Ergodic maps of the additive circle #

This file contains proofs of ergodicity for maps of the additive circle.

Main definitions: #

• AddCircle.ergodic_zsmul: given n : ℤ such that 1 < |n|, the self map y ↦ n • y on the additive circle is ergodic (wrt the Haar measure).
• AddCircle.ergodic_nsmul: given n : ℕ such that 1 < n, the self map y ↦ n • y on the additive circle is ergodic (wrt the Haar measure).
• AddCircle.ergodic_zsmul_add: given n : ℤ such that 1 < |n| and x : AddCircle T, the self map y ↦ n • y + x on the additive circle is ergodic (wrt the Haar measure).
• AddCircle.ergodic_nsmul_add: given n : ℕ such that 1 < n and x : AddCircle T, the self map y ↦ n • y + x on the additive circle is ergodic (wrt the Haar measure).
theorem AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self {T : } [hT : Fact (0 < T)] {s : Set ()} (hs : MeasureTheory.NullMeasurableSet s MeasureTheory.volume) {ι : Type u_1} {l : } [l.NeBot] {u : ι} (hu₁ : ∀ (i : ι), MeasureTheory.volume.ae.EventuallyEq (u i +ᵥ s) s) (hu₂ : Filter.Tendsto (addOrderOf u) l Filter.atTop) :
MeasureTheory.volume.ae.EventuallyEq s MeasureTheory.volume.ae.EventuallyEq s Set.univ

If a null-measurable subset of the circle is almost invariant under rotation by a family of rational angles with denominators tending to infinity, then it must be almost empty or almost full.

theorem AddCircle.ergodic_zsmul {T : } [hT : Fact (0 < T)] {n : } (hn : 1 < |n|) :
Ergodic (fun (y : ) => n y) MeasureTheory.volume
theorem AddCircle.ergodic_nsmul {T : } [hT : Fact (0 < T)] {n : } (hn : 1 < n) :
Ergodic (fun (y : ) => n y) MeasureTheory.volume
theorem AddCircle.ergodic_zsmul_add {T : } [hT : Fact (0 < T)] (x : ) {n : } (h : 1 < |n|) :
Ergodic (fun (y : ) => n y + x) MeasureTheory.volume
theorem AddCircle.ergodic_nsmul_add {T : } [hT : Fact (0 < T)] (x : ) {n : } (h : 1 < n) :
Ergodic (fun (y : ) => n y + x) MeasureTheory.volume