Résonances de Ruelle en quantification géométrique

M0
M
R2

ˆ
PH
T?R2
Amˆ Am
ˆ
PH
~→0
tr(e−iˆ
Ht/~) = Pne−iEnt/~
ρ(E)ˆ H

ρ(E) = Pnδ(E−En)
∆E=~/∆t
t∼1/~α;α > 0

Mo=2 1
1 1 R2T2
M
Mo=2 1
1 1 H1(q, p) = acos(2πq); a= 0.01
M0
x= (q, p)∈R2
ω=dq ∧dp
H0(q, p) = 1
2αq2+1 2βp2+γqp, α, β, γ ∈R.
x(t)dq/dt =γq +βp
dp/dt =−αq −γp x(1) = M0x(0)
Mo:= A B
C D =exp γ β −α−γ∈SL(2,R), i.e. det(M0) = 1
γ2> αβ M0
e±λ0λ0=pγ2+αβ
u0, s0∈R2
A, B, C, D ∈Z
M0∈SL(2,Z)∀x∈R2, n ∈Z2,
M0(x+n) = M0(x) + M0(n)≡M0(x)mod1
M0T2=R2/Z2

M
C∞H1:T2→RH1
R2M1R2H1
M:= M1.M0
M
H(t) = H0pour 0≤t < 1mod2
H1pour 1≤t < 2mod2
M(x+n) = M(x) + M0(n),∀x∈R2, n ∈Z2.
M0(x+n) = M0(x) + M0(n)M1(x+n) = M1(x) + n,
M(x+n) = M1(M0(x) + M0(n)) = M(x) + M0(n), M0(n)∈Z2
M
M0
C∞C1
 > 0,||H1||C2< , M M0:M=
HM0H−1H:T2→T2M
DMx:TxT2→TMxT2x∈T2
x
TxT2=Eu(x)⊕Es(x)
Eu⊕Es∈TT2M
λx
||DMx(ξ)|| ≥ eλx||ξ||,∀ξ∈Eu(x)et ||DMx(η)|| ≤ e−λx||η||,∀η∈Es(x).
M=M0λx=λ0∀x