Semiconjugate by sSup

In this file we prove two facts about semiconjugate (families of) functions.

First, if an order isomorphism fa : α → α is semiconjugate to an order embedding fb : β → β by g : α → β, then fb is semiconjugate to fa by y ↦ sSup {x | g x ≤ y}, see Semiconj.symm_adjoint.

Second, consider two actions f₁ f₂ : G → α → α of a group on a complete lattice by order isomorphisms. Then the map x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x) semiconjugates each f₁ g' to f₂ g', see Function.sSup_div_semiconj. In the case of a conditionally complete lattice, a similar statement holds true under an additional assumption that each set {(f₁ g)⁻¹ (f₂ g x) | g : G} is bounded above, see Function.csSup_div_semiconj.

The lemmas come from Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée, Proposition 2.1 and 5.4 respectively. In the paper they are formulated for homeomorphisms of the circle, so in order to apply results from this file one has to lift these homeomorphisms to the real line first.

def IsOrderRightAdjoint {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α → β) (g : β → α) : Prop

We say that g : β → α is an order right adjoint function for f : α → β if it sends each y to a least upper bound for {x | f x ≤ y}. If α is a partial order, and f : α → β has a right adjoint, then this right adjoint is unique.

Equations

• IsOrderRightAdjoint f g = ∀ (y : β), IsLUB {x : α | f x ≤ y} (g y)

theorem isOrderRightAdjoint_sSup {α : Type u_1} {β : Type u_2} [CompleteLattice α] [Preorder β] (f : α → β) : IsOrderRightAdjoint f fun (y : β) => sSup {x : α | f x ≤ y}

theorem isOrderRightAdjoint_csSup {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [Preorder β] (f : α → β) (hne : ∀ (y : β), ∃ (x : α), f x ≤ y) (hbdd : ∀ (y : β), BddAbove {x : α | f x ≤ y}) : IsOrderRightAdjoint f fun (y : β) => sSup {x : α | f x ≤ y}

theorem IsOrderRightAdjoint.unique {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {f : α → β} {g₁ g₂ : β → α} (h₁ : IsOrderRightAdjoint f g₁) (h₂ : IsOrderRightAdjoint f g₂) : g₁ = g₂

theorem IsOrderRightAdjoint.right_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {g : β → α} (h : IsOrderRightAdjoint f g) : Monotone g

theorem IsOrderRightAdjoint.orderIso_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] {f : α → β} {g : β → α} (h : IsOrderRightAdjoint f g) (e : β ≃o γ) : IsOrderRightAdjoint (⇑e ∘ f) (g ∘ ⇑e.symm)

theorem IsOrderRightAdjoint.comp_orderIso {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] {f : α → β} {g : β → α} (h : IsOrderRightAdjoint f g) (e : γ ≃o α) : IsOrderRightAdjoint (f ∘ ⇑e) (⇑e.symm ∘ g)

theorem Function.Semiconj.symm_adjoint {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {fa : α ≃o α} {fb : β ↪o β} {g : α → β} (h : Function.Semiconj g ⇑fa ⇑fb) {g' : β → α} (hg' : IsOrderRightAdjoint g g') : Function.Semiconj g' ⇑fb ⇑fa

If an order automorphism fa is semiconjugate to an order embedding fb by a function g and g' is an order right adjoint of g (i.e. g' y = sSup {x | f x ≤ y}), then fb is semiconjugate to fa by g'.

This is a version of Proposition 2.1 from Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée.

theorem Function.semiconj_of_isLUB {α : Type u_1} {G : Type u_4} [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃o α) {h : α → α} (H : ∀ (x : α), IsLUB (Set.range fun (g' : G) => (f₁ g')⁻¹ ((f₂ g') x)) (h x)) (g : G) : Function.Semiconj h ⇑(f₂ g) ⇑(f₁ g)

theorem Function.sSup_div_semiconj {α : Type u_1} {G : Type u_4} [CompleteLattice α] [Group G] (f₁ f₂ : G →* α ≃o α) (g : G) : Function.Semiconj (fun (x : α) => ⨆ (g' : G), (f₁ g')⁻¹ ((f₂ g') x)) ⇑(f₂ g) ⇑(f₁ g)

Consider two actions f₁ f₂ : G → α → α of a group on a complete lattice by order isomorphisms. Then the map x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x) semiconjugates each f₁ g' to f₂ g'.

This is a version of Proposition 5.4 from Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée.

theorem Function.csSup_div_semiconj {α : Type u_1} {G : Type u_4} [ConditionallyCompleteLattice α] [Group G] (f₁ f₂ : G →* α ≃o α) (hbdd : ∀ (x : α), BddAbove (Set.range fun (g : G) => (f₁ g)⁻¹ ((f₂ g) x))) (g : G) : Function.Semiconj (fun (x : α) => ⨆ (g' : G), (f₁ g')⁻¹ ((f₂ g') x)) ⇑(f₂ g) ⇑(f₁ g)

Consider two actions f₁ f₂ : G → α → α of a group on a conditionally complete lattice by order isomorphisms. Suppose that each set $s(x)=\{f_1(g)^{-1} (f_2(g)(x)) | g \in G\}$ is bounded above. Then the map x ↦ sSup s(x) semiconjugates each f₁ g' to f₂ g'.

This is a version of Proposition 5.4 from Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée.