File size: 2,794 Bytes
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 | import OAI.Dynamics.EntropyCounterexample.AngularDerivative
namespace OAI
noncomputable section
open Filter Topology
open scoped OnePoint
namespace Problem340
theorem eventually_zero_comp_inv {f : ℂ → ℂ} (hf : HasCompactSupport f)
(hzero : f 0 = 0) :
(fun z : ℂ => f z⁻¹) =ᶠ[𝓝 0] (fun _ => 0) := by
change ∀ᶠ z : ℂ in 𝓝 0, f z⁻¹ = 0
rw [← nhdsNE_sup_pure (0 : ℂ), eventually_sup]
constructor
· have hinv : Tendsto (Inv.inv : ℂ → ℂ) (𝓝[≠] 0) (coclosedCompact ℂ) := by
simpa only [Filter.coclosedCompact_eq_cocompact,
← Metric.cobounded_eq_cocompact] using
(tendsto_inv₀_nhdsNE_zero (α := ℂ))
exact hinv.eventually (hasCompactSupport_iff_eventuallyEq.mp hf)
· simpa using hzero
theorem contDiff_comp_inv_of_compactSupport {f : ℂ → ℂ} {n : WithTop ℕ∞}
(hf : ContDiff ℝ n f) (hcomp : HasCompactSupport f) (hzero : f 0 = 0) :
ContDiff ℝ n (fun z : ℂ => f z⁻¹) := by
rw [contDiff_iff_contDiffAt]
intro z
by_cases hz : z = 0
· subst z
exact (contDiffAt_const : ContDiffAt ℝ n (fun _ : ℂ => (0 : ℂ)) 0).congr_of_eventuallyEq (eventually_zero_comp_inv hcomp hzero)
· exact hf.contDiffAt.comp z (contDiffAt_id.inv hz)
def planeZeroExtension (f : ℂ → ℂ) (z : OnePoint ℂ) : ℂ := z.elim 0 f
@[simp] theorem planeZeroExtension_infty (f : ℂ → ℂ) :
planeZeroExtension f ∞ = 0 := rfl
@[simp] theorem planeZeroExtension_coe (f : ℂ → ℂ) (z : ℂ) :
planeZeroExtension f (z : OnePoint ℂ) = f z := rfl
theorem continuous_planeZeroExtension {f : ℂ → ℂ} (hf : Continuous f)
(hcomp : HasCompactSupport f) : Continuous (planeZeroExtension f) := by
rw [OnePoint.continuous_iff]
constructor
· change Tendsto f (coclosedCompact ℂ) (𝓝 (0 : ℂ))
exact Tendsto.congr' (hasCompactSupport_iff_eventuallyEq.mp hcomp).symm
tendsto_const_nhds
· exact hf
theorem continuous_cutoff_angular_onePoint {χ : ℂ → ℂ} {n : WithTop ℕ∞}
(hχ : ContDiff ℝ n χ) (hcomp : HasCompactSupport χ)
(hzero : ∀ᶠ z in 𝓝 (0 : ℂ), χ z = 0) (k : ℕ) :
Continuous (planeZeroExtension (fun z => χ z * angularPhase z ^ k)) := by
apply continuous_planeZeroExtension (contDiff_cutoff_angular_power hχ hzero k).continuous
exact hcomp.mul_right
theorem contDiff_cutoff_angular_inv {χ : ℂ → ℂ} {n : WithTop ℕ∞}
(hχ : ContDiff ℝ n χ) (hcomp : HasCompactSupport χ)
(hzero : ∀ᶠ z in 𝓝 (0 : ℂ), χ z = 0) (k : ℕ) :
ContDiff ℝ n (fun z : ℂ => χ z⁻¹ * angularPhase z⁻¹ ^ k) := by
apply contDiff_comp_inv_of_compactSupport (contDiff_cutoff_angular_power hχ hzero k)
hcomp.mul_right
simp [hzero.self_of_nhds]
end Problem340
end
end OAI
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