mathlib documentation

dynamics.omega_limit

ω-limits #

For a function ϕ : τ → α → β where β is a topological space, we define the ω-limit under ϕ of a set s in α with respect to filter f on τ: an element y : β is in the ω-limit of s if the forward images of s intersect arbitrarily small neighbourhoods of y frequently "in the direction of f".

In practice ϕ is often a continuous monoid-act, but the definition requires only that ϕ has a coercion to the appropriate function type. In the case where τ is or and f is at_top, we recover the usual definition of the ω-limit set as the set of all y such that there exist sequences (tₙ), (xₙ) such that ϕ tₙ xₙ ⟶ y as n ⟶ ∞.

Notations #

The omega_limit locale provides the localised notation ω for omega_limit, as well as ω⁺ and ω⁻ for omega_limit at_top and omega_limit at_bot respectively for when the acting monoid is endowed with an order.

Definition and notation #

def omega_limit {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) :
set β

The ω-limit of a set s under ϕ with respect to a filter f is ⋂ u ∈ f, cl (ϕ u s).

Equations

Elementary properties #

theorem omega_limit_def {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) :
ω f ϕ s = ⋂ (u : set τ) (H : u f), closure (set.image2 ϕ u s)
theorem omega_limit_subset_of_tendsto {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (ϕ : τ → α → β) (s : set α) {m : τ → τ} {f₁ f₂ : filter τ} (hf : filter.tendsto m f₁ f₂) :
ω f₁ (λ (t : τ) (x : α), ϕ (m t) x) s ω f₂ ϕ s
theorem omega_limit_mono_left {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (ϕ : τ → α → β) (s : set α) {f₁ f₂ : filter τ} (hf : f₁ f₂) :
ω f₁ ϕ s ω f₂ ϕ s
theorem omega_limit_mono_right {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) {s₁ s₂ : set α} (hs : s₁ s₂) :
ω f ϕ s₁ ω f ϕ s₂
theorem is_closed_omega_limit {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) :
is_closed (ω f ϕ s)
theorem maps_to_omega_limit' {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (s : set α) {α' : Type u_4} {β' : Type u_5} [topological_space β'] {f : filter τ} {ϕ : τ → α → β} {ϕ' : τ → α' → β'} {ga : α → α'} {s' : set α'} (hs : set.maps_to ga s s') {gb : β → β'} (hg : ∀ᶠ (t : τ) in f, set.eq_on (gb ϕ t) (ϕ' t ga) s) (hgc : continuous gb) :
set.maps_to gb (ω f ϕ s) (ω f ϕ' s')
theorem maps_to_omega_limit {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (s : set α) {α' : Type u_4} {β' : Type u_5} [topological_space β'] {f : filter τ} {ϕ : τ → α → β} {ϕ' : τ → α' → β'} {ga : α → α'} {s' : set α'} (hs : set.maps_to ga s s') {gb : β → β'} (hg : ∀ (t : τ) (x : α), gb (ϕ t x) = ϕ' t (ga x)) (hgc : continuous gb) :
set.maps_to gb (ω f ϕ s) (ω f ϕ' s')
theorem omega_limit_image_eq {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (s : set α) {α' : Type u_4} (ϕ : τ → α' → β) (f : filter τ) (g : α → α') :
ω f ϕ (g '' s) = ω f (λ (t : τ) (x : α), ϕ t (g x)) s
theorem omega_limit_preimage_subset {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] {α' : Type u_4} (ϕ : τ → α' → β) (s : set α') (f : filter τ) (g : α → α') :
ω f (λ (t : τ) (x : α), ϕ t (g x)) (g ⁻¹' s) ω f ϕ s

Equivalent definitions of the omega limit #

The next few lemmas are various versions of the property characterising ω-limits:

theorem mem_omega_limit_iff_frequently {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) (y : β) :
y ω f ϕ s ∀ (n : set β), n 𝓝 y(∃ᶠ (t : τ) in f, (s ϕ t ⁻¹' n).nonempty)

An element y is in the ω-limit set of s w.r.t. f if the preimages of an arbitrary neighbourhood of y frequently (w.r.t. f) intersects of s.

theorem mem_omega_limit_iff_frequently₂ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) (y : β) :
y ω f ϕ s ∀ (n : set β), n 𝓝 y(∃ᶠ (t : τ) in f, (ϕ t '' s n).nonempty)

An element y is in the ω-limit set of s w.r.t. f if the forward images of s frequently (w.r.t. f) intersect arbitrary neighbourhoods of y.

theorem mem_omega_limit_singleton_iff_map_cluster_point {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (x : α) (y : β) :
y ω f ϕ {x} map_cluster_pt y f (λ (t : τ), ϕ t x)

An element y is in the ω-limit of x w.r.t. f if the forward images of x frequently (w.r.t. f) falls within an arbitrary neighbourhood of y.

Set operations and omega limits #

theorem omega_limit_inter {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s₁ s₂ : set α) :
ω f ϕ (s₁ s₂) ω f ϕ s₁ ω f ϕ s₂
theorem omega_limit_Inter {τ : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_4} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (p : ι → set α) :
ω f ϕ (⋂ (i : ι), p i) ⋂ (i : ι), ω f ϕ (p i)
theorem omega_limit_union {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s₁ s₂ : set α) :
ω f ϕ (s₁ s₂) = ω f ϕ s₁ ω f ϕ s₂
theorem omega_limit_Union {τ : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_4} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (p : ι → set α) :
(⋃ (i : ι), ω f ϕ (p i)) ω f ϕ (⋃ (i : ι), p i)

Different expressions for omega limits, useful for rewrites. In particular, one may restrict the intersection to sets in f which are subsets of some set v also in f.

