# 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 ℕ, ℤ, ℝ⁺, or ℝ, 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) :
∀ (t : τ), ϕ t '' s s
def IsFwInvariant {τ : Type u_1} {α : Type u_2} [] [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} [] [Zero τ] {ϕ : ταα} {s : Set α} (h : ) :
theorem IsFwInvariant.isInvariant {τ : Type u_1} {α : Type u_2} {ϕ : ταα} {s : Set α} (h : ) :

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

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

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

### Flows #

structure Flow (τ : Type u_1) [] [] [] (α : Type u_2) [] :
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.instInhabitedFlow {τ : Type u_1} [] [] [] {α : Type u_2} [] :
Inhabited (Flow τ α)
instance Flow.instCoeFunFlowForAll {τ : Type u_1} [] [] [] {α : Type u_2} [] :
CoeFun (Flow τ α) fun x => ταα
theorem Flow.ext {τ : Type u_1} [] [] [] {α : Type u_2} [] {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} [] [] [] {α : Type u_2} [] (ϕ : Flow τ α) {β : Type u_3} [] {t : βτ} (ht : ) {f : βα} (hf : ) :
Continuous fun x => Flow.toFun ϕ (t x) (f x)
theorem Continuous.flow {τ : Type u_1} [] [] [] {α : Type u_2} [] (ϕ : Flow τ α) {β : Type u_3} [] {t : βτ} (ht : ) {f : βα} (hf : ) :
Continuous fun x => Flow.toFun ϕ (t x) (f x)

Alias of Flow.continuous.

theorem Flow.map_add {τ : Type u_1} [] [] [] {α : Type u_2} [] (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} [] [] [] {α : Type u_2} [] (ϕ : Flow τ α) :
= id
theorem Flow.map_zero_apply {τ : Type u_1} [] [] [] {α : Type u_2} [] (ϕ : Flow τ α) (x : α) :
Flow.toFun ϕ 0 x = x
def Flow.fromIter {α : Type u_2} [] {g : αα} (h : ) :

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

Instances For
def Flow.restrict {τ : Type u_1} [] [] [] {α : Type u_2} [] (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} [] [] {α : Type u_2} [] (ϕ : Flow τ α) (s : Set α) :
IsInvariant ϕ.toFun s ∀ (t : τ), '' s = s
def Flow.reverse {τ : Type u_1} [] [] {α : Type u_2} [] (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} [] [] {α : Type u_2} [] (ϕ : Flow τ α) (t : τ) :
def Flow.toHomeomorph {τ : Type u_1} [] [] {α : Type u_2} [] (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} [] [] {α : Type u_2} [] (ϕ : Flow τ α) (t : τ) (s : Set α) :
'' s = Flow.toFun ϕ (-t) ⁻¹' s