Minimal action of a group

In this file we define an action of a monoid M on a topological space α to be minimal if the M-orbit of every point x : α is dense. We also provide an additive version of this definition and prove some basic facts about minimal actions.

TODO

• Define a minimal set of an action.

Tags

group action, minimal

class AddAction.IsMinimal (M : Type u_1) (α : Type u_2) [AddMonoid M] [TopologicalSpace α] [AddAction M α] : Prop

An action of an additive monoid M on a topological space is called minimal if the M-orbit of every point x : α is dense.

• dense_orbit : ∀ (x : α), Dense (AddAction.orbit M x)

class MulAction.IsMinimal (M : Type u_1) (α : Type u_2) [Monoid M] [TopologicalSpace α] [MulAction M α] : Prop

An action of a monoid M on a topological space is called minimal if the M-orbit of every point x : α is dense.

• dense_orbit : ∀ (x : α), Dense (MulAction.orbit M x)

theorem MulAction.dense_orbit (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α) : Dense (MulAction.orbit M x)

theorem AddAction.dense_orbit (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α) : Dense (AddAction.orbit M x)

theorem denseRange_smul (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α) : DenseRange fun (c : M) => c • x

theorem denseRange_vadd (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α) : DenseRange fun (c : M) => c +ᵥ x

@[instance 100]

instance MulAction.isMinimal_of_pretransitive (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsPretransitive M α] : MulAction.IsMinimal M α

Equations

• ⋯ = ⋯

@[instance 100]

instance AddAction.isMinimal_of_pretransitive (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsPretransitive M α] : AddAction.IsMinimal M α

Equations

• ⋯ = ⋯

theorem IsOpen.exists_smul_mem (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ∃ (c : M), c • x ∈ U

theorem IsOpen.exists_vadd_mem (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ∃ (c : M), c +ᵥ x ∈ U

theorem IsOpen.iUnion_preimage_smul (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ⋃ (c : M), (fun (x : α) => c • x) ⁻¹' U = Set.univ

theorem IsOpen.iUnion_preimage_vadd (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ⋃ (c : M), (fun (x : α) => c +ᵥ x) ⁻¹' U = Set.univ

theorem IsOpen.iUnion_smul (G : Type u_2) {α : Type u_3} [Group G] [TopologicalSpace α] [MulAction G α] [MulAction.IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ⋃ (g : G), g • U = Set.univ

theorem IsOpen.iUnion_vadd (G : Type u_2) {α : Type u_3} [AddGroup G] [TopologicalSpace α] [AddAction G α] [AddAction.IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) : ⋃ (g : G), g +ᵥ U = Set.univ

theorem IsCompact.exists_finite_cover_smul (G : Type u_2) {α : Type u_3} [Group G] [TopologicalSpace α] [MulAction G α] [MulAction.IsMinimal G α] [ContinuousConstSMul G α] {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) : ∃ (I : Finset G), K ⊆ ⋃ g ∈ I, g • U

theorem IsCompact.exists_finite_cover_vadd (G : Type u_2) {α : Type u_3} [AddGroup G] [TopologicalSpace α] [AddAction G α] [AddAction.IsMinimal G α] [ContinuousConstVAdd G α] {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) : ∃ (I : Finset G), K ⊆ ⋃ g ∈ I, g +ᵥ U

theorem dense_of_nonempty_smul_invariant (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {s : Set α} (hne : s.Nonempty) (hsmul : ∀ (c : M), c • s ⊆ s) : Dense s

theorem dense_of_nonempty_vadd_invariant (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {s : Set α} (hne : s.Nonempty) (hsmul : ∀ (c : M), c +ᵥ s ⊆ s) : Dense s

theorem eq_empty_or_univ_of_smul_invariant_closed (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {s : Set α} (hs : IsClosed s) (hsmul : ∀ (c : M), c • s ⊆ s) : s = ∅ ∨ s = Set.univ

theorem eq_empty_or_univ_of_vadd_invariant_closed (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {s : Set α} (hs : IsClosed s) (hsmul : ∀ (c : M), c +ᵥ s ⊆ s) : s = ∅ ∨ s = Set.univ

theorem isMinimal_iff_isClosed_smul_invariant (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] : MulAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = Set.univ

theorem isMinimal_iff_isClosed_vadd_invariant (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [ContinuousConstVAdd M α] : AddAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c +ᵥ s ⊆ s) → s = ∅ ∨ s = Set.univ

@[deprecated isMinimal_iff_isClosed_smul_invariant]

theorem isMinimal_iff_closed_smul_invariant (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] : MulAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = Set.univ

Alias of isMinimal_iff_isClosed_smul_invariant.