theorem omega_limit_eq_Inter {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) :
ω f ϕ s = ⋂ (u : (f.sets)), closure (set.image2 ϕ u s)
theorem omega_limit_eq_bInter_inter {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) {v : set τ} (hv : v f) :
ω f ϕ s = ⋂ (u : set τ) (H : u f), closure (set.image2 ϕ (u v) s)
theorem omega_limit_eq_Inter_inter {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) {v : set τ} (hv : v f) :
ω f ϕ s = ⋂ (u : (f.sets)), closure (set.image2 ϕ (u v) s)
theorem omega_limit_subset_closure_fw_image {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) {u : set τ} (hu : u f) :
ω f ϕ s closure (set.image2 ϕ u s)

`ω-limits and compactness #

theorem eventually_closure_subset_of_is_compact_absorbing_of_is_open_of_omega_limit_subset' {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) {c : set β} (hc₁ : is_compact c) (hc₂ : ∃ (v : set τ) (H : v f), closure (set.image2 ϕ v s) c) {n : set β} (hn₁ : is_open n) (hn₂ : ω f ϕ s n) :
∃ (u : set τ) (H : u f), closure (set.image2 ϕ u s) n

A set is eventually carried into any open neighbourhood of its ω-limit: if c is a compact set such that closure {ϕ t x | t ∈ v, x ∈ s} ⊆ c for some v ∈ f and n is an open neighbourhood of ω f ϕ s, then for some u ∈ f we have closure {ϕ t x | t ∈ u, x ∈ s} ⊆ n.

theorem eventually_closure_subset_of_is_compact_absorbing_of_is_open_of_omega_limit_subset {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [t2_space β] {c : set β} (hc₁ : is_compact c) (hc₂ : ∀ᶠ (t : τ) in f, set.maps_to (ϕ t) s c) {n : set β} (hn₁ : is_open n) (hn₂ : ω f ϕ s n) :
∃ (u : set τ) (H : u f), closure (set.image2 ϕ u s) n

A set is eventually carried into any open neighbourhood of its ω-limit: if c is a compact set such that closure {ϕ t x | t ∈ v, x ∈ s} ⊆ c for some v ∈ f and n is an open neighbourhood of ω f ϕ s, then for some u ∈ f we have closure {ϕ t x | t ∈ u, x ∈ s} ⊆ n.

theorem eventually_maps_to_of_is_compact_absorbing_of_is_open_of_omega_limit_subset {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [t2_space β] {c : set β} (hc₁ : is_compact c) (hc₂ : ∀ᶠ (t : τ) in f, set.maps_to (ϕ t) s c) {n : set β} (hn₁ : is_open n) (hn₂ : ω f ϕ s n) :
∀ᶠ (t : τ) in f, set.maps_to (ϕ t) s n
theorem eventually_closure_subset_of_is_open_of_omega_limit_subset {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [compact_space β] {v : set β} (hv₁ : is_open v) (hv₂ : ω f ϕ s v) :
∃ (u : set τ) (H : u f), closure (set.image2 ϕ u s) v
theorem eventually_maps_to_of_is_open_of_omega_limit_subset {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [compact_space β] {v : set β} (hv₁ : is_open v) (hv₂ : ω f ϕ s v) :
∀ᶠ (t : τ) in f, set.maps_to (ϕ t) s v
theorem nonempty_omega_limit_of_is_compact_absorbing {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [f.ne_bot] {c : set β} (hc₁ : is_compact c) (hc₂ : ∃ (v : set τ) (H : v f), closure (set.image2 ϕ v s) c) (hs : s.nonempty) :
(ω f ϕ s).nonempty

The ω-limit of a nonempty set w.r.t. a nontrivial filter is nonempty.

theorem nonempty_omega_limit {τ : Type u_1} {α : Type u_2} {β : Type u_3} [topological_space β] (f : filter τ) (ϕ : τ → α → β) (s : set α) [compact_space β] [f.ne_bot] (hs : s.nonempty) :
(ω f ϕ s).nonempty

ω-limits of Flows by a Monoid #

theorem flow.is_invariant_omega_limit {τ : Type u_1} [topological_space τ] [add_monoid τ] [has_continuous_add τ] {α : Type u_2} [topological_space α] (f : filter τ) (ϕ : flow τ α) (s : set α) (hf : ∀ (t : τ), filter.tendsto (has_add.add t) f f) :
theorem flow.omega_limit_image_subset {τ : Type u_1} [topological_space τ] [add_monoid τ] [has_continuous_add τ] {α : Type u_2} [topological_space α] (f : filter τ) (ϕ : flow τ α) (s : set α) (t : τ) (ht : filter.tendsto (λ (_x : τ), _x + t) f f) :
ω f ϕ (ϕ t '' s) ω f ϕ s

ω-limits of Flows by a Group #

@[simp]
theorem flow.omega_limit_image_eq {τ : Type u_1} [topological_space τ] [add_comm_group τ] [topological_add_group τ] {α : Type u_2} [topological_space α] (f : filter τ) (ϕ : flow τ α) (s : set α) (hf : ∀ (t : τ), filter.tendsto (λ (_x : τ), _x + t) f f) (t : τ) :
ω f ϕ (ϕ t '' s) = ω f ϕ s

the ω-limit of a forward image of s is the same as the ω-limit of s.

theorem flow.omega_limit_omega_limit {τ : Type u_1} [topological_space τ] [add_comm_group τ] [topological_add_group τ] {α : Type u_2} [topological_space α] (f : filter τ) (ϕ : flow τ α) (s : set α) (hf : ∀ (t : τ), filter.tendsto (has_add.add t) f f) :
ω f ϕ (ω f ϕ s) ω f ϕ s