| import OAI.Dynamics.StandardMap.BridgeWeights |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| 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] |
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| 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 |
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| end OAI |
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