| import Mathlib.Topology.Homeomorph.Lemmas |
| import Mathlib.Topology.Instances.Real.Lemmas |
| import Mathlib.Analysis.Normed.Lp.lpSpace |
| import Mathlib.Analysis.Normed.Operator.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped ENNReal |
|
|
| universe uG |
|
|
| namespace ThompsonNonamenability |
|
|
| abbrev UnitInterval := Set.Icc (0 : ℝ) 1 |
|
|
| def unitIntervalZero : UnitInterval := ⟨0, le_rfl, zero_le_one⟩ |
| def unitIntervalOne : UnitInterval := ⟨1, zero_le_one, le_rfl⟩ |
|
|
| structure IntervalHomeomorph where |
| toHomeomorph : UnitInterval ≃ₜ UnitInterval |
|
|
| namespace IntervalHomeomorph |
|
|
| instance : CoeFun IntervalHomeomorph (fun _ => UnitInterval → UnitInterval) := |
| ⟨fun f => f.toHomeomorph⟩ |
|
|
| end IntervalHomeomorph |
|
|
| def IsDyadic (x : ℝ) : Prop := |
| ∃ (k : ℤ) (n : ℕ), x = (k : ℝ) / (2 : ℝ) ^ n |
|
|
| structure DyadicPLWitness (f : IntervalHomeomorph) where |
| pieceCount : ℕ |
| positive : 0 < pieceCount |
| knots : Fin (pieceCount + 1) → UnitInterval |
| strictMono_knots : StrictMono knots |
| first : knots 0 = unitIntervalZero |
| last : knots (Fin.last pieceCount) = unitIntervalOne |
| dyadic : ∀ i, IsDyadic (knots i : ℝ) |
| cover : ∀ x : UnitInterval, ∃ i : Fin pieceCount, |
| knots i.castSucc ≤ x ∧ x ≤ knots i.succ |
| exponent : Fin pieceCount → ℤ |
| affine : ∀ (i : Fin pieceCount) (x : UnitInterval), |
| knots i.castSucc ≤ x → x ≤ knots i.succ → |
| (f x : ℝ) = (f (knots i.castSucc) : ℝ) + |
| (2 : ℝ) ^ exponent i * ((x : ℝ) - (knots i.castSucc : ℝ)) |
|
|
| def HasDyadicPLPieces (f : IntervalHomeomorph) : Prop := Nonempty (DyadicPLWitness f) |
|
|
| def F := {f : IntervalHomeomorph // StrictMono f ∧ HasDyadicPLPieces f} |
|
|
| abbrev BoundedReal (G : Type uG) := lp (fun _ : G => ℝ) ∞ |
|
|
| variable {G : Type uG} [Group G] |
|
|
| def leftPull (h : G) : BoundedReal G →ₗ[ℝ] BoundedReal G where |
| toFun f := ⟨fun g => f (h * g), memℓp_infty ⟨‖f‖, by |
| rintro y ⟨g, rfl⟩ |
| exact lp.norm_apply_le_norm ENNReal.top_ne_zero f (h * g)⟩⟩ |
| map_add' f g := by |
| apply lp.ext |
| funext x |
| rfl |
| map_smul' r f := by |
| apply lp.ext |
| funext x |
| rfl |
|
|
| structure InvariantMean (G : Type uG) [Group G] where |
| toLinearMap : BoundedReal G →ₗ[ℝ] ℝ |
| positive : ∀ f : BoundedReal G, (∀ g, 0 ≤ f g) → 0 ≤ toLinearMap f |
| normalized : toLinearMap 1 = 1 |
| left_invariant : ∀ (h : G) (f : BoundedReal G), |
| toLinearMap (leftPull h f) = toLinearMap f |
|
|
| end ThompsonNonamenability |
|
|
| end |
|
|
| end OAI |
|
|
