# mathlib3documentation

dynamics.ergodic.ergodic

# Ergodic maps and measures #

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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: #

• pre_ergodic: the ergodicity condition without the measure preserving condition. This exists to share code between the ergodic and quasi_ergodic definitions.
• ergodic: the definition of ergodic maps / measures.
• quasi_ergodic: the definition of quasi ergodic maps / measures.
• ergodic.quasi_ergodic: an ergodic map / measure is quasi ergodic.
• quasi_ergodic.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 pre_ergodic {α : Type u_1} {m : measurable_space α} (f : α α) (μ : . "volume_tac") :
Prop

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.

@[nolint]
structure ergodic {α : Type u_1} {m : measurable_space α} (f : α α) (μ : . "volume_tac") :
Prop
• to_measure_preserving :
• to_pre_ergodic : μ

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

@[nolint]
structure quasi_ergodic {α : Type u_1} {m : measurable_space α} (f : α α) (μ : . "volume_tac") :
Prop
• to_quasi_measure_preserving :
• to_pre_ergodic : μ

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.

theorem pre_ergodic.measure_self_or_compl_eq_zero {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f ⁻¹' s = s) :
μ s = 0 μ s = 0
theorem pre_ergodic.prob_eq_zero_or_one {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (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 pre_ergodic.of_iterate {α : Type u_1} {m : measurable_space α} {f : α α} {μ : measure_theory.measure α} (n : ) (hf : μ) :
μ
theorem measure_theory.measure_preserving.pre_ergodic_of_pre_ergodic_conjugate {α : Type u_1} {m : measurable_space α} {f : α α} {μ : measure_theory.measure α} {β : Type u_2} {m' : measurable_space β} {μ' : measure_theory.measure β} {g : α β} (hg : μ') (hf : μ) {f' : β β} (h_comm : g f = f' g) :
μ'
theorem measure_theory.measure_preserving.pre_ergodic_conjugate_iff {α : Type u_1} {m : measurable_space α} {f : α α} {μ : measure_theory.measure α} {β : Type u_2} {m' : measurable_space β} {μ' : measure_theory.measure β} {e : α ≃ᵐ β} (h : μ') :
pre_ergodic (e f (e.symm)) μ' μ
theorem measure_theory.measure_preserving.ergodic_conjugate_iff {α : Type u_1} {m : measurable_space α} {f : α α} {μ : measure_theory.measure α} {β : Type u_2} {m' : measurable_space β} {μ' : measure_theory.measure β} {e : α ≃ᵐ β} (h : μ') :
ergodic (e f (e.symm)) μ' μ
theorem quasi_ergodic.ae_empty_or_univ' {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f ⁻¹' s =ᵐ[μ] s) :

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

theorem ergodic.quasi_ergodic {α : Type u_1} {m : measurable_space α} {f : α α} {μ : measure_theory.measure α} (hf : μ) :

An ergodic map is quasi ergodic.

theorem ergodic.ae_empty_or_univ_of_preimage_ae_le' {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ) :

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 : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ) :

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 : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ) :

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 : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f ⁻¹' s ≤ᵐ[μ] s) :
theorem ergodic.ae_empty_or_univ_of_ae_le_preimage {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : s ≤ᵐ[μ] f ⁻¹' s) :
theorem ergodic.ae_empty_or_univ_of_image_ae_le {α : Type u_1} {m : measurable_space α} {f : α α} {s : set α} {μ : measure_theory.measure α} (hf : μ) (hs : measurable_set s) (hs' : f '' s ≤ᵐ[μ] s) :