# Ergodic maps and measures #

Let f : α → α be measure preserving with respect to a measure μ. We say f is ergodic with respect to μ (or μ is ergodic with respect to f) if the only measurable sets s such that f⁻¹' s = s are either almost empty or full.

In this file we define ergodic maps / measures together with quasi-ergodic maps / measures and provide some basic API. Quasi-ergodicity is a weaker condition than ergodicity for which the measure preserving condition is relaxed to quasi measure preserving.

# Main definitions: #

• PreErgodic: the ergodicity condition without the measure preserving condition. This exists to share code between the Ergodic and QuasiErgodic definitions.
• Ergodic: the definition of ergodic maps / measures.
• QuasiErgodic: the definition of quasi ergodic maps / measures.
• Ergodic.quasiErgodic: an ergodic map / measure is quasi ergodic.
• QuasiErgodic.ae_empty_or_univ': when the map is quasi measure preserving, one may relax the strict invariance condition to almost invariance in the ergodicity condition.
structure PreErgodic {α : Type u_1} {m : } (f : αα) (μ : ) :

A map f : α → α is said to be pre-ergodic with respect to a measure μ if any measurable strictly invariant set is either almost empty or full.

Instances For
theorem PreErgodic.ae_empty_or_univ {α : Type u_1} {m : } {f : αα} {μ : } (self : ) ⦃s : Set α :
f ⁻¹' s = ss =ᵐ[μ] s =ᵐ[μ] Set.univ
structure Ergodic {α : Type u_1} {m : } (f : αα) (μ : ) extends , :

A map f : α → α is said to be ergodic with respect to a measure μ if it is measure preserving and pre-ergodic.

Instances For
structure QuasiErgodic {α : Type u_1} {m : } (f : αα) (μ : ) extends , :

A map f : α → α is said to be quasi ergodic with respect to a measure μ if it is quasi measure preserving and pre-ergodic.

Instances For
theorem PreErgodic.measure_self_or_compl_eq_zero {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hs : ) (hs' : f ⁻¹' s = s) :
μ s = 0 μ s = 0
theorem PreErgodic.ae_mem_or_ae_nmem {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hsm : ) (hs : f ⁻¹' s = s) :
(∀ᵐ (x : α) ∂μ, x s) ∀ᵐ (x : α) ∂μ, xs
theorem PreErgodic.prob_eq_zero_or_one {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hs : ) (hs' : f ⁻¹' s = s) :
μ s = 0 μ s = 1

On a probability space, the (pre)ergodicity condition is a zero one law.

theorem PreErgodic.of_iterate {α : Type u_1} {m : } {f : αα} {μ : } (n : ) (hf : ) :
theorem MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate {α : Type u_1} {m : } {f : αα} {μ : } {β : Type u_2} {m' : } {μ' : } {g : αβ} (hg : ) (hf : ) {f' : ββ} (h_comm : g f = f' g) :
PreErgodic f' μ'
theorem MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff {α : Type u_1} {m : } {f : αα} {μ : } {β : Type u_2} {m' : } {μ' : } {e : α ≃ᵐ β} (h : ) :
PreErgodic (e f e.symm) μ'
theorem MeasureTheory.MeasurePreserving.ergodic_conjugate_iff {α : Type u_1} {m : } {f : αα} {μ : } {β : Type u_2} {m' : } {μ' : } {e : α ≃ᵐ β} (h : ) :
Ergodic (e f e.symm) μ' Ergodic f μ
theorem QuasiErgodic.ae_empty_or_univ' {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hs : ) (hs' : f ⁻¹' s =ᵐ[μ] s) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ

For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are still either almost empty or full.

theorem QuasiErgodic.ae_empty_or_univ₀ {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hsm : ) (hs : f ⁻¹' s =ᵐ[μ] s) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ

For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are still either almost empty or full.

theorem QuasiErgodic.ae_mem_or_ae_nmem₀ {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : ) (hsm : ) (hs : f ⁻¹' s =ᵐ[μ] s) :
(∀ᵐ (x : α) ∂μ, x s) ∀ᵐ (x : α) ∂μ, xs

For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are still either almost empty or full.

theorem Ergodic.quasiErgodic {α : Type u_1} {m : } {f : αα} {μ : } (hf : Ergodic f μ) :

An ergodic map is quasi ergodic.

theorem Ergodic.ae_empty_or_univ_of_preimage_ae_le' {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ

See also Ergodic.ae_empty_or_univ_of_preimage_ae_le.

theorem Ergodic.ae_empty_or_univ_of_ae_le_preimage' {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ

See also Ergodic.ae_empty_or_univ_of_ae_le_preimage.

theorem Ergodic.ae_empty_or_univ_of_image_ae_le' {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ

See also Ergodic.ae_empty_or_univ_of_image_ae_le.

theorem Ergodic.ae_empty_or_univ_of_preimage_ae_le {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : f ⁻¹' s ≤ᵐ[μ] s) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ
theorem Ergodic.ae_empty_or_univ_of_ae_le_preimage {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : s ≤ᵐ[μ] f ⁻¹' s) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ
theorem Ergodic.ae_empty_or_univ_of_image_ae_le {α : Type u_1} {m : } {f : αα} {s : Set α} {μ : } (hf : Ergodic f μ) (hs : ) (hs' : f '' s ≤ᵐ[μ] s) :
s =ᵐ[μ] s =ᵐ[μ] Set.univ