| import OAI.Dynamics.EntropyCounterexample.ClockPeriodizationLocal |
| import OAI.Dynamics.EntropyCounterexample.ClockLocalExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Function |
| open scoped ContDiff |
|
|
| namespace Problem340.ClockQuotient |
| local instance : Fact (0 < (100 : ℝ)) := ⟨by norm_num⟩ |
|
|
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| omit [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] in |
|
|
| theorem liftIco_clockHomeomorph_periodic {f : ℝ × E → F} |
| (hp : ∀ z, Periodic (fun t => f (t, z)) 100) (p : ℝ × E) : |
| AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, p.2)) |
| (clockHomeomorph.symm (clockHomeomorph (p.1 : AddCircle (100 : ℝ)))) = f p := by |
| rw [clockHomeomorph.symm_apply_apply] |
| exact (liftPeriodic_eq_liftIco f hp ((p.1 : AddCircle (100 : ℝ)), p.2)).symm.trans |
| (liftPeriodic_coe f hp p.1 p.2) |
|
|
| theorem hasLocalC1Extension_liftIco_of_periodic {f : ℝ × E → F} |
| (hp : ∀ z, Periodic (fun t => f (t, z)) 100) (hf : ContDiff ℝ 1 f) : |
| HasLocalC1Extension (fun x : StandardClock × E => (x.1.1, x.2)) |
| (fun x => AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, x.2)) |
| (clockHomeomorph.symm x.1)) := by |
| apply hasLocalC1Extension_of_clock_lift_with_parameters |
| simpa only [liftIco_clockHomeomorph_periodic hp] using hf |
|
|
| theorem hasLocalC1Extension_liftIco_of_periodic_on {V : Set E} (hV : IsOpen V) |
| {f : ℝ × E → F} (hp : ∀ z, Periodic (fun t => f (t, z)) 100) |
| (hf : ContDiffOn ℝ 1 f (univ ×ˢ V)) : |
| HasLocalC1Extension (fun x : StandardClock × V => (x.1.1, x.2.1)) |
| (fun x => AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, x.2.1)) |
| (clockHomeomorph.symm x.1)) := by |
| refine hasLocalC1Extension_of_clock_lift_on hV |
| (fun x : StandardClock × E => AddCircle.liftIco (100 : ℝ) 0 |
| (fun t => f (t, x.2)) (clockHomeomorph.symm x.1)) ?_ |
| simpa only [liftIco_clockHomeomorph_periodic hp] using hf |
|
|
| omit [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace ℝ F] in |
|
|
| theorem liftIco_clockHomeomorph_strip {f : ℝ × E → F} |
| (hs : ∀ x, f x ≠ 0 → x.1 ∈ Ioo 0 100) (p : ℝ × E) : |
| AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, p.2)) |
| (clockHomeomorph.symm (clockHomeomorph (p.1 : AddCircle (100 : ℝ)))) = |
| periodize 100 f p := by |
| rw [clockHomeomorph.symm_apply_apply] |
| exact (liftPeriodize_eq_liftIco hs ((p.1 : AddCircle (100 : ℝ)), p.2)).symm.trans |
| (liftPeriodize_coe 100 f p.1 p.2) |
|
|
| theorem hasLocalC1Extension_liftIco_of_strip_support {f : ℝ × E → F} |
| (hf : ContDiff ℝ 1 f) (hs : ∀ x, f x ≠ 0 → x.1 ∈ Ioo 0 100) : |
| HasLocalC1Extension (fun x : StandardClock × E => (x.1.1, x.2)) |
| (fun x => AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, x.2)) |
| (clockHomeomorph.symm x.1)) := by |
| apply hasLocalC1Extension_of_clock_lift_with_parameters |
| have hs' : ∀ x, f x ≠ 0 → x.1 ∈ Icc 0 100 := |
| fun x hx => ⟨(hs x hx).1.le, (hs x hx).2.le⟩ |
| simpa only [liftIco_clockHomeomorph_strip hs] using |
| contDiff_periodize (by norm_num : (0 : ℝ) < 100) hf hs' |
|
|
| theorem hasLocalC1Extension_liftIco_of_strip_support_on {V : Set E} (hV : IsOpen V) |
| {f : ℝ × E → F} (hf : ContDiffOn ℝ 1 f (univ ×ˢ V)) |
| (hs : ∀ x, f x ≠ 0 → x.1 ∈ Ioo 0 100) : |
| HasLocalC1Extension (fun x : StandardClock × V => (x.1.1, x.2.1)) |
| (fun x => AddCircle.liftIco (100 : ℝ) 0 (fun t => f (t, x.2.1)) |
| (clockHomeomorph.symm x.1)) := by |
| refine hasLocalC1Extension_of_clock_lift_on hV |
| (fun x : StandardClock × E => AddCircle.liftIco (100 : ℝ) 0 |
| (fun t => f (t, x.2)) (clockHomeomorph.symm x.1)) ?_ |
| have hs' : ∀ x, f x ≠ 0 → x.1 ∈ Icc 0 100 := |
| fun x hx => ⟨(hs x hx).1.le, (hs x hx).2.le⟩ |
| simpa only [liftIco_clockHomeomorph_strip hs] using |
| contDiffOn_periodize (by norm_num : (0 : ℝ) < 100) hf hs' |
|
|
| end Problem340.ClockQuotient |
|
|
| end |
|
|
| end OAI |
|
|
