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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 | 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
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