| import OAI.Dynamics.EntropyCounterexample.ClockCharts |
| import OAI.Geometry.SingularHomology.HomologyConjugacy |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340.CircleClockDoubling |
|
|
| abbrev AngularCircle := AddCircle (1 : ℝ) |
|
|
| def homeomorph : AngularCircle ≃ₜ StandardClock := |
| (AddCircle.homeomorphCircle (by norm_num : (1 : ℝ) ≠ 0)).trans |
| circleClockHomeomorph |
|
|
| @[simp] theorem homeomorph_apply (t : AngularCircle) : |
| homeomorph t = circleClockHomeomorph (AddCircle.toCircle t) := by |
| simp only [homeomorph, Homeomorph.trans_apply, AddCircle.homeomorphCircle_apply] |
| |
| def angularDouble (t : AngularCircle) : AngularCircle := t + t |
|
|
| theorem continuous_angularDouble : Continuous angularDouble := |
| continuous_id.add continuous_id |
|
|
| def clockDouble (x : StandardClock) : StandardClock := |
| circleClockHomeomorph ((circleClockHomeomorph.symm x) ^ 2) |
|
|
| theorem continuous_clockDouble : Continuous clockDouble := |
| circleClockHomeomorph.continuous.comp (circleClockHomeomorph.symm.continuous.pow 2) |
|
|
| @[simp] theorem homeomorph_angularDouble (t : AngularCircle) : |
| homeomorph (angularDouble t) = clockDouble (homeomorph t) := by |
| simp only [homeomorph_apply, angularDouble, AddCircle.toCircle_add, |
| clockDouble, Homeomorph.symm_apply_apply, pow_two] |
|
|
| theorem clockDouble_eq_conjugateMap : |
| clockDouble = conjugateMap homeomorph angularDouble := by |
| funext x |
| rw [conjugateMap, homeomorph_angularDouble, Homeomorph.apply_symm_apply] |
|
|
| @[simp] theorem clockDouble_coe (x : StandardClock) : |
| (clockDouble x).1 = complexClockLinearIsometry |
| ((complexClockLinearIsometry.symm x.1) ^ 2) := by |
| change complexClockLinearIsometry (((circleClockHomeomorph.symm x : Circle) ^ 2 : Circle) : ℂ) = _ |
| simp only [Circle.coe_pow] |
| rfl |
|
|
| theorem clockDouble_coordinates (x : StandardClock) : |
| (clockDouble x).1 = WithLp.toLp 2 ![x.1 0 ^ 2 - x.1 1 ^ 2, 2 * x.1 0 * x.1 1] := by |
| rw [clockDouble_coe] |
| let z := complexClockLinearIsometry.symm x.1 |
| have hx : complexClockLinearIsometry z = x.1 := |
| complexClockLinearIsometry.apply_symm_apply x.1 |
| have hx0 : z.re = x.1 0 := congrArg (fun y : ClockAmbient => y 0) hx |
| have hx1 : z.im = x.1 1 := congrArg (fun y : ClockAmbient => y 1) hx |
| change complexClockLinearIsometry (z ^ 2) = _ |
| ext i |
| fin_cases i < |
| Complex.mul_re, Complex.mul_im, hx0, hx1] |
|
|
| def homologyEquiv (n : ℕ) : |
| RealSingularHomology n AngularCircle ≃ₗ[ℝ] RealSingularHomology n StandardClock := |
| realHomologyEquiv n homeomorph |
|
|
| @[simp] theorem homologyEquiv_apply (n : ℕ) (v : RealSingularHomology n AngularCircle) : |
| homologyEquiv n v = realHomologyMap n homeomorph homeomorph.continuous v := rfl |
|
|
| theorem homologyEquiv_intertwines (n : ℕ) (v : RealSingularHomology n AngularCircle) : |
| realHomologyMap n clockDouble continuous_clockDouble (homologyEquiv n v) = |
| homologyEquiv n (realHomologyMap n angularDouble continuous_angularDouble v) := by |
| simpa only [clockDouble_eq_conjugateMap, homologyEquiv] using |
| realHomologyEquiv_conjugateMap n homeomorph continuous_angularDouble v |
|
|
| theorem eigenvector_transport (n : ℕ) (c : ℝ) |
| (v : RealSingularHomology n AngularCircle) (hv : v ≠ 0) |
| (heigen : realHomologyMap n angularDouble continuous_angularDouble v = c • v) : |
| ∃ w : RealSingularHomology n StandardClock, w ≠ 0 ∧ |
| realHomologyMap n clockDouble continuous_clockDouble w = c • w := by |
| refine ⟨homologyEquiv n v, ?_, ?_⟩ |
| · exact (map_ne_zero_iff (homologyEquiv n) (homologyEquiv n).injective).2 hv |
| · rw [homologyEquiv_intertwines, heigen, map_smul] |
|
|
| end Problem340.CircleClockDoubling |
|
|
| end |
|
|
| end OAI |
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|
