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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Dynamics.StandardMap.BridgeWeights
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped Topology ENNReal
namespace StandardMapEntropy
lemma tSolution_abs_le_pow (v : ℕ → ℝ) (M : ℝ) (p : ℕ) (hp : 1 ≤ p)
(hM : 1 ≤ M) (hv : ∀ i, 1 ≤ i → i < p → |v i|+1 ≤ M) :
|tSolution v p| ≤ M^((p:ℝ)-1) := by
obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : p ≠ 0)
have hh := (transferProduct_norm_bounds v M n (by linarith)
(fun i hi hin => hv i hi (by omega))).2
have hp := (transferProduct v n).le_opNorm (1:ℂ)
have he : (transferProduct v n 1).re=tSolution v (n+1) := by
change (transferProduct v n ⟨1,0⟩).re=tSolution v (n+1)
rw [transferProduct_pair]; rfl
calc
_ = |(transferProduct v n 1).re| := congrArg abs he.symm
_ ≤ ‖transferProduct v n 1‖ := Complex.abs_re_le_norm _
_ ≤ M^n := by simpa only [norm_one,mul_one] using hp.trans (by simpa using hh)
_ = _ := by simp only [Nat.cast_succ,add_sub_cancel_right,Real.rpow_natCast]
lemma nonlinear_dirichlet_fixedPoint (k q a r : ℝ) (n : ℕ)
(hk : 0 ≤ k) (hr : |r| ≤ 1/4)
(hq : growthBase k^(-(3/5:ℝ)) ≤ 1/2)
(hsmall : (384*Real.pi)*growthBase k^(-(7/10:ℝ)) ≤ 1/2)
(hU : ∀ p, 1 ≤ p → p ≤ n →
|dirichletSolution (orbitCoefficient k q a) (n+1) p| ≤
12*growthBase k^(-(9/10:ℝ)*(p:ℝ))) :
∃ ξ : Fin n → ℝ,
(∀ i, |ξ i| ≤ 8/growthBase k^((4/5:ℝ)*((i:ℝ)+1))) ∧
∀ i, ξ i=dirichletSolution (orbitCoefficient k q a) (n+1) (i+1)*r+
∑ j, greenMatrix (orbitCoefficient k q a) n i j *
(-(phi k (liftedOrbit k q a (j+1)+ξ j)-phi k (liftedOrbit k q a (j+1))-
potential k (liftedOrbit k q a (j+1))*ξ j)) := by
let M := growthBase k
have hMm : 4 ≤ M := growthBase_ge_four k hk
have hm : 1 < M := by linarith
have hp : 0 < M := by linarith
let w : Fin n → ℝ := fun i => M^((4/5:ℝ)*((i:ℝ)+1))
let U : Fin n → ℝ := fun i => dirichletSolution (orbitCoefficient k q a) (n+1) (i+1)
let F : Fin n → ℝ → ℝ := fun j z => -(phi k (liftedOrbit k q a (j+1)+z)-
phi k (liftedOrbit k q a (j+1))-potential k (liftedOrbit k q a (j+1))*z)
let L : Fin n → ℝ := fun i => (2*Real.pi*M)*8/w i
have hw (i : Fin n) : 0 < w i := Real.rpow_pos_of_pos hp _
have hlin (i : Fin n) : |w i*U i*r| ≤ (8:ℝ)/2 := by
have hU':=hU (i+1) (by omega) (by omega)
have hexp : M^((4/5:ℝ)*((i:ℝ)+1))*M^(-(9/10:ℝ)*((i:ℝ)+1)) ≤ 1 := by
rw [← Real.rpow_add hp]
exact Real.rpow_le_one_of_one_le_of_nonpos hm.le (by have : (0:ℝ) ≤ (i:ℕ) := Nat.cast_nonneg _; linarith)
rw [abs_mul,abs_mul,abs_of_pos (hw i)]
calc
_ ≤ w i*(12*M^(-(9/10:ℝ)*((i:ℝ)+1)))*(1/4) := by
gcongr
simpa only [U,Nat.cast_add,Nat.cast_one] using hU'
_ ≤ 3 := by dsimp only [w]; nlinarith
_ ≤ _ := by norm_num
have hrow (i : Fin n) : ∑ j, |w i*greenMatrix (orbitCoefficient k q a) n i j| *L j/w j ≤ 1/2 := by
have hh:=green_weighted_rows M 12 8 n (tSolution (orbitCoefficient k q a))
(dirichletSolution (orbitCoefficient k q a) (n+1)) hm (by norm_num) (by norm_num) hq
(fun p hp' hn => tSolution_abs_le_pow _ M p hp' hm.le
(fun i hi hip => potential_bound k (liftedOrbit k q a i) hk)) hU i
have he : ∀ i j : Fin n, greenMatrix (orbitCoefficient k q a) n i j=
tSolution (orbitCoefficient k q a) (min (i:ℕ) (j:ℕ)+1)*
dirichletSolution (orbitCoefficient k q a) (n+1) (max (i:ℕ) (j:ℕ)+1) := by
intro i j
simp only [greenMatrix,greenKernel,min_add_add_right,max_add_add_right]
simp_rw [he]
exact hh.trans (by nlinarith only [hsmall])
exact finite_weighted_dirichlet w U (greenMatrix (orbitCoefficient k q a) n) F L r 8 hw
(by norm_num) (fun i => by dsimp [L]; positivity)
(by intro i; simp [F])
(by
intro i x y hx hy
dsimp only [F]
rw [← neg_sub,abs_neg]
have hh:=phi_remainder_lipschitz k (liftedOrbit k q a (i+1)) x y (8/w i) hk hx hy
dsimp only [L,M]
convert hh using 1 <;> ring_nf)
hlin hrow
end StandardMapEntropy
end OAI
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