Documentation

Mathlib.Dynamics.Ergodic.Conservative

Conservative systems #

In this file we define f : α → α to be a conservative system w.r.t a measure μ if f is non-singular (MeasureTheory.QuasiMeasurePreserving) and for every measurable set s of positive measure at least one point x ∈ s returns back to s after some number of iterations of f. There are several properties that look like they are stronger than this one but actually follow from it:

We also prove the topological Poincaré recurrence theorem MeasureTheory.Conservative.ae_frequently_mem_of_mem_nhds. Let f : α → α be a conservative dynamical system on a topological space with second countable topology and measurable open sets. Then almost every point x : α is recurrent: it visits every neighborhood s ∈ 𝓝 x infinitely many times.

Tags #

conservative dynamical system, Poincare recurrence theorem

We say that a non-singular (MeasureTheory.QuasiMeasurePreserving) self-map is conservative if for any measurable set s of positive measure there exists x ∈ s such that x returns back to s under some iteration of f.

  • measurable : Measurable f
  • absolutelyContinuous : (MeasureTheory.Measure.map f μ).AbsolutelyContinuous μ
  • exists_mem_iterate_mem' : ∀ ⦃s : Set α⦄, MeasurableSet sμ s 0xs, ∃ (m : ), m 0 f^[m] x s

    If f is a conservative self-map and s is a measurable set of nonzero measure, then there exists a point x ∈ s that returns to s under a non-zero iteration of f.

Instances For
    theorem MeasureTheory.Conservative.exists_mem_iterate_mem' {α : Type u_2} [MeasurableSpace α] {f : αα} {μ : MeasureTheory.Measure α} (self : MeasureTheory.Conservative f μ) ⦃s : Set α :
    MeasurableSet sμ s 0xs, ∃ (m : ), m 0 f^[m] x s

    If f is a conservative self-map and s is a measurable set of nonzero measure, then there exists a point x ∈ s that returns to s under a non-zero iteration of f.

    A self-map preserving a finite measure is conservative.

    The identity map is conservative w.r.t. any measure.

    theorem MeasureTheory.Conservative.measureRestrict {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Conservative f μ) (hs : Set.MapsTo f s s) :
    MeasureTheory.Conservative f (μ.restrict s)

    Restriction of a conservative system to an invariant set is a conservative system, formulated in terms of the restriction of the measure.

    theorem MeasureTheory.Conservative.exists_mem_iterate_mem {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hsm : MeasureTheory.NullMeasurableSet s μ) (hs₀ : μ s 0) :
    xs, ∃ (m : ), m 0 f^[m] x s

    If f is a conservative self-map and s is a null measurable set of nonzero measure, then there exists a point x ∈ s that returns to s under a non-zero iteration of f.

    theorem MeasureTheory.Conservative.frequently_measure_inter_ne_zero {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (h0 : μ s 0) :
    ∃ᶠ (m : ) in Filter.atTop, μ (s f^[m] ⁻¹' s) 0

    If f is a conservative map and s is a measurable set of nonzero measure, then for infinitely many values of m a positive measure of points x ∈ s returns back to s after m iterations of f.

    theorem MeasureTheory.Conservative.exists_gt_measure_inter_ne_zero {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (h0 : μ s 0) (N : ) :
    m > N, μ (s f^[m] ⁻¹' s) 0

    If f is a conservative map and s is a measurable set of nonzero measure, then for an arbitrarily large m a positive measure of points x ∈ s returns back to s after m iterations of f.

    theorem MeasureTheory.Conservative.measure_mem_forall_ge_image_not_mem_eq_zero {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (n : ) :
    μ {x : α | x s mn, f^[m] xs} = 0

    Poincaré recurrence theorem: given a conservative map f and a measurable set s, the set of points x ∈ s such that x does not return to s after ≥ n iterations has measure zero.

    theorem MeasureTheory.Conservative.ae_mem_imp_frequently_image_mem {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) :
    ∀ᵐ (x : α) ∂μ, x s∃ᶠ (n : ) in Filter.atTop, f^[n] x s

    Poincaré recurrence theorem: given a conservative map f and a measurable set s, almost every point x ∈ s returns back to s infinitely many times.

    theorem MeasureTheory.Conservative.inter_frequently_image_mem_ae_eq {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) :
    s {x : α | ∃ᶠ (n : ) in Filter.atTop, f^[n] x s} =ᵐ[μ] s
    theorem MeasureTheory.Conservative.measure_inter_frequently_image_mem_eq {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) :
    μ (s {x : α | ∃ᶠ (n : ) in Filter.atTop, f^[n] x s}) = μ s
    theorem MeasureTheory.Conservative.ae_forall_image_mem_imp_frequently_image_mem {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) :
    ∀ᵐ (x : α) ∂μ, ∀ (k : ), f^[k] x s∃ᶠ (n : ) in Filter.atTop, f^[n] x s

    Poincaré recurrence theorem: if f is a conservative dynamical system and s is a measurable set, then for μ-a.e. x, if the orbit of x visits s at least once, then it visits s infinitely many times.

    theorem MeasureTheory.Conservative.frequently_ae_mem_and_frequently_image_mem {α : Type u_2} [MeasurableSpace α] {f : αα} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (h0 : μ s 0) :
    ∃ᵐ (x : α) ∂μ, x s ∃ᶠ (n : ) in Filter.atTop, f^[n] x s

    If f is a conservative self-map and s is a measurable set of positive measure, then ae μ-frequently we have x ∈ s and s returns to s under infinitely many iterations of f.

    theorem MeasureTheory.Conservative.ae_frequently_mem_of_mem_nhds {α : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [SecondCountableTopology α] [OpensMeasurableSpace α] {f : αα} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Conservative f μ) :
    ∀ᵐ (x : α) ∂μ, snhds x, ∃ᶠ (n : ) in Filter.atTop, f^[n] x s

    Poincaré recurrence theorem. Let f : α → α be a conservative dynamical system on a topological space with second countable topology and measurable open sets. Then almost every point x : α is recurrent: it visits every neighborhood s ∈ 𝓝 x infinitely many times.

    Iteration of a conservative system is a conservative system.