Theorems · Theorem · dynamical systems

ergodic_vadd_of_denseRange_zsmul

∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : TopologicalSpace G] [ContinuousNeg G] {X : Type u_2}
  [inst_3 : TopologicalSpace X] [R1Space X] [inst_5 : MeasurableSpace X] [BorelSpace X] [inst_7 : AddAction G X]
  [ContinuousVAdd G X] {g : G},
  (DenseRange fun x => x • g) →
    ∀ (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicVAdd G X μ],
      Ergodic (fun x => g +ᵥ x) μ

If an additive monoid M continuously acts on an R₁ topological space X, g is an element of M such that its integer multiples are dense in M, and μ is a finite inner regular measure on X which is ergodic with respect to the action of M, then the vector addition of g is an ergodic map.

Defined in

Mathlib.Dynamics.Ergodic.Action.OfMinimal

Cited by

1 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupTopologicalSpaceContinuousNegTopologicalSpaceR1SpaceMeasurableSpaceBorelSpaceAddActionContinuousVAddMeasureTheory.IsFiniteMeasureMeasureTheory.Measure.InnerRegularErgodicVAdd