| import OAI.Dynamics.StandardMap.BridgeKernel |
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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 PlaneAreaPreserving.eligible_row {A : ℂ →L[ℝ] ℂ} (hA : PlaneAreaPreserving A) |
| (s : ℂ) (hu : ‖s‖=1) (hs : |s.re| ≤ 1/4) (hstable : ‖A s‖=‖A‖⁻¹) : |
| ‖A‖/3 ≤ |(A 1).re| ∨ ‖A‖/3 ≤ |(A 1).im| := by |
| have him := vertical_im_lower s hu hs |
| have hD : 0 < ‖A‖ := by obtain ⟨a,ha,hb,hc,hd,he⟩:=hA.singular_pair; linarith |
| have hw := abs_wedge_le_norm_mul (A 1) (A s) |
| rw [hA,hstable] at hw |
| have haxes : wedge 1 s=s.im := by simp [wedge] |
| rw [haxes] at hw |
| have hlow : (3/4)*‖A‖ ≤ ‖A 1‖ := by |
| have hh:=mul_le_mul_of_nonneg_right (him.trans hw) hD.le |
| simpa only [mul_assoc,inv_mul_cancel₀ hD.ne',mul_one] using hh |
| have hup:=Complex.norm_le_abs_re_add_abs_im (A 1) |
| by_contra hh |
| push Not at hh |
| linarith |
| lemma prefix_stable_vertical (v : ℕ → ℝ) (M : ℝ) (B : ℕ) |
| (hB : 1 ≤ B) (hM : 1 < M) (hq : M^(-(49/25:ℝ)) ≤ 1/2) |
| (hsmall : 3*M^(-(47/50:ℝ)) ≤ 1/4) |
| (hv : ∀ i, 1 ≤ i → i ≤ B → |v i|+1 ≤ M) |
| (hg : ∀ i, 1 ≤ i → i ≤ B → M^((49/50:ℝ)*(i:ℝ)) ≤ ‖transferProduct v i‖) : |
| ∃ s : ℂ, ‖s‖=1 ∧ |s.re| ≤ 1/4 ∧ ‖transferProduct v B s‖=‖transferProduct v B‖⁻¹ ∧ |
| ∀ l, 1 ≤ l → l ≤ B → ‖transferProduct v l s‖ ≤ 3*M^(-(47/50:ℝ)*(l:ℝ)) := by |
| obtain ⟨s,hu,hs,hpref⟩:=prefix_stable_vector v M B hB hM hq hv hg |
| refine ⟨s,hu,?_,hs,hpref⟩ |
| have hh:=hpref 1 le_rfl hB |
| have hc : (transferProduct v 1 s).im=s.re := rfl |
| calc |
| _ = |(transferProduct v 1 s).im| := congrArg abs hc.symm |
| _ ≤ ‖transferProduct v 1 s‖ := Complex.abs_im_le_norm _ |
| _ ≤ 3*M^(-(47/50:ℝ)) := by simpa using hh |
| _ ≤ _ := hsmall |
| lemma initial_kernel_dirichlet (v : ℕ → ℝ) (J : ℕ) (u : ℂ) |
| (_hui : u.im=1) (_hJ : 1 ≤ J) (huj : linearSolution v 1 u.re J=0) |
| (ht : tSolution v J ≠ 0) : |
| ∀ p, linearSolution v 1 u.re p=dirichletSolution v J p := by |
| have hb:=boundaryValue_unique v J ht u.re huj |
| intro p |
| rw [hb] |
| rfl |
| end StandardMapEntropy |
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| end OAI |
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