[Merged by Bors] - feat: von Neumann Mean Ergodic Theorem#6053
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[Merged by Bors] - feat: von Neumann Mean Ergodic Theorem#6053
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- Add an instance saying that `LinearMap.eqLocus f g` is a complete space. - Drop `priority := 100` in `completeSpace_ker`: this is the main way to prove that the kernel is a complete space.
Rename: - `tendsto_iff_norm_tendsto_one` → `tendsto_iff_norm_div_tendsto_zero`; - `tendsto_iff_norm_tendsto_zero` → `tendsto_iff_norm_sub_tendsto_zero`; - `tendsto_one_iff_norm_tendsto_one` → `tendsto_one_iff_norm_tendsto_zero`; - `Filter.Tendsto.continuous_of_equicontinuous_at` → `Filter.Tendsto.continuous_of_equicontinuousAt`.
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
sgouezel
reviewed
Sep 8, 2023
| (hg_proj : ∀ x : LinearMap.eqLocus f 1, g x = x) | ||
| (hg_ker : (LinearMap.ker g : Set E) ⊆ closure (LinearMap.range (f - 1))) (x : E) : | ||
| Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop (𝓝 (g x)) := by | ||
| obtain ⟨y, hy, z, hz, rfl⟩ : ∃ y, g y = 0 ∧ ∃ z, IsFixedPt f z ∧ x = y + z := |
Contributor
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could you sketch informally the proof in a short comment?
| dist (g (f^[n] x)) (g x) / n := by | ||
| simp [dist_birkhoffAverage_birkhoffAverage, dist_birkhoffSum_apply_birkhoffSum] | ||
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| theorem tendsto_birkhoffAverage_apply_sub_birkhoffAverage {f : α → α} {g : α → E} {x : α} |
Contributor
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can you add a version assuming that the range of g is globally bounded (not only along the orbit)?
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Julian
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Sep 11, 2023
* fix-lint: (463 commits) chore: lint-style.py was calling str.replace incorrectly chore: module doc for #find_home, #minimize_imports, import early (#7095) chore: reduce imports to Data.Rat.Cast.CharZero (#7091) chore: cleanup imports of Data/Rat/Cast/Defs (#7092) chore: linarith only needs ring1 (#7090) refactor(Data/Int/Bitwise): use `<<<` and `>>>` notation (#6789) chore: delete redundant commented-out positivity test (#7085) chore: fix docstrings, names and aligns about paracompacity of emetric spaces (#7064) feat(Data/Finsupp): add notation (#6367) refactor: re-home some meta code (#6921) fix: don't use `False` as a bool, use `false` (#7059) chore: fix reference to `compactCovering` in docstring (#7065) fix: fix link in docstring of IsWellFounded (#7063) chore: tidy various files (#7041) feat: roots in an algebra (#6740) chore: revert ProofWidgets bump in #7044 and #7056 (#7069) feat: super factorial (#6768) feat(LinearAlgebra/Matrix/Trace): add Matrix.trace_diagonal (#7061) chore: complete ProofWidgets bump (#7056) feat: More `Finset.sup'` lemmas (#7021) feat: self_lt_pow (#7058) chore: move some files to `MeasureTheory/MeasurableSpace/` (#7045) chore(RingTheory/Nilpotent): untwine two universes (#7057) feat: von Neumann Mean Ergodic Theorem (#6053) feat: (sSup, Iic) and (Ici, sInf) are Galois connections (#6951) feat: derivative along a vector (#7038) feat: check for some common git problems in CI (#7043) chore: bump ProofWidgets (#7044) feat(Topology/Algebra): add `DiscreteTopology Mˣ` (#7028) chore: simplify `by rfl` (#7039) chore: tidy various files (#7035) refactor(LinearAlgebra/CliffordAlgebra/Conjugation): expose implementation details of 'reverse' (#6783) feat: add `DiscreteTopology.of_continuous_injective` (#7029) feat: restore the module docstring code snippet (#7030) feat: flesh out the API for `realPart` and `imaginaryPart` (#7023) chore: missing simps lemmas for `Multiplicative` defs (#7016) feat: characterize isLittleO on principal filters (#7008) doc: cleanup documentation on Basis.constr (#7007) feat: cleanup API around differentiable functions (#7004) feat: add Nat.digits_append_digits (#6999) feat: definition of the homology of a short complex (#6994) chore: implement porting notes about Polish spaces (#6991) feat(Algebra/Category/ModuleCat): composition of restriction of scalar functors (#6915) feat: compute the integral of sqrt (1 - x ^ 2) (#6905) chore(*/TensorProduct): missing functorial lemmas (#6781) feat: a functor from a small category to a filtered category factors through a small filtered category (#6212) feat: expand API on locally integrable functions (#7006) chore: split Tactic.NormNum.Basic (#7002) feat: a few lemmas on continuous functions (#7005) feat: ZMod.castHom_self (#7013) ...
kodyvajjha
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Sep 22, 2023
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orthogonalProjection KisKᗮ#6943