently affect BRICS abnormal share returns depending to whether the country is oil importer or oil exporter; while it exerted a positive effect on Russian market (exporter), its effect on the rest of BRICS (importers) stock returns seems negative. The gold and silver prices have negative influence on the abnormal cumulative returns for all the countries under study. Thus, these metals had not lost their usefulness as a safe haven to protect against deal with uncertainty over Trump's presidential win. The negative influence of Bitcoin on BRICS share returns implies that the investors in the considered countries turn to the digital currency as a refuge from weaker fiat currencies.

Regression results-based on the intention votes

Considering the intention votes through social media, search queries and public opinion polls as indicators of markets' beliefs regarding the US election (Table 3), we show that the results are still robust. In particular, Google Trends statistically and negatively affect Brazilian, Indian, Chinese and South African shares, while they exert a positive impact on the Russian stocks. Similar results are found when using Twitter hashtags and polling data (with the exception of Brazil). The results of the effect of intention votes on the stock returns while considering the control variables are reported in Table 4. Whatever the public opinions proxies used (Google Trends, Twitter or polls), we often show that the only winner from the US election outcome. The additional explanatory variables still exert similar effects. WTI impact positively the oil exporting country (Russia), while its effect on the oil importing countries seems negative which is yet highly expected. Gold and Silver affect negatively the BRICS stock returns, highlighting their viability to serve as a safe haven in this period of upheaval. Bitcoin has been

shown to be negatively correlated with stock returns, pointing toward its safe haven and hedging capabilities.

Remarkably, the use of polls seems less appropriate than social media and search queries since more significant results are found for the second cases. In particular, we show that market sentiment reflected in search queries and individual text messages matters for assessing the responses of BRICS stock markets to US election event. In light of the ubiquity of social media data and the ability to deal with a large data volume, the use of this kind of data appears a quite interesting field for future studies on the effects of economic and political events. On the contrary, some polls' coefficients seem insignificant (Brazil for unconditional analysis and South Africa for conditional investigation). The polls usually report only the results while leaving out the "don't knows", and directly transform answers into opinions. Moreover, the pollsters report the beliefs of a random sample of the entire population, and thus it is not the best representative of the full public opinion. This may explain why the 2016 US election prediction were flawed. In fact, the majority of projections gave Hillary Clinton more chance of winning the US presidential election (see Appendix A). In this context, Sociologist Herbert Gans asserted that "polls are answers to questions rather than opinions". To be more effective, the pollsters should pose accurate questions, telling the politicians and the public how exactly respondents feel about such event, and if they have been politically active in behalf of these feelings. percent tariffs on some products imported could be counterproductive (risk of exacerbation of currencies competition, strong appreciation of the dollar, inflation pressures, etc.). The hope for these countries is that the rise in US interest rates will be gradual. But the inflationary agenda of the new US president may force the US Central Bank to accelerate the move. Furthermore, the United States is one of South African biggest export destination and to achieve a hike in import costs will threaten South Africa's economy. However, the uncertainty arising from a Trump victory is good for the gold price, as investors turn to this yellow metal in period of upheaval. As one of the world's dominant gold producers, South Africa will benefit from the confidence in gold as a hedge or safe haven. Brazil -as commoditydependent country-seems also poised to emerge from recession due to the surge of oil and commodity prices.

The Trump's win has also caught India's stock market off-guard. It is expected that high import tariffs would affect adversely its economy, especially with the resulted extreme volatility of its currency against dollar. But what works in India's favor is that it has relatively low external financing needs and is not largely dependent on exports, in addition to its macro-economic parameters which seem relatively stronger. This makes it insulated from the untoward shocks that may harm flows into the rest of BRICS markets.

Exceptionally, the Russian shares benefited remarkably from the announcement of Trump's victory. The positive market reaction may partly reflect hopes for sanctions against Russian companies over Crimean crisis to be eased or lifted. With Trump in the White House, Russian investors are betting that the iced US-Russia relationship may start melting due to the president-elect's affinity towards president Vladimir Putin. Besides, Trump has been keen to stimulate US commodity production, such as oil, gas and coal, so some can anticipate that the US presidential election will constrain a rise in commodity prices, benefiting the biggest energy producers like Russia.

