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a44267b | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import Mathlib.Topology.Instances.AddCircle.DenseSubgroup
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.LinearAlgebra.Pi
import Mathlib.Topology.Algebra.Group.Subgroup
import Mathlib.Topology.Algebra.Constructions.DomMulAct
namespace OAI
/-!
# A finite dense generating family for diagonal parameters
In each real logarithmic coordinate, shifts by one and by square root
of two generate a dense additive subgroup. Thus the diagonal action
has a concrete finite topologically generating family; no one of its
elements is asserted to be ergodic.
-/
noncomputable section
open Set
open scoped BigOperators
namespace DukePrimeDegree
variable {I : Type*} [Fintype I] [DecidableEq I]
def diagonalParameterGenerators (p : I × Bool) : I → ℝ :=
Pi.single p.1 (if p.2 then Real.sqrt 2 else 1)
def singleCoordinateAddHom (i : I) : ℝ →+ (I → ℝ) where
toFun := fun t : ℝ => (Pi.single i t : I → ℝ)
map_zero' := by ext j; simp
map_add' s t := by ext j; by_cases h : i = j <;> simp [h]
omit [Fintype I] in
lemma continuous_singleCoordinateAddHom (i : I) :
Continuous (singleCoordinateAddHom i : ℝ → I → ℝ) := by
apply continuous_pi
intro j
by_cases h : i = j
· subst j
have he : (fun t : ℝ => singleCoordinateAddHom i t i) = id := by
funext t
simp [singleCoordinateAddHom]
change Continuous (fun t : ℝ => singleCoordinateAddHom i t i)
rw [he]
exact continuous_id
· simpa [singleCoordinateAddHom, h] using
(continuous_const : Continuous (fun _ : ℝ => (0 : ℝ)))
theorem dense_diagonalParameterGenerators :
Dense (AddSubgroup.closure (range (diagonalParameterGenerators (I := I))) : Set (I → ℝ)) := by
let H := AddSubgroup.closure (range (diagonalParameterGenerators (I := I)))
have hline (i : I) (t : ℝ) : Pi.single i t ∈ H.topologicalClosure := by
let S : AddSubgroup ℝ := H.topologicalClosure.comap (singleCoordinateAddHom i)
have hSclosed : IsClosed (S : Set ℝ) :=
H.isClosed_topologicalClosure.preimage (continuous_singleCoordinateAddHom i)
have h1 : (1 : ℝ) ∈ S := by
apply H.le_topologicalClosure
apply AddSubgroup.subset_closure
exact ⟨(i, false), by simp [diagonalParameterGenerators, singleCoordinateAddHom]⟩
have h2 : Real.sqrt 2 ∈ S := by
apply H.le_topologicalClosure
apply AddSubgroup.subset_closure
exact ⟨(i, true), by simp [diagonalParameterGenerators, singleCoordinateAddHom]⟩
have hd : Dense (AddSubgroup.closure {Real.sqrt 2, 1} : Set ℝ) :=
dense_addSubgroupClosure_pair_iff.mpr (by simpa using irrational_sqrt_two)
have hle : AddSubgroup.closure {Real.sqrt 2, 1} ≤ S :=
(AddSubgroup.closure_le _).mpr (by rw [pair_subset_iff]; exact ⟨h2, h1⟩)
have hdS : Dense (S : Set ℝ) := hd.mono hle
have ht : t ∈ (S : Set ℝ) := by
rw [← hSclosed.closure_eq]
exact hdS t
exact ht
intro v
change v ∈ closure (H : Set (I → ℝ))
change v ∈ H.topologicalClosure
have he : (∑ i, Pi.single i (v i)) = v := by
ext j
simp
rw [← he]
exact H.topologicalClosure.sum_mem (fun i _ => hline i (v i))
def multiplicativeParameterGenerators (p : I × Bool) : Multiplicative (I → ℝ) :=
Multiplicative.ofAdd (diagonalParameterGenerators p)
lemma dense_multiplicativeParameterGenerators :
Dense (Subgroup.closure (range (multiplicativeParameterGenerators (I := I))) :
Set (Multiplicative (I → ℝ))) := by
have he : (AddSubgroup.closure (range (diagonalParameterGenerators (I := I)))).toSubgroup =
Subgroup.closure (range (multiplicativeParameterGenerators (I := I))) := by
rw [AddSubgroup.toSubgroup_closure]
congr 1
rw [← he]
exact dense_diagonalParameterGenerators
/-- On an abelian group the domain-action type tag is also a group
homomorphism, so dense generators transfer to the Koopman action. -/
def parameterDomainHom : Multiplicative (I → ℝ) →*
(Multiplicative (I → ℝ))ᵈᵐᵃ where
toFun := DomMulAct.mk
map_one' := rfl
map_mul' g h := by rw [DomMulAct.mk_mul, mul_comm]
def domainParameterGenerators (p : I × Bool) : (Multiplicative (I → ℝ))ᵈᵐᵃ :=
parameterDomainHom (multiplicativeParameterGenerators p)
theorem dense_domainParameterGenerators :
Dense (Subgroup.closure (range (domainParameterGenerators (I := I))) :
Set (Multiplicative (I → ℝ))ᵈᵐᵃ) := by
have he : (Subgroup.closure (range (multiplicativeParameterGenerators (I := I)))).map
parameterDomainHom = Subgroup.closure (range (domainParameterGenerators (I := I))) := by
rw [MonoidHom.map_closure, ← Set.range_comp]
rfl
rw [← he, Subgroup.coe_map]
have hd : DenseRange (parameterDomainHom (I := I)) := DomMulAct.mk.surjective.denseRange
exact hd.dense_image DomMulAct.continuous_mk dense_multiplicativeParameterGenerators
end DukePrimeDegree
end
end OAI
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