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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: #

structure pre_ergodic {α : Type u_1} {m : measurable_space α} (f : α α) (μ : measure_theory.measure α . "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 : measure_theory.measure α . "volume_tac") :
Prop

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 : measure_theory.measure α . "volume_tac") :
Prop

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 : pre_ergodic f 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} [measure_theory.is_probability_measure μ] (hf : pre_ergodic f 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 : pre_ergodic f^[n] 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 : measure_theory.measure_preserving g μ μ') (hf : pre_ergodic f μ) {f' : β β} (h_comm : g f = f' g) :
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 : measure_theory.measure_preserving e μ μ') :
ergodic (e f (e.symm)) μ' ergodic f 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 : quasi_ergodic f 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 : ergodic f 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 : ergodic f 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 : ergodic f 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 : ergodic f 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} [measure_theory.is_finite_measure μ] (hf : ergodic f 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} [measure_theory.is_finite_measure μ] (hf : ergodic f 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} [measure_theory.is_finite_measure μ] (hf : ergodic f μ) (hs : measurable_set s) (hs' : f '' s ≤ᵐ[μ] s) :