Conclusions

Since the Trump's win in the US presidential election, analysts all over the world start asking questions on how and to what extent the uncertainty surrounding this unexpected outcome will affect the world markets, and which markets will suffer and benefit under a Trump's administration. This paper aims at offering some answers to these questions by delving into the BRICS stock markets.

  et al. 2012; Bouoiyour and Selmi 2015, among others) and the economic and financial costs of Brexit(Bouoiyour and Selmi 2016 b). We attempt, in the following, to demonstrate that social media discussion and search related queries for the 2016 US election help us to track the evolution of markets' beliefs about US presidential election outcome. Twitter is becoming very popular among financial professionals. It permits them to comment on economic and political events and to distribute their view to either their followers or even a wider audience in an extremely speedy way. Many people use their Twitter accounts to express and disseminate their opinions on the US election. The advantage of using Twitter data for research purposes is that (1) users not only receive information but can actively share information, (2) tweets can be used to extract not only a consensus view on such event, but also the degree of agreement or disagreement.A further task of this study is to use public opinion polls to measure the intention votes toward Trump. The pollsters' reports and press releases often start with asking a specific question and then present tables with the statistical proportions of poll respondents giving all the answers. For the case of US presidential election, the question was: "If the general election were held today, and the candidates were Hillary Clinton for the Democrats and Donald Trump for the Republicans, for whom would you vote? If not sure, or would not vote, ask: Toward which do you lean?" The polls report the results used here to explain the variation of BRICS stock returns.In brief, OLS regressions of the stock market return (STR) for each BRICS country on three intention votes' indicators (Google Trends, Twitter searches and polling data transformed in log) are estimated. STR is calculated by considering the

  is a scenario the Obama administration has been watchful to circumvent. Chinese leaders and the propaganda machines they control have yet begun using Donald Trump's to pressurize a nationalist agenda. Last but not least, Brazil, India and South Africa should also carefully assess what new geopolitical risks may emerge with the more confrontational Trump foreign policy towards countries like China or Russia, with which these countries have strong economic commitments.

  ratio stock price (in log) at time t and the lagged stock price, i.e.,

			t STR		log(	1  t P t P	)
	where t P is the stock price.
	t STR	t ds GoogleTren	i

Table 1 . Trump's impacts on BRICS abnormal returns: Unconditional OLS regression results

 1

	t STR	t ds GoogleTren	WTI	t	t Gold	t Silver	t Bitcoin	i

Real Clear Politics at http://www.realclearpolitics.com/epolls/latest_polls/.

includes Brazil, India, China and South Africa where a sharp drop of stock values is found during the election day and over [+1; +5] event window. The second group is formed by Russia where we note a marked increase in the abnormal stock returns over [0; +5] event window).

The results of the stock event study without considering potential control variables (i.e., unconditional analysis) are displayed in Table

1

. We find that the announcement of Trump's win (the event day [0; 0]) resulted in statistically significant negative CARs for all the BRICS (except Russia where we note a positive response), being somewhat stronger for China and Brazil than for India and South Africa. The BRICS-market reactions do not change in terms of sign during the [+ 1; +5] event window, but the strength of the responses appear more pronounced during the post-election period. The Russian share market, by contrast, gained markedly from this unexpected election outcome either for [0; 0] event day or over [+1; +5] event widow.

Notes: All regressions are controlled for heteroskedasticity and the p-values are given in parentheses.

* , * * , * * * denote statistical significance at the 10%, 5% and 1% levels, respectively.

Table 2 . Trump's impacts on BRICS abnormal returns: Conditional OLS regression results

