File size: 2,465 Bytes
0dd3af4 | 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 | 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
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