Documentation

Mathlib.Dynamics.Flow

Flows and invariant sets #

This file defines a flow on a topological space α by a topological monoid τ as a continuous monoid-act of τ on α. Anticipating the cases where τ is one of , , ℝ⁺, or , we use additive notation for the monoids, though the definition does not require commutativity.

A subset s of α is invariant under a family of maps ϕₜ : α → α if ϕₜ s ⊆ s for all t. In many cases ϕ will be a flow on α. For the cases where ϕ is a flow by an ordered (additive, commutative) monoid, we additionally define forward invariance, where t ranges over those elements which are nonnegative.

Additionally, we define such constructions as the restriction of a flow onto an invariant subset, and the time-reversal of a flow by a group.

Invariant sets #

def IsInvariant {τ : Type u_1} {α : Type u_2} (ϕ : ταα) (s : Set α) :

A set s ⊆ α is invariant under ϕ : τ → α → α if ϕ t s ⊆ s for all t in τ.

Instances For
    theorem isInvariant_iff_image {τ : Type u_1} {α : Type u_2} (ϕ : ταα) (s : Set α) :
    IsInvariant ϕ s ∀ (t : τ), ϕ t '' s s
    def IsFwInvariant {τ : Type u_1} {α : Type u_2} [Preorder τ] [Zero τ] (ϕ : ταα) (s : Set α) :

    A set s ⊆ α is forward-invariant under ϕ : τ → α → α if ϕ t s ⊆ s for all t ≥ 0.

    Instances For
      theorem IsInvariant.isFwInvariant {τ : Type u_1} {α : Type u_2} [Preorder τ] [Zero τ] {ϕ : ταα} {s : Set α} (h : IsInvariant ϕ s) :
      theorem IsFwInvariant.isInvariant {τ : Type u_1} {α : Type u_2} [CanonicallyOrderedAddMonoid τ] {ϕ : ταα} {s : Set α} (h : IsFwInvariant ϕ s) :

      If τ is a CanonicallyOrderedAddMonoid (e.g., or ℝ≥0), then the notions IsFwInvariant and IsInvariant are equivalent.

      theorem isFwInvariant_iff_isInvariant {τ : Type u_1} {α : Type u_2} [CanonicallyOrderedAddMonoid τ] {ϕ : ταα} {s : Set α} :

      If τ is a CanonicallyOrderedAddMonoid (e.g., or ℝ≥0), then the notions IsFwInvariant and IsInvariant are equivalent.

      Flows #

      structure Flow (τ : Type u_1) [TopologicalSpace τ] [AddMonoid τ] [ContinuousAdd τ] (α : Type u_2) [TopologicalSpace α] :
      Type (max u_1 u_2)

      A flow on a topological space α by an additive topological monoid τ is a continuous monoid action of τ on α.

      Instances For
        instance Flow.instCoeFunFlowForAll {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] :
        CoeFun (Flow τ α) fun x => ταα
        theorem Flow.ext {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] {ϕ₁ : Flow τ α} {ϕ₂ : Flow τ α} :
        (∀ (t : τ) (x : α), Flow.toFun ϕ₁ t x = Flow.toFun ϕ₂ t x) → ϕ₁ = ϕ₂
        theorem Flow.continuous {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) {β : Type u_3} [TopologicalSpace β] {t : βτ} (ht : Continuous t) {f : βα} (hf : Continuous f) :
        Continuous fun x => Flow.toFun ϕ (t x) (f x)
        theorem Continuous.flow {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) {β : Type u_3} [TopologicalSpace β] {t : βτ} (ht : Continuous t) {f : βα} (hf : Continuous f) :
        Continuous fun x => Flow.toFun ϕ (t x) (f x)

        Alias of Flow.continuous.

        theorem Flow.map_add {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (t₁ : τ) (t₂ : τ) (x : α) :
        Flow.toFun ϕ (t₁ + t₂) x = Flow.toFun ϕ t₁ (Flow.toFun ϕ t₂ x)
        @[simp]
        theorem Flow.map_zero {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) :
        Flow.toFun ϕ 0 = id
        theorem Flow.map_zero_apply {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (x : α) :
        Flow.toFun ϕ 0 x = x
        def Flow.fromIter {α : Type u_2} [TopologicalSpace α] {g : αα} (h : Continuous g) :

        Iterations of a continuous function from a topological space α to itself defines a semiflow by on α.

        Instances For
          def Flow.restrict {τ : Type u_1} [AddMonoid τ] [TopologicalSpace τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) {s : Set α} (h : IsInvariant ϕ.toFun s) :
          Flow τ s

          Restriction of a flow onto an invariant set.

          Instances For
            theorem Flow.isInvariant_iff_image_eq {τ : Type u_1} [AddCommGroup τ] [TopologicalSpace τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (s : Set α) :
            IsInvariant ϕ.toFun s ∀ (t : τ), Flow.toFun ϕ t '' s = s
            def Flow.reverse {τ : Type u_1} [AddCommGroup τ] [TopologicalSpace τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) :
            Flow τ α

            The time-reversal of a flow ϕ by a (commutative, additive) group is defined ϕ.reverse t x = ϕ (-t) x.

            Instances For
              theorem Flow.continuous_toFun {τ : Type u_1} [AddCommGroup τ] [TopologicalSpace τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (t : τ) :
              def Flow.toHomeomorph {τ : Type u_1} [AddCommGroup τ] [TopologicalSpace τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (t : τ) :
              α ≃ₜ α

              The map ϕ t as a homeomorphism.

              Instances For
                theorem Flow.image_eq_preimage {τ : Type u_1} [AddCommGroup τ] [TopologicalSpace τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (ϕ : Flow τ α) (t : τ) (s : Set α) :
                Flow.toFun ϕ t '' s = Flow.toFun ϕ (-t) ⁻¹' s