 2

		Brazil	Russia	India	China	South Africa
			Event day [0 ; 0]
	Constant	1.32445***	-0.026138	-0.018209	0.015787	0.050083
		(0.0009)	(0.1171)	(0.2281)	(0.1891)	(0.1549)
	Event	-0.133970*	0.121378**	-0.07356*	-0.193872**	-0.044113*
		(0.0620)	(0.0043)	(0.0339)	(0.0029)	(0.0546)
	WTI	-0.031881*	0.10128***	-0.01578**	-0.068994*	-0.049743**
		(0.0202)	(0.0003)	(0.0083)	(0.0304)	(0.0056)
	GOLD	-0.023951*	-0.013544**	-0.074435*	-0.062891*	-0.04439*
		(0.0256)	(0.0073)	(0.0486)	(0.0380)	(0.0967)
	Silver		-0.063511*	-0.064773*	-0.074992*	-0.062508**
		(0.0035)	(0.0405)	(0.0968)	(0.0924)	(0.0043)
	Bitcoin	-0.13417**	-0.1146*	-0.128721*	-0.19142*	-0.106724**
		(0.0015)	(0.0456)	(0.0462)	(0.0215)	(0.0095)
	Adjusted R 2	0.89	0.88	0.93	0.92	0.91
			Event window [+1; +5]
	Constant Event WTI GOLD Silver Bitcoin	1.668467* (0.0077) -0.169456* (0.0391) -0.059222** (0.0067) -0.059454* (0.0279) -0.03145* (0.0139) -0.119422 (0.3617)	1.581424* (0.0218) 0.141723** (0.0020) 0.100776* (0.0638) -0.075213* (0.0955) -0.236306* (0.0140) -0.098422** (0.0014)	1.26723** (0.0015) -0.0687*** (0.0007) -0.08012** (0.0023) -0.0684*** (0.0001) -0.064791* (0.0577) -0.143359* (0.0140)	1.14096* (0.0456) -0.18282* (0.0367) -0.037125* (0.0282) -0.110881* (0.0782) -0.05489** (0.0096) -0.14763** (0.0064)	1.32945 * (0.0871) -0.069619* (0.0707) -0.02473** (0.0043) -0.12243** (0.0079) -0.020562* (0.0351) -0.066735* (0.0875)
	Adjusted R 2	0.91	0.94	0.90	0.91	0.89
	Notes: All regressions are controlled for heteroskedasticity and the p-values are given in
	parentheses. * , * * , * * * denote statistical significance at the 10%, 5% and 1% levels, respectively.

Table 3 . The impacts of the intention votes on BRICS stock returns: Unconditional OLS regression results

 3

		Brazil	Russia	India	China	South Africa
			STR and Google Trends
	Constant	0.763241**	0.662156**	0.853594*	0.271307	0.00345***
		(0.0065)	(0.0059)	(0.0739)	(0.1680)	(0.0007)
	Google Trends	-0.13456	0.176446**	-0.108786*	-0.180459	-0.01234*
		(0.2451)	(0.0052)	(0.0400)	(0.2558)	(0.0156)
	Adjusted R 2	0.83	0.81	0.82	0.85	0.83
			STR and Twitter
	Constant	1.116414*	1.347377*	1.19710*	1.565629**	1.491338*
		(0.0425)	(0.0905)	(0.0819)	(0.0096)	(0.0315)
	Twitter	-0.168191*	0.153365**	-0.077745*	-0.14438***	-0.085861*
		(0.0556)	(0.0091)	(0.0806)	(0.0001)	(0.0527)
	Adjusted R 2	0.89	0.86	0.85	0.79	0.85
			STR and polls
	Constant	0.141563*	0.175537**	0.110998	0.033970	0.021178
		(0.0749)	(0.0091)	(0.8754)	(0.1620)	(0.2743)
	polls	0.119329	0.127439*	-0.07988**	-0.16188*	-0.09128***
		(0.2670)	(0.0425)	(0.0082)	(0.0202)	(0.003)
	Adjusted R 2	0.85	0.82	0.84	0.88	0.86
	Notes: All regressions are controlled for heteroskedasticity and the p-values are given in
	parentheses. * , * * , * * * denote statistical significance at the 10%, 5% and 1% levels, respectively.

Table 4 . The impacts of the intention votes on BRICS stock returns: Conditional OLS regression results

 4 All regressions are controlled for heteroskedasticity and the p-values are given in parentheses. * , * * , * * * denote statistical significance at the 10%, 5% and 1% levels, respectively. How can we explain these heterogeneous reactions of BRICS markets? China's stock market seems the most damaged by the victory of Donald Trump. The nervousness was fueled by Trump's provocative words on the campaign trail about how China is a currency manipulator, coupled with its fierce protectionist stance much of it directed toward China. His protectionist approach could undoubtedly harm the capital and trade flows between the United States and China. It must be stressed that the United States is the largest market for Chinese exports, accounting for approximately 20 percent of the global exports. In this way, imposing a 45 percent tariff on Chinese imports into the US as Trump proclaimed in his campaign would constitute a serious risk for Chinese economy. This aggressive US trade policy could result in a substantial China's growth slowmoving coupled with a loss of manufacturing jobs. The fact that Trump's economic agenda seeks to slash China's huge trade surplus with the US would damage shares involved in Chinese exports.Some emerging countries often indebted in greenback (including Brazil and South Africa) are heavily dependent on foreign capital. The strength of the dollar and the rise in the interest rate on the bond market are likely to prompt massive capital outflows to the United States. To this we must add that the developed countries tend to become more protectionists. The uncertainty is greater as no one knows whether the US elect-president will transform his protectionist promises into action. In any case, Trump's anti-trade rhetoric aimed at imposing a 35 and 45

