Documentation

Mathlib.Dynamics.Minimal

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 #

Tags #

group action, minimal

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

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.

Instances
    theorem AddAction.IsMinimal.dense_orbit {M : Type u_1} {α : Type u_2} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [self : AddAction.IsMinimal M α] (x : α) :
    class MulAction.IsMinimal (M : Type u_1) (α : Type u_2) [Monoid M] [TopologicalSpace α] [MulAction M α] :

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

    Instances
      theorem MulAction.IsMinimal.dense_orbit {M : Type u_1} {α : Type u_2} [Monoid M] [TopologicalSpace α] [MulAction M α] [self : MulAction.IsMinimal M α] (x : α) :
      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) :
      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) :
      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
      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
      @[instance 100]
      Equations
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      @[instance 100]
      Equations
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      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.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.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_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
      abbrev IsOpen.iUnion_vadd.match_1 (G : Type u_1) {α : Type u_2} [AddGroup G] [AddAction G α] {U : Set α} (x : α) (motive : (∃ (c : G), c +ᵥ x U)Prop) :
      ∀ (x_1 : ∃ (c : G), c +ᵥ x U), (∀ (g : G) (hg : g +ᵥ x U), motive )motive x_1
      Equations
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      Instances For
        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 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 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 : Set α} {U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
        ∃ (I : Finset G), K gI, g +ᵥ U
        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 : Set α} {U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
        ∃ (I : Finset G), K gI, g U
        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) :
        abbrev dense_of_nonempty_vadd_invariant.match_1 {α : Type u_1} {s : Set α} (motive : s.NonemptyProp) :
        ∀ (hne : s.Nonempty), (∀ (x : α) (hx : x s), motive )motive hne
        Equations
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        Instances For
          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) :
          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 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 isMinimal_iff_closed_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
          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