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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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.Dynamics.StandardMap.FourPoint
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped ENNReal Topology
namespace StandardMapEntropy
noncomputable def transferStepEquiv (v : ℝ) : ℂ ≃L[ℝ] ℂ :=
{ toLinearEquiv :=
{ toLinearMap := (transferStep v).toLinearMap
invFun := transferStepInv v
left_inv := transferStepInv_inv v
right_inv := transferStep_inv v }
continuous_toFun := (transferStep v).continuous
continuous_invFun := (transferStepInv v).continuous }
@[simp] lemma transferStepEquiv_apply (v : ℝ) (u : ℂ) : transferStepEquiv v u=transferStep v u := rfl
@[simp] lemma transferStepEquiv_symm_apply (v : ℝ) (u : ℂ) : (transferStepEquiv v).symm u=transferStepInv v u := rfl
noncomputable def positivePrefix (G : ℤ → (ℂ ≃L[ℝ] ℂ)) : ℕ → (ℂ ≃L[ℝ] ℂ)
| 0 => ContinuousLinearEquiv.refl ℝ ℂ
| n+1 => (positivePrefix G n).trans (G (n:ℤ))
noncomputable def negativePrefix (G : ℤ → (ℂ ≃L[ℝ] ℂ)) : ℕ → (ℂ ≃L[ℝ] ℂ)
| 0 => ContinuousLinearEquiv.refl ℝ ℂ
| n+1 => (negativePrefix G n).trans (G (-((n:ℤ)+1))).symm
noncomputable def integerPrefix (G : ℤ → (ℂ ≃L[ℝ] ℂ)) : ℤ → (ℂ ≃L[ℝ] ℂ)
| .ofNat n => positivePrefix G n
| .negSucc n => negativePrefix G (n+1)
@[simp] lemma integerPrefix_nat (G : ℤ → (ℂ ≃L[ℝ] ℂ)) (n : ℕ) : integerPrefix G (n:ℤ)=positivePrefix G n := rfl
@[simp] lemma integerPrefix_neg_nat (G : ℤ → (ℂ ≃L[ℝ] ℂ)) (n : ℕ) : integerPrefix G (-(n:ℤ))=negativePrefix G n := by
cases n <;> rfl
lemma integerPrefix_step (G : ℤ → (ℂ ≃L[ℝ] ℂ)) (i : ℤ) (u : ℂ) :
integerPrefix G (i+1) u=G i (integerPrefix G i u) := by
cases i with
| ofNat n => rfl
| negSucc n =>
change integerPrefix G (-((n:ℤ)+1)+1) u=G (-((n:ℤ)+1)) (negativePrefix G (n+1) u)
rw [show -((n:ℤ)+1)+1=-(n:ℤ) by ring,integerPrefix_neg_nat]
simp only [negativePrefix,ContinuousLinearEquiv.trans_apply,ContinuousLinearEquiv.apply_symm_apply]
lemma equiv_area_symm (A : ℂ ≃L[ℝ] ℂ) (hA : PlaneAreaPreserving A.toContinuousLinearMap) :
PlaneAreaPreserving A.symm.toContinuousLinearMap := by
intro u v
have h := hA (A.symm u) (A.symm v)
simpa using h.symm
lemma equiv_area_trans (A B : ℂ ≃L[ℝ] ℂ)
(hA : PlaneAreaPreserving A.toContinuousLinearMap) (hB : PlaneAreaPreserving B.toContinuousLinearMap) :
PlaneAreaPreserving (A.trans B).toContinuousLinearMap := by
intro u v
exact (hB (A u) (A v)).trans (hA u v)
lemma integerPrefix_area (G : ℤ → (ℂ ≃L[ℝ] ℂ))
(hG : ∀ i, PlaneAreaPreserving (G i).toContinuousLinearMap) (i : ℤ) :
PlaneAreaPreserving (integerPrefix G i).toContinuousLinearMap := by
have hp : ∀ n, PlaneAreaPreserving (positivePrefix G n).toContinuousLinearMap := by
intro n
induction n with
| zero => exact fun _ _ => rfl
| succ n ih => exact equiv_area_trans _ _ ih (hG _)
have hm : ∀ n, PlaneAreaPreserving (negativePrefix G n).toContinuousLinearMap := by
intro n
induction n with
| zero => exact fun _ _ => rfl
| succ n ih => exact equiv_area_trans _ _ ih (equiv_area_symm _ (hG _))
cases i with
| ofNat n => exact hp n
| negSucc n => exact hm (n+1)
noncomputable def torusPrefix (k : ℝ) (z : Torus) (i : ℤ) : ℂ ≃L[ℝ] ℂ :=
integerPrefix (fun a => transferStepEquiv (torusPotential k (torusIter k a z).1)) i
noncomputable def torusTransfer (k : ℝ) (z : Torus) (i j : ℤ) : ℂ ≃L[ℝ] ℂ :=
(torusPrefix k z i).symm.trans (torusPrefix k z j)
lemma torusTransfer_apply (k : ℝ) (z : Torus) (i j : ℤ) (u : ℂ) :
torusTransfer k z i j u=torusPrefix k z j ((torusPrefix k z i).symm u) := rfl
lemma torusTransfer_comp (k : ℝ) (z : Torus) (i j l : ℤ) (u : ℂ) :
torusTransfer k z j l (torusTransfer k z i j u)=torusTransfer k z i l u := by
simp only [torusTransfer_apply,ContinuousLinearEquiv.symm_apply_apply]
lemma torusTransfer_self (k : ℝ) (z : Torus) (i : ℤ) :
(torusTransfer k z i i).toContinuousLinearMap=ContinuousLinearMap.id ℝ ℂ := by
ext u
exact ContinuousLinearEquiv.apply_symm_apply _ _
lemma torusTransfer_area (k : ℝ) (z : Torus) (i j : ℤ) :
PlaneAreaPreserving (torusTransfer k z i j).toContinuousLinearMap := by
apply equiv_area_trans
· apply equiv_area_symm
exact integerPrefix_area (fun a => transferStepEquiv (torusPotential k (torusIter k a z).1))
(fun a => transferStep_area _) i
· exact integerPrefix_area (fun a => transferStepEquiv (torusPotential k (torusIter k a z).1))
(fun a => transferStep_area _) j
lemma torusTransfer_forward (k : ℝ) (z : Torus) (a : ℤ) (n : ℕ) :
(torusTransfer k z a (a+(n:ℤ))).toContinuousLinearMap=torusSegmentTransfer k z a n := by
induction n with
| zero => simpa only [Nat.cast_zero,add_zero,torusSegmentTransfer,transferProduct] using torusTransfer_self k z a
| succ n ih =>
ext u
change torusPrefix k z (a+((n+1:ℕ):ℤ)) ((torusPrefix k z a).symm u)=_
rw [show a+((n+1:ℕ):ℤ)=(a+(n:ℤ))+1 by omega]
change integerPrefix _ ((a+(n:ℤ))+1) _=_
rw [integerPrefix_step]
change transferStep (torusPotential k (torusIter k (a+(n:ℤ)) z).1)
((torusTransfer k z a (a+(n:ℤ))).toContinuousLinearMap u)=_
rw [ih]
simp only [torusSegmentTransfer,transferProduct,ContinuousLinearMap.comp_apply,torusSegmentCoefficient]
congr 1
exact congrArg (fun t : ℤ => transferStep (torusPotential k (torusIter k t z).1)) (by omega)
end StandardMapEntropy
end OAI
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