		Brazil	Russia	India	China	South Africa
			STR and Google Trends
	Constant Google Trends WTI GOLD Silver	1.19873* (0.0200) -0.163564* (0.0621) -0.168227* (0.0599) -0.092015** (0.0091) -0.04321***	1.166422* (0.0111) 0.135711** (0.0058) 0.101875** (0.0086) -0.083335* (0.0116) -0.03214**	1.531872 (0.2447) -0.10499* (0.0330) -0.050940* (0.0465) -0.1162*** (0.0008) -0.054678*	1.133039* (0.0309) -0.158649** (0.0025) -0.102084** (0.0014) -0.142460* (0.0497) -0.072341**	1.896641** (0.0030) -0.098390* (0.0835) -0.054893* (0.0216) -0.199722* (0.0343) -0.034521**
	Bitcoin	(0.0009) -0.10543*	(0.0054) -0.09653**	(-0.0311) -0.132452*	(0.0064) -0.142456*	(0.0055) -0.097632*
		(-0.0674)	(0.0081)	(-0.0510)	(0.0431)	(0.0389)
	Adjusted R 2	0.88	0.90	0.87	0.86	0.89
			STR and Twitter
	Constant Twitter WTI GOLD	1.25881** (0.0097) -0.160209* (0.0616) -0.150977* (0.0142) -0.128905*	1.49428* (0.0187) 0.121423* (0.0138) 0.101423* (0.0356) -0.063101**	1.53943** (0.0081) -0.086845* (0.0527) -0.027995* (0.6996) -0.11304**	1.626058** (0.0017) -0.174548* (0.0019) -0.079679** (0.0011) -0.101694*	1.702818* (0.0185) -0.09235* (0.0886) -0.05778*** (0.0009) -0.181309*
	Silver	(0.0474) -0.06432**	(0.0079) -0.057234*	(0.0012) -0.0467***	(0.0428) -0.07625**	(0.0556) -0.04693**
	Bitcoin	(0.0038) -0.089972*	(0.0679) -0.069432	(0.0000) -0.11789**	(0.0049) -0.14698***	(0.0062) -0.08721***
		(0.0164)	(0.1520)	(0.0013)	(0.0004)	(0.0009)
	Adjusted R 2	0.87	0.84	0.88	0.90	0.92
			STR and polls

Notes:

outcome. Trump's win divided the BRICS into highly damaged (China and Brazil), moderately threatened (India and South Africa) and benefiting (Russia) markets.

Daily stock returns are calculated as the first natural logarithmic difference of the stock price.

Using an event-study methodology and the voter intentions in US electionsbased social media, search queries and public opinion polls, we robustly find that the BRICS equities are not equally exposed to the Trump's victory. Two main groups are derived from our regression analyses; although some markets emerged losers (China, Brazil, India and South Africa, in this order), others unfolded winners (Russia).

In general, the worst-performing markets are those which (a) borrowed dollars expecting the greenback to depreciate over time; and (b) will suffer more intensely from Donald Trump's neo-mercantilist attitude and protectionist rhetoric, or more precisely its promises revolving around slapping 45 percent duty on Chinese imports into the US to make it easier for US companies to compete, stirring fears of a currency war with China and heavily punishing all companies that have sent US works overseas. Add to this the Trump's inflammatory words on the campaign trail into several issues (especially by dubbing China as a "currency manipulator") compounded investors' uncertainty across emerging markets.

The Russian market, by contrast, benefited from the unexpected US election outcome due to the Republican president's warm tone towards Putin over the campaign and the Trump's suggestions to meaningfully improve the US-Russia relationship. Regardless of the favorable reading of the US election outcome for Russian case, the country remains facing huge challenges blighting its economy such as the lack of diversification (in particular, the great dependency to volatile and speculative commodities). Under such circumstances, the ascendancy of Donald Trump to the White House as of 20 January 2017 will not be a magic bullet for the raft of Russia's serious economic problems.

Whether the US president-elect makes good on those threats, and whether his extreme rhetoric turns into actual policies, Trump's promises have varying economic and geopolitical implications. For instance, the Trump's negative stance towards China might be used politically by Chinese leaders to stoke